%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : ITP277^3 : 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 : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Feb 28 09:16:29 AM UTC 2026
% Result : Theorem 62.96s 13.45s
% Output : Refutation 62.96s
% Verified :
% SZS Type : Refutation
% Derivation depth : 14
% Number of leaves : 11
% Syntax : Number of formulae : 56 ( 18 unt; 0 typ; 4 def)
% Number of atoms : 196 ( 75 equ; 0 cnn)
% Maximal formula atoms : 4 ( 3 avg)
% Number of connectives : 314 ( 54 ~; 46 |; 6 &; 192 @)
% ( 4 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Number of types : 128 ( 127 usr)
% Number of type conns : 0 ( 0 >; 0 *; 0 +; 0 <<)
% Number of symbols : 1414 (1411 usr; 111 con; 0-8 aty)
% Number of variables : 47 ( 0 ^; 47 !; 0 ?; 47 :)
% Comments :
%------------------------------------------------------------------------------
thf(type_def_5,type,
rat: $tType ).
thf(type_def_6,type,
int: $tType ).
thf(type_def_8,type,
real: $tType ).
thf(type_def_9,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(type_def_10,type,
nat: $tType ).
thf(type_def_11,type,
product_prod_nat_nat: $tType ).
thf(type_def_12,type,
product_prod_int_int: $tType ).
thf(type_def_13,type,
complex: $tType ).
thf(type_def_14,type,
code_integer: $tType ).
thf(type_def_15,type,
num: $tType ).
thf(type_def_16,type,
option_num: $tType ).
thf(type_def_17,type,
produc6271795597528267376eger_o: $tType ).
thf(type_def_18,type,
produc8923325533196201883nteger: $tType ).
thf(type_def_19,type,
set_Extended_enat: $tType ).
thf(type_def_20,type,
extended_enat: $tType ).
thf(type_def_21,type,
set_fi4554929511873752355omplex: $tType ).
thf(type_def_22,type,
filter6041513312241820739omplex: $tType ).
thf(type_def_23,type,
set_fi7789364187291644575l_real: $tType ).
thf(type_def_24,type,
filter2146258269922977983l_real: $tType ).
thf(type_def_25,type,
set_nat: $tType ).
thf(type_def_26,type,
set_set_nat: $tType ).
thf(type_def_27,type,
set_int: $tType ).
thf(type_def_28,type,
set_real: $tType ).
thf(type_def_29,type,
filter_real: $tType ).
thf(type_def_30,type,
option_nat: $tType ).
thf(type_def_31,type,
filter_nat: $tType ).
thf(type_def_32,type,
filter1242075044329608583at_nat: $tType ).
thf(type_def_33,type,
set_Pr5085853215250843933omplex: $tType ).
thf(type_def_34,type,
set_Pr6218003697084177305l_real: $tType ).
thf(type_def_35,type,
set_o: $tType ).
thf(type_def_36,type,
set_complex: $tType ).
thf(type_def_37,type,
set_list_o: $tType ).
thf(type_def_38,type,
set_list_complex: $tType ).
thf(type_def_39,type,
set_li5464603477888414924d_enat: $tType ).
thf(type_def_40,type,
set_list_int: $tType ).
thf(type_def_41,type,
set_list_list_nat: $tType ).
thf(type_def_42,type,
set_list_nat: $tType ).
thf(type_def_43,type,
set_li5450038453877631591at_nat: $tType ).
thf(type_def_44,type,
set_list_set_nat: $tType ).
thf(type_def_45,type,
set_list_VEBT_VEBT: $tType ).
thf(type_def_46,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(type_def_47,type,
set_Pr4329608150637261639at_nat: $tType ).
thf(type_def_48,type,
set_Product_unit: $tType ).
thf(type_def_49,type,
set_VEBT_VEBT: $tType ).
thf(type_def_50,type,
set_num: $tType ).
thf(type_def_51,type,
set_rat: $tType ).
thf(type_def_52,type,
set_set_complex: $tType ).
thf(type_def_53,type,
set_se7270636423289371942d_enat: $tType ).
thf(type_def_54,type,
set_set_int: $tType ).
thf(type_def_55,type,
set_se7855581050983116737at_nat: $tType ).
thf(type_def_56,type,
char: $tType ).
thf(type_def_57,type,
produc1319942482725812455at_nat: $tType ).
thf(type_def_58,type,
list_nat: $tType ).
thf(type_def_59,type,
literal: $tType ).
thf(type_def_60,type,
produc859450856879609959at_nat: $tType ).
thf(type_def_61,type,
produc3843707927480180839at_nat: $tType ).
thf(type_def_62,type,
list_o: $tType ).
thf(type_def_63,type,
list_int: $tType ).
thf(type_def_64,type,
vEBT_VEBT: $tType ).
thf(type_def_65,type,
set_Pr958786334691620121nt_int: $tType ).
thf(type_def_66,type,
set_Code_integer: $tType ).
thf(type_def_67,type,
list_list_o: $tType ).
thf(type_def_68,type,
list_list_nat: $tType ).
thf(type_def_69,type,
list_list_VEBT_VEBT: $tType ).
thf(type_def_70,type,
list_VEBT_VEBT: $tType ).
thf(type_def_71,type,
list_P7333126701944960589_nat_o: $tType ).
thf(type_def_72,type,
list_P3521021558325789923at_int: $tType ).
thf(type_def_73,type,
list_P6011104703257516679at_nat: $tType ).
thf(type_def_74,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(type_def_75,type,
option_o: $tType ).
thf(type_def_76,type,
option_int: $tType ).
thf(type_def_77,type,
list_num: $tType ).
thf(type_def_78,type,
option4927543243414619207at_nat: $tType ).
thf(type_def_79,type,
list_real: $tType ).
thf(type_def_80,type,
option_real: $tType ).
thf(type_def_81,type,
list_set_nat: $tType ).
thf(type_def_82,type,
option_set_nat: $tType ).
thf(type_def_83,type,
option_VEBT_VEBT: $tType ).
thf(type_def_84,type,
list_complex: $tType ).
thf(type_def_85,type,
list_Extended_enat: $tType ).
thf(type_def_86,type,
list_P3795440434834930179_o_int: $tType ).
thf(type_def_87,type,
product_prod_o_int: $tType ).
thf(type_def_88,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(type_def_89,type,
produc2504756804600209347T_VEBT: $tType ).
thf(type_def_90,type,
list_P5087981734274514673_int_o: $tType ).
thf(type_def_91,type,
product_prod_int_o: $tType ).
thf(type_def_92,type,
list_P5707943133018811711nt_int: $tType ).
thf(type_def_93,type,
list_P8198026277950538467nt_nat: $tType ).
thf(type_def_94,type,
product_prod_int_nat: $tType ).
thf(type_def_95,type,
list_P7524865323317820941T_VEBT: $tType ).
thf(type_def_96,type,
produc1531783533982839933T_VEBT: $tType ).
thf(type_def_97,type,
product_prod_nat_o: $tType ).
thf(type_def_98,type,
product_prod_nat_int: $tType ).
thf(type_def_99,type,
produc8025551001238799321T_VEBT: $tType ).
thf(type_def_100,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(type_def_101,type,
produc334124729049499915VEBT_o: $tType ).
thf(type_def_102,type,
list_P4547456442757143711BT_int: $tType ).
thf(type_def_103,type,
produc4894624898956917775BT_int: $tType ).
thf(type_def_104,type,
list_P7037539587688870467BT_nat: $tType ).
thf(type_def_105,type,
produc9072475918466114483BT_nat: $tType ).
thf(type_def_106,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(type_def_107,type,
produc8243902056947475879T_VEBT: $tType ).
thf(type_def_108,type,
set_Pr6588086440996610945on_nat: $tType ).
thf(type_def_109,type,
set_Pr7459493094073627847at_nat: $tType ).
thf(type_def_110,type,
produc4953844613479565601on_nat: $tType ).
thf(type_def_111,type,
set_char: $tType ).
thf(type_def_112,type,
produc2233624965454879586on_nat: $tType ).
thf(type_def_113,type,
produc7248412053542808358at_nat: $tType ).
thf(type_def_114,type,
produc4471711990508489141at_nat: $tType ).
thf(type_def_115,type,
produc8306885398267862888on_nat: $tType ).
thf(type_def_116,type,
produc3447558737645232053on_num: $tType ).
thf(type_def_117,type,
produc7036089656553540234on_num: $tType ).
thf(type_def_118,type,
produc1193250871479095198on_num: $tType ).
thf(type_def_119,type,
produc6121120109295599847at_nat: $tType ).
thf(type_def_120,type,
produc5491161045314408544at_nat: $tType ).
thf(type_def_121,type,
produc5542196010084753463at_nat: $tType ).
thf(type_def_122,type,
produc8064648209034914857omplex: $tType ).
thf(type_def_123,type,
produc1621487020699730983d_enat: $tType ).
thf(type_def_124,type,
produc2115011035271226405et_int: $tType ).
thf(type_def_125,type,
produc7819656566062154093et_nat: $tType ).
thf(type_def_126,type,
produc4411394909380815293omplex: $tType ).
thf(type_def_127,type,
produc2422161461964618553l_real: $tType ).
thf(type_def_128,type,
product_unit: $tType ).
thf(type_def_129,type,
set_Pr6308028481084910985omplex: $tType ).
thf(type_def_130,type,
set_Pr2112562347474612743d_enat: $tType ).
thf(type_def_131,type,
set_Pr2522554150109002629et_int: $tType ).
thf(type_def_132,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(func_def_0,type,
set_Pr7459493094073627847at_nat: $tType ).
thf(func_def_1,type,
produc1319942482725812455at_nat: $tType ).
thf(func_def_2,type,
produc5542196010084753463at_nat: $tType ).
thf(func_def_3,type,
produc5491161045314408544at_nat: $tType ).
thf(func_def_4,type,
set_Pr4329608150637261639at_nat: $tType ).
thf(func_def_5,type,
produc1193250871479095198on_num: $tType ).
thf(func_def_6,type,
produc8306885398267862888on_nat: $tType ).
thf(func_def_7,type,
produc6121120109295599847at_nat: $tType ).
thf(func_def_8,type,
produc4471711990508489141at_nat: $tType ).
thf(func_def_9,type,
produc7036089656553540234on_num: $tType ).
thf(func_def_10,type,
produc2233624965454879586on_nat: $tType ).
thf(func_def_11,type,
produc3843707927480180839at_nat: $tType ).
thf(func_def_12,type,
set_Pr8693737435421807431at_nat: $tType ).
thf(func_def_13,type,
produc859450856879609959at_nat: $tType ).
thf(func_def_14,type,
set_Pr2112562347474612743d_enat: $tType ).
thf(func_def_15,type,
set_Pr6308028481084910985omplex: $tType ).
thf(func_def_16,type,
set_Pr6588086440996610945on_nat: $tType ).
thf(func_def_17,type,
produc1621487020699730983d_enat: $tType ).
thf(func_def_18,type,
set_fi4554929511873752355omplex: $tType ).
thf(func_def_19,type,
list_P7413028617227757229T_VEBT: $tType ).
thf(func_def_20,type,
set_Pr5488025237498180813et_nat: $tType ).
thf(func_def_21,type,
set_Pr2522554150109002629et_int: $tType ).
thf(func_def_22,type,
produc8064648209034914857omplex: $tType ).
thf(func_def_23,type,
produc3447558737645232053on_num: $tType ).
thf(func_def_24,type,
produc4953844613479565601on_nat: $tType ).
thf(func_def_25,type,
produc7248412053542808358at_nat: $tType ).
thf(func_def_26,type,
set_fi7789364187291644575l_real: $tType ).
thf(func_def_27,type,
filter6041513312241820739omplex: $tType ).
thf(func_def_28,type,
list_P7037539587688870467BT_nat: $tType ).
thf(func_def_29,type,
list_P4547456442757143711BT_int: $tType ).
thf(func_def_30,type,
list_P5647936690300460905T_VEBT: $tType ).
thf(func_def_31,type,
list_P7524865323317820941T_VEBT: $tType ).
thf(func_def_32,type,
set_li5450038453877631591at_nat: $tType ).
thf(func_def_33,type,
produc8243902056947475879T_VEBT: $tType ).
thf(func_def_34,type,
set_se7855581050983116737at_nat: $tType ).
thf(func_def_35,type,
set_Pr5085853215250843933omplex: $tType ).
thf(func_def_36,type,
produc7819656566062154093et_nat: $tType ).
thf(func_def_37,type,
produc2115011035271226405et_int: $tType ).
thf(func_def_38,type,
produc8923325533196201883nteger: $tType ).
thf(func_def_39,type,
list_P3126845725202233233VEBT_o: $tType ).
thf(func_def_40,type,
list_P7495141550334521929T_VEBT: $tType ).
thf(func_def_41,type,
filter2146258269922977983l_real: $tType ).
thf(func_def_42,type,
option4927543243414619207at_nat: $tType ).
thf(func_def_43,type,
filter1242075044329608583at_nat: $tType ).
thf(func_def_44,type,
set_Pr6218003697084177305l_real: $tType ).
thf(func_def_45,type,
list_P6011104703257516679at_nat: $tType ).
thf(func_def_46,type,
list_P3521021558325789923at_int: $tType ).
thf(func_def_47,type,
list_P8198026277950538467nt_nat: $tType ).
thf(func_def_48,type,
list_P5707943133018811711nt_int: $tType ).
thf(func_def_49,type,
produc9072475918466114483BT_nat: $tType ).
thf(func_def_50,type,
produc4894624898956917775BT_int: $tType ).
thf(func_def_51,type,
produc8025551001238799321T_VEBT: $tType ).
thf(func_def_52,type,
produc1531783533982839933T_VEBT: $tType ).
thf(func_def_53,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(func_def_54,type,
set_Pr958786334691620121nt_int: $tType ).
thf(func_def_55,type,
produc4411394909380815293omplex: $tType ).
thf(func_def_56,type,
list_list_VEBT_VEBT: $tType ).
thf(func_def_57,type,
set_list_list_nat: $tType ).
thf(func_def_58,type,
list_P7333126701944960589_nat_o: $tType ).
thf(func_def_59,type,
list_P5087981734274514673_int_o: $tType ).
thf(func_def_60,type,
list_P3795440434834930179_o_int: $tType ).
thf(func_def_61,type,
set_list_VEBT_VEBT: $tType ).
thf(func_def_62,type,
produc334124729049499915VEBT_o: $tType ).
thf(func_def_63,type,
produc2504756804600209347T_VEBT: $tType ).
thf(func_def_64,type,
set_list_set_nat: $tType ).
thf(func_def_65,type,
produc6271795597528267376eger_o: $tType ).
thf(func_def_66,type,
set_li5464603477888414924d_enat: $tType ).
thf(func_def_67,type,
produc2422161461964618553l_real: $tType ).
thf(func_def_68,type,
set_se7270636423289371942d_enat: $tType ).
thf(func_def_69,type,
product_prod_nat_nat: $tType ).
thf(func_def_70,type,
product_prod_nat_int: $tType ).
thf(func_def_71,type,
product_prod_int_nat: $tType ).
thf(func_def_72,type,
product_prod_int_int: $tType ).
thf(func_def_73,type,
set_list_complex: $tType ).
thf(func_def_74,type,
set_set_complex: $tType ).
thf(func_def_75,type,
option_VEBT_VEBT: $tType ).
thf(func_def_76,type,
option_set_nat: $tType ).
thf(func_def_77,type,
list_list_nat: $tType ).
thf(func_def_78,type,
list_VEBT_VEBT: $tType ).
thf(func_def_79,type,
set_list_nat: $tType ).
thf(func_def_80,type,
set_list_int: $tType ).
thf(func_def_81,type,
product_prod_nat_o: $tType ).
thf(func_def_82,type,
product_prod_int_o: $tType ).
thf(func_def_83,type,
product_prod_o_int: $tType ).
thf(func_def_84,type,
list_set_nat: $tType ).
thf(func_def_85,type,
set_VEBT_VEBT: $tType ).
thf(func_def_86,type,
set_set_nat: $tType ).
thf(func_def_87,type,
set_set_int: $tType ).
thf(func_def_88,type,
set_Code_integer: $tType ).
thf(func_def_89,type,
list_Extended_enat: $tType ).
thf(func_def_90,type,
set_Product_unit: $tType ).
thf(func_def_91,type,
set_Extended_enat: $tType ).
thf(func_def_92,type,
list_list_o: $tType ).
thf(func_def_93,type,
list_complex: $tType ).
thf(func_def_94,type,
set_list_o: $tType ).
thf(func_def_95,type,
set_complex: $tType ).
thf(func_def_96,type,
option_real: $tType ).
thf(func_def_97,type,
filter_real: $tType ).
thf(func_def_98,type,
option_num: $tType ).
thf(func_def_99,type,
option_nat: $tType ).
thf(func_def_100,type,
option_int: $tType ).
thf(func_def_101,type,
filter_nat: $tType ).
thf(func_def_102,type,
set_char: $tType ).
thf(func_def_103,type,
list_real: $tType ).
thf(func_def_104,type,
set_real: $tType ).
thf(func_def_105,type,
list_num: $tType ).
thf(func_def_106,type,
list_nat: $tType ).
thf(func_def_107,type,
list_int: $tType ).
thf(func_def_108,type,
vEBT_VEBT: $tType ).
thf(func_def_109,type,
set_rat: $tType ).
thf(func_def_110,type,
set_num: $tType ).
thf(func_def_111,type,
set_nat: $tType ).
thf(func_def_112,type,
set_int: $tType ).
thf(func_def_113,type,
code_integer: $tType ).
thf(func_def_114,type,
product_unit: $tType ).
thf(func_def_115,type,
option_o: $tType ).
thf(func_def_116,type,
extended_enat: $tType ).
thf(func_def_117,type,
list_o: $tType ).
thf(func_def_118,type,
complex: $tType ).
thf(func_def_119,type,
literal: $tType ).
thf(func_def_120,type,
set_o: $tType ).
thf(func_def_121,type,
char: $tType ).
thf(func_def_122,type,
real: $tType ).
thf(func_def_123,type,
rat: $tType ).
thf(func_def_124,type,
num: $tType ).
thf(func_def_125,type,
nat: $tType ).
thf(func_def_126,type,
int: $tType ).
thf(func_def_127,type,
archim2889992004027027881ng_rat: rat > int ).
thf(func_def_128,type,
archim7802044766580827645g_real: real > int ).
thf(func_def_129,type,
archim3151403230148437115or_rat: rat > int ).
thf(func_def_130,type,
archim6058952711729229775r_real: real > int ).
thf(func_def_131,type,
archimedean_frac_rat: rat > rat ).
thf(func_def_132,type,
archim2898591450579166408c_real: real > real ).
thf(func_def_133,type,
archim7778729529865785530nd_rat: rat > int ).
thf(func_def_134,type,
archim8280529875227126926d_real: real > int ).
thf(func_def_135,type,
bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).
thf(func_def_136,type,
bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).
thf(func_def_137,type,
bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).
thf(func_def_138,type,
bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $o ).
thf(func_def_139,type,
bNF_re3403563459893282935_int_o: ( int > int > $o ) > ( ( int > $o ) > ( int > $o ) > $o ) > ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_140,type,
bNF_re5089333283451836215nt_o_o: ( int > int > $o ) > ( $o > $o > $o ) > ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_141,type,
bNF_re578469030762574527_nat_o: ( nat > nat > $o ) > ( ( nat > $o ) > ( nat > $o ) > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_142,type,
bNF_re4705727531993890431at_o_o: ( nat > nat > $o ) > ( $o > $o > $o ) > ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_143,type,
bNF_re6830278522597306478at_int: ( nat > nat > $o ) > ( product_prod_nat_nat > int > $o ) > ( nat > product_prod_nat_nat ) > ( nat > int ) > $o ).
thf(func_def_144,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_145,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_146,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_147,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_148,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_149,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_150,type,
bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_151,type,
binomial: nat > nat > nat ).
thf(func_def_152,type,
gbinomial_complex: complex > nat > complex ).
thf(func_def_153,type,
gbinomial_int: int > nat > int ).
thf(func_def_154,type,
gbinomial_nat: nat > nat > nat ).
thf(func_def_155,type,
gbinomial_rat: rat > nat > rat ).
thf(func_def_156,type,
gbinomial_real: real > nat > real ).
thf(func_def_157,type,
bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_158,type,
bit_concat_bit: nat > int > int > int ).
thf(func_def_159,type,
bit_ri7919022796975470100ot_int: int > int ).
thf(func_def_160,type,
bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).
thf(func_def_161,type,
bit_ri631733984087533419it_int: nat > int > int ).
thf(func_def_162,type,
bit_se725231765392027082nd_int: int > int > int ).
thf(func_def_163,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(func_def_164,type,
bit_se8568078237143864401it_int: nat > int > int ).
thf(func_def_165,type,
bit_se8570568707652914677it_nat: nat > nat > nat ).
thf(func_def_166,type,
bit_se2159334234014336723it_int: nat > int > int ).
thf(func_def_167,type,
bit_se2000444600071755411sk_int: nat > int ).
thf(func_def_168,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(func_def_169,type,
bit_se1409905431419307370or_int: int > int > int ).
thf(func_def_170,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(func_def_171,type,
bit_se545348938243370406it_int: nat > int > int ).
thf(func_def_172,type,
bit_se547839408752420682it_nat: nat > nat > nat ).
thf(func_def_173,type,
bit_se2793503036327961859nteger: nat > code_integer > code_integer ).
thf(func_def_174,type,
bit_se7879613467334960850it_int: nat > int > int ).
thf(func_def_175,type,
bit_se7882103937844011126it_nat: nat > nat > nat ).
thf(func_def_176,type,
bit_se2923211474154528505it_int: nat > int > int ).
thf(func_def_177,type,
bit_se2925701944663578781it_nat: nat > nat > nat ).
thf(func_def_178,type,
bit_se8260200283734997820nteger: nat > code_integer > code_integer ).
thf(func_def_179,type,
bit_se4203085406695923979it_int: nat > int > int ).
thf(func_def_180,type,
bit_se4205575877204974255it_nat: nat > nat > nat ).
thf(func_def_181,type,
bit_se6526347334894502574or_int: int > int > int ).
thf(func_def_182,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(func_def_183,type,
bit_se1146084159140164899it_int: int > nat > $o ).
thf(func_def_184,type,
bit_se1148574629649215175it_nat: nat > nat > $o ).
thf(func_def_185,type,
bit_take_bit_num: nat > num > option_num ).
thf(func_def_186,type,
code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).
thf(func_def_187,type,
code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_188,type,
code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_189,type,
code_int_of_integer: code_integer > int ).
thf(func_def_190,type,
code_integer_of_int: int > code_integer ).
thf(func_def_191,type,
code_nat_of_integer: code_integer > nat ).
thf(func_def_192,type,
code_num_of_integer: code_integer > num ).
thf(func_def_193,type,
comple2295165028678016749d_enat: set_Extended_enat > extended_enat ).
thf(func_def_194,type,
comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).
thf(func_def_195,type,
comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).
thf(func_def_196,type,
complete_Inf_Inf_nat: set_nat > nat ).
thf(func_def_197,type,
comple7806235888213564991et_nat: set_set_nat > set_nat ).
thf(func_def_198,type,
comple4398354569131411667d_enat: set_Extended_enat > extended_enat ).
thf(func_def_199,type,
complete_Sup_Sup_int: set_int > int ).
thf(func_def_200,type,
complete_Sup_Sup_nat: set_nat > nat ).
thf(func_def_201,type,
comple1385675409528146559p_real: set_real > real ).
thf(func_def_202,type,
comple7399068483239264473et_nat: set_set_nat > set_nat ).
thf(func_def_203,type,
arg: complex > real ).
thf(func_def_204,type,
cis: real > complex ).
thf(func_def_205,type,
cnj: complex > complex ).
thf(func_def_206,type,
complex2: real > real > complex ).
thf(func_def_207,type,
im: complex > real ).
thf(func_def_208,type,
re: complex > real ).
thf(func_def_209,type,
csqrt: complex > complex ).
thf(func_def_210,type,
imaginary_unit: complex ).
thf(func_def_211,type,
condit2214826472909112428ve_nat: set_nat > $o ).
thf(func_def_212,type,
differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).
thf(func_def_213,type,
has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).
thf(func_def_214,type,
has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).
thf(func_def_215,type,
divmod_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_216,type,
eucl_rel_int: int > int > product_prod_int_int > $o ).
thf(func_def_217,type,
unique6319869463603278526ux_int: product_prod_int_int > $o ).
thf(func_def_218,type,
unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).
thf(func_def_219,type,
unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).
thf(func_def_220,type,
unique5052692396658037445od_int: num > num > product_prod_int_int ).
thf(func_def_221,type,
unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).
thf(func_def_222,type,
unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_223,type,
unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).
thf(func_def_224,type,
unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_225,type,
euclid3395696857347342551nt_int: int > int ).
thf(func_def_226,type,
extended_eSuc: extended_enat > extended_enat ).
thf(func_def_227,type,
extended_enat2: nat > extended_enat ).
thf(func_def_228,type,
extended_Abs_enat: option_nat > extended_enat ).
thf(func_def_229,type,
extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).
thf(func_def_230,type,
extend3600170679010898289d_enat: ( nat > extended_enat ) > extended_enat > extended_enat > extended_enat ).
thf(func_def_231,type,
extend5688581933313929465d_enat: extended_enat ).
thf(func_def_232,type,
semiri1408675320244567234ct_nat: nat > nat ).
thf(func_def_233,type,
semiri2265585572941072030t_real: nat > real ).
thf(func_def_234,type,
inverse_inverse_real: real > real ).
thf(func_def_235,type,
at_bot_real: filter_real ).
thf(func_def_236,type,
at_top_nat: filter_nat ).
thf(func_def_237,type,
at_top_real: filter_real ).
thf(func_def_238,type,
cofinite_nat: filter_nat ).
thf(func_def_239,type,
eventually_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_240,type,
eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).
thf(func_def_241,type,
eventually_real: ( real > $o ) > filter_real > $o ).
thf(func_def_242,type,
filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
thf(func_def_243,type,
filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
thf(func_def_244,type,
filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
thf(func_def_245,type,
frequently_nat: ( nat > $o ) > filter_nat > $o ).
thf(func_def_246,type,
princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).
thf(func_def_247,type,
princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).
thf(func_def_248,type,
prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
thf(func_def_249,type,
finite_card_o: set_o > nat ).
thf(func_def_250,type,
finite_card_complex: set_complex > nat ).
thf(func_def_251,type,
finite121521170596916366d_enat: set_Extended_enat > nat ).
thf(func_def_252,type,
finite_card_int: set_int > nat ).
thf(func_def_253,type,
finite_card_list_o: set_list_o > nat ).
thf(func_def_254,type,
finite5120063068150530198omplex: set_list_complex > nat ).
thf(func_def_255,type,
finite7441382602597825044d_enat: set_li5464603477888414924d_enat > nat ).
thf(func_def_256,type,
finite_card_list_int: set_list_int > nat ).
thf(func_def_257,type,
finite7325466520557071688st_nat: set_list_list_nat > nat ).
thf(func_def_258,type,
finite_card_list_nat: set_list_nat > nat ).
thf(func_def_259,type,
finite249151656366948015at_nat: set_li5450038453877631591at_nat > nat ).
thf(func_def_260,type,
finite5631907774883551598et_nat: set_list_set_nat > nat ).
thf(func_def_261,type,
finite5915292604075114978T_VEBT: set_list_VEBT_VEBT > nat ).
thf(func_def_262,type,
finite_card_nat: set_nat > nat ).
thf(func_def_263,type,
finite711546835091564841at_nat: set_Pr1261947904930325089at_nat > nat ).
thf(func_def_264,type,
finite1207074278014112911at_nat: set_Pr8693737435421807431at_nat > nat ).
thf(func_def_265,type,
finite3771342082235030671at_nat: set_Pr4329608150637261639at_nat > nat ).
thf(func_def_266,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(func_def_267,type,
finite_card_real: set_real > nat ).
thf(func_def_268,type,
finite_card_set_nat: set_set_nat > nat ).
thf(func_def_269,type,
finite7802652506058667612T_VEBT: set_VEBT_VEBT > nat ).
thf(func_def_270,type,
finite_finite_o: set_o > $o ).
thf(func_def_271,type,
finite3207457112153483333omplex: set_complex > $o ).
thf(func_def_272,type,
finite4001608067531595151d_enat: set_Extended_enat > $o ).
thf(func_def_273,type,
finite_finite_int: set_int > $o ).
thf(func_def_274,type,
finite_finite_list_o: set_list_o > $o ).
thf(func_def_275,type,
finite8712137658972009173omplex: set_list_complex > $o ).
thf(func_def_276,type,
finite1862508098717546133d_enat: set_li5464603477888414924d_enat > $o ).
thf(func_def_277,type,
finite3922522038869484883st_int: set_list_int > $o ).
thf(func_def_278,type,
finite8100373058378681591st_nat: set_list_nat > $o ).
thf(func_def_279,type,
finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).
thf(func_def_280,type,
finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).
thf(func_def_281,type,
finite_finite_nat: set_nat > $o ).
thf(func_def_282,type,
finite_finite_num: set_num > $o ).
thf(func_def_283,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_284,type,
finite4392333629123659920at_nat: set_Pr8693737435421807431at_nat > $o ).
thf(func_def_285,type,
finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o ).
thf(func_def_286,type,
finite_finite_rat: set_rat > $o ).
thf(func_def_287,type,
finite_finite_real: set_real > $o ).
thf(func_def_288,type,
finite6551019134538273531omplex: set_set_complex > $o ).
thf(func_def_289,type,
finite5468666774076196335d_enat: set_se7270636423289371942d_enat > $o ).
thf(func_def_290,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(func_def_291,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(func_def_292,type,
finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).
thf(func_def_293,type,
finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).
thf(func_def_294,type,
bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).
thf(func_def_295,type,
bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).
thf(func_def_296,type,
bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).
thf(func_def_297,type,
comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).
thf(func_def_298,type,
comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).
thf(func_def_299,type,
comp_nat_real_nat: ( nat > real ) > ( nat > nat ) > nat > real ).
thf(func_def_300,type,
id_o: $o > $o ).
thf(func_def_301,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_302,type,
inj_on_nat_char: ( nat > char ) > set_nat > $o ).
thf(func_def_303,type,
inj_on_real_real: ( real > real ) > set_real > $o ).
thf(func_def_304,type,
inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).
thf(func_def_305,type,
map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).
thf(func_def_306,type,
map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).
thf(func_def_307,type,
map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).
thf(func_def_308,type,
map_fu1856342031159181835at_o_o: ( real > nat > rat ) > ( $o > $o ) > ( ( nat > rat ) > $o ) > real > $o ).
thf(func_def_309,type,
strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).
thf(func_def_310,type,
the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).
thf(func_def_311,type,
fun_max_strict: set_Pr4329608150637261639at_nat ).
thf(func_def_312,type,
fun_max_weak: set_Pr4329608150637261639at_nat ).
thf(func_def_313,type,
fun_min_strict: set_Pr4329608150637261639at_nat ).
thf(func_def_314,type,
fun_min_weak: set_Pr4329608150637261639at_nat ).
thf(func_def_315,type,
fun_pair_leq: set_Pr8693737435421807431at_nat ).
thf(func_def_316,type,
fun_pair_less: set_Pr8693737435421807431at_nat ).
thf(func_def_317,type,
fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).
thf(func_def_318,type,
gcd_Gcd_int: set_int > int ).
thf(func_def_319,type,
gcd_Gcd_nat: set_nat > nat ).
thf(func_def_320,type,
gcd_Lcm_int: set_int > int ).
thf(func_def_321,type,
gcd_Lcm_nat: set_nat > nat ).
thf(func_def_322,type,
bezw: nat > nat > product_prod_int_int ).
thf(func_def_323,type,
bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_324,type,
gcd_gcd_int: int > int > int ).
thf(func_def_325,type,
gcd_gcd_nat: nat > nat > nat ).
thf(func_def_326,type,
gcd_lcm_int: int > int > int ).
thf(func_def_327,type,
gcd_lcm_nat: nat > nat > nat ).
thf(func_def_328,type,
gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_329,type,
abs_abs_Code_integer: code_integer > code_integer ).
thf(func_def_330,type,
abs_abs_int: int > int ).
thf(func_def_331,type,
abs_abs_rat: rat > rat ).
thf(func_def_332,type,
abs_abs_real: real > real ).
thf(func_def_333,type,
minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(func_def_334,type,
minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_335,type,
minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_336,type,
minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_337,type,
minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_338,type,
minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(func_def_339,type,
minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).
thf(func_def_340,type,
minus_minus_complex: complex > complex > complex ).
thf(func_def_341,type,
minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_342,type,
minus_minus_int: int > int > int ).
thf(func_def_343,type,
minus_minus_nat: nat > nat > nat ).
thf(func_def_344,type,
minus_minus_rat: rat > rat > rat ).
thf(func_def_345,type,
minus_minus_real: real > real > real ).
thf(func_def_346,type,
minus_minus_set_o: set_o > set_o > set_o ).
thf(func_def_347,type,
minus_811609699411566653omplex: set_complex > set_complex > set_complex ).
thf(func_def_348,type,
minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
thf(func_def_349,type,
minus_minus_set_int: set_int > set_int > set_int ).
thf(func_def_350,type,
minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_351,type,
minus_minus_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_352,type,
minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_353,type,
minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_354,type,
minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_355,type,
minus_minus_set_real: set_real > set_real > set_real ).
thf(func_def_356,type,
minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_357,type,
minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_358,type,
one_one_Code_integer: code_integer ).
thf(func_def_359,type,
one_one_complex: complex ).
thf(func_def_360,type,
one_on7984719198319812577d_enat: extended_enat ).
thf(func_def_361,type,
one_one_int: int ).
thf(func_def_362,type,
one_one_nat: nat ).
thf(func_def_363,type,
one_one_rat: rat ).
thf(func_def_364,type,
one_one_real: real ).
thf(func_def_365,type,
plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).
thf(func_def_366,type,
plus_plus_complex: complex > complex > complex ).
thf(func_def_367,type,
plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_368,type,
plus_plus_int: int > int > int ).
thf(func_def_369,type,
plus_plus_nat: nat > nat > nat ).
thf(func_def_370,type,
plus_plus_num: num > num > num ).
thf(func_def_371,type,
plus_plus_rat: rat > rat > rat ).
thf(func_def_372,type,
plus_plus_real: real > real > real ).
thf(func_def_373,type,
plus_plus_literal: literal > literal > literal ).
thf(func_def_374,type,
sgn_sgn_Code_integer: code_integer > code_integer ).
thf(func_def_375,type,
sgn_sgn_complex: complex > complex ).
thf(func_def_376,type,
sgn_sgn_int: int > int ).
thf(func_def_377,type,
sgn_sgn_rat: rat > rat ).
thf(func_def_378,type,
sgn_sgn_real: real > real ).
thf(func_def_379,type,
times_3573771949741848930nteger: code_integer > code_integer > code_integer ).
thf(func_def_380,type,
times_times_complex: complex > complex > complex ).
thf(func_def_381,type,
times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_382,type,
times_times_int: int > int > int ).
thf(func_def_383,type,
times_times_nat: nat > nat > nat ).
thf(func_def_384,type,
times_times_rat: rat > rat > rat ).
thf(func_def_385,type,
times_times_real: real > real > real ).
thf(func_def_386,type,
uminus_uminus_o_o: ( $o > $o ) > $o > $o ).
thf(func_def_387,type,
uminus_uminus_int_o: ( int > $o ) > int > $o ).
thf(func_def_388,type,
uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).
thf(func_def_389,type,
uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).
thf(func_def_390,type,
uminus_uminus_real_o: ( real > $o ) > real > $o ).
thf(func_def_391,type,
uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).
thf(func_def_392,type,
uminus1351360451143612070nteger: code_integer > code_integer ).
thf(func_def_393,type,
uminus1482373934393186551omplex: complex > complex ).
thf(func_def_394,type,
uminus_uminus_int: int > int ).
thf(func_def_395,type,
uminus_uminus_rat: rat > rat ).
thf(func_def_396,type,
uminus_uminus_real: real > real ).
thf(func_def_397,type,
uminus_uminus_set_o: set_o > set_o ).
thf(func_def_398,type,
uminus8566677241136511917omplex: set_complex > set_complex ).
thf(func_def_399,type,
uminus417252749190364093d_enat: set_Extended_enat > set_Extended_enat ).
thf(func_def_400,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(func_def_401,type,
uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).
thf(func_def_402,type,
uminus5710092332889474511et_nat: set_nat > set_nat ).
thf(func_def_403,type,
uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_404,type,
uminus4384627049435823934at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_405,type,
uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_406,type,
uminus612125837232591019t_real: set_real > set_real ).
thf(func_def_407,type,
uminus613421341184616069et_nat: set_set_nat > set_set_nat ).
thf(func_def_408,type,
zero_z3403309356797280102nteger: code_integer ).
thf(func_def_409,type,
zero_zero_complex: complex ).
thf(func_def_410,type,
zero_z5237406670263579293d_enat: extended_enat ).
thf(func_def_411,type,
zero_zero_int: int ).
thf(func_def_412,type,
zero_zero_nat: nat ).
thf(func_def_413,type,
zero_zero_rat: rat ).
thf(func_def_414,type,
zero_zero_real: real ).
thf(func_def_415,type,
zero_zero_literal: literal ).
thf(func_def_416,type,
groups4406642042086082107nteger: ( $o > code_integer ) > set_o > code_integer ).
thf(func_def_417,type,
groups5328290441151304332omplex: ( $o > complex ) > set_o > complex ).
thf(func_def_418,type,
groups8505340233167759370_o_int: ( $o > int ) > set_o > int ).
thf(func_def_419,type,
groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).
thf(func_def_420,type,
groups7872700643590313910_o_rat: ( $o > rat ) > set_o > rat ).
thf(func_def_421,type,
groups8691415230153176458o_real: ( $o > real ) > set_o > real ).
thf(func_def_422,type,
groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).
thf(func_def_423,type,
groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).
thf(func_def_424,type,
groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).
thf(func_def_425,type,
groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).
thf(func_def_426,type,
groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).
thf(func_def_427,type,
groups6818542070133387226omplex: ( extended_enat > complex ) > set_Extended_enat > complex ).
thf(func_def_428,type,
groups2025484359314973016at_int: ( extended_enat > int ) > set_Extended_enat > int ).
thf(func_def_429,type,
groups2027974829824023292at_nat: ( extended_enat > nat ) > set_Extended_enat > nat ).
thf(func_def_430,type,
groups1392844769737527556at_rat: ( extended_enat > rat ) > set_Extended_enat > rat ).
thf(func_def_431,type,
groups4148127829035722712t_real: ( extended_enat > real ) > set_Extended_enat > real ).
thf(func_def_432,type,
groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).
thf(func_def_433,type,
groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).
thf(func_def_434,type,
groups4538972089207619220nt_int: ( int > int ) > set_int > int ).
thf(func_def_435,type,
groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).
thf(func_def_436,type,
groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).
thf(func_def_437,type,
groups8778361861064173332t_real: ( int > real ) > set_int > real ).
thf(func_def_438,type,
groups6529277132148336714omplex: ( list_nat > complex ) > set_list_nat > complex ).
thf(func_def_439,type,
groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).
thf(func_def_440,type,
groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).
thf(func_def_441,type,
groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).
thf(func_def_442,type,
groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).
thf(func_def_443,type,
groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_444,type,
groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).
thf(func_def_445,type,
groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).
thf(func_def_446,type,
groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).
thf(func_def_447,type,
groups1900718384385340925at_nat: ( produc859450856879609959at_nat > nat ) > set_Pr8693737435421807431at_nat > nat ).
thf(func_def_448,type,
groups3860910324918113789at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > nat ).
thf(func_def_449,type,
groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).
thf(func_def_450,type,
groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).
thf(func_def_451,type,
groups1932886352136224148al_int: ( real > int ) > set_real > int ).
thf(func_def_452,type,
groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).
thf(func_def_453,type,
groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).
thf(func_def_454,type,
groups8097168146408367636l_real: ( real > real ) > set_real > real ).
thf(func_def_455,type,
groups8255218700646806128omplex: ( set_nat > complex ) > set_set_nat > complex ).
thf(func_def_456,type,
groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).
thf(func_def_457,type,
groups5107569545109728110t_real: ( set_nat > real ) > set_set_nat > real ).
thf(func_def_458,type,
groups1705073143266064639nt_int: ( int > int ) > set_int > int ).
thf(func_def_459,type,
groups705719431365010083at_int: ( nat > int ) > set_nat > int ).
thf(func_def_460,type,
groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).
thf(func_def_461,type,
groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).
thf(func_def_462,type,
groups4561878855575611511st_nat: list_nat > nat ).
thf(func_def_463,type,
the_int: ( int > $o ) > int ).
thf(func_def_464,type,
the_Pr4378521158711661632nt_int: ( product_prod_int_int > $o ) > product_prod_int_int ).
thf(func_def_465,type,
the_real: ( real > $o ) > real ).
thf(func_def_466,type,
if_Code_integer: $o > code_integer > code_integer > code_integer ).
thf(func_def_467,type,
if_complex: $o > complex > complex > complex ).
thf(func_def_468,type,
if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).
thf(func_def_469,type,
if_int: $o > int > int > int ).
thf(func_def_470,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(func_def_471,type,
if_list_nat: $o > list_nat > list_nat > list_nat ).
thf(func_def_472,type,
if_nat: $o > nat > nat > nat ).
thf(func_def_473,type,
if_num: $o > num > num > num ).
thf(func_def_474,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(func_def_475,type,
if_option_num: $o > option_num > option_num > option_num ).
thf(func_def_476,type,
if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).
thf(func_def_477,type,
if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_478,type,
if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).
thf(func_def_479,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_480,type,
if_rat: $o > rat > rat > rat ).
thf(func_def_481,type,
if_real: $o > real > real > real ).
thf(func_def_482,type,
if_set_int: $o > set_int > set_int > set_int ).
thf(func_def_483,type,
if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_484,type,
infini7641415182203889163d_enat: set_Extended_enat > nat > extended_enat ).
thf(func_def_485,type,
infini8530281810654367211te_nat: set_nat > nat > nat ).
thf(func_def_486,type,
abs_Integ: product_prod_nat_nat > int ).
thf(func_def_487,type,
rep_Integ: int > product_prod_nat_nat ).
thf(func_def_488,type,
int_ge_less_than: int > set_Pr958786334691620121nt_int ).
thf(func_def_489,type,
int_ge_less_than2: int > set_Pr958786334691620121nt_int ).
thf(func_def_490,type,
intrel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_491,type,
nat2: int > nat ).
thf(func_def_492,type,
pcr_int: product_prod_nat_nat > int > $o ).
thf(func_def_493,type,
power_int_real: real > int > real ).
thf(func_def_494,type,
ring_11222124179247155820nteger: set_Code_integer ).
thf(func_def_495,type,
ring_1_Ints_complex: set_complex ).
thf(func_def_496,type,
ring_1_Ints_int: set_int ).
thf(func_def_497,type,
ring_1_Ints_rat: set_rat ).
thf(func_def_498,type,
ring_1_Ints_real: set_real ).
thf(func_def_499,type,
ring_18347121197199848620nteger: int > code_integer ).
thf(func_def_500,type,
ring_17405671764205052669omplex: int > complex ).
thf(func_def_501,type,
ring_1_of_int_int: int > int ).
thf(func_def_502,type,
ring_1_of_int_rat: int > rat ).
thf(func_def_503,type,
ring_1_of_int_real: int > real ).
thf(func_def_504,type,
inf_inf_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(func_def_505,type,
inf_inf_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_506,type,
inf_inf_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_507,type,
inf_inf_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_508,type,
inf_in5163264567034779214_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_509,type,
inf_inf_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_510,type,
inf_inf_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(func_def_511,type,
inf_inf_int: int > int > int ).
thf(func_def_512,type,
inf_inf_nat: nat > nat > nat ).
thf(func_def_513,type,
inf_inf_rat: rat > rat > rat ).
thf(func_def_514,type,
inf_inf_real: real > real > real ).
thf(func_def_515,type,
inf_inf_set_o: set_o > set_o > set_o ).
thf(func_def_516,type,
inf_inf_set_complex: set_complex > set_complex > set_complex ).
thf(func_def_517,type,
inf_in8357106775501769908d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
thf(func_def_518,type,
inf_inf_set_int: set_int > set_int > set_int ).
thf(func_def_519,type,
inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_520,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_521,type,
inf_inf_set_num: set_num > set_num > set_num ).
thf(func_def_522,type,
inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_523,type,
inf_in4302113700860409141at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_524,type,
inf_in7913087082777306421at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_525,type,
inf_inf_set_rat: set_rat > set_rat > set_rat ).
thf(func_def_526,type,
inf_inf_set_real: set_real > set_real > set_real ).
thf(func_def_527,type,
inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_528,type,
semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_529,type,
sup_sup_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).
thf(func_def_530,type,
sup_sup_int_o: ( int > $o ) > ( int > $o ) > int > $o ).
thf(func_def_531,type,
sup_sup_list_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).
thf(func_def_532,type,
sup_sup_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).
thf(func_def_533,type,
sup_su8986005896011022210_nat_o: ( produc859450856879609959at_nat > $o ) > ( produc859450856879609959at_nat > $o ) > produc859450856879609959at_nat > $o ).
thf(func_def_534,type,
sup_su2080679670758317954_nat_o: ( produc3843707927480180839at_nat > $o ) > ( produc3843707927480180839at_nat > $o ) > produc3843707927480180839at_nat > $o ).
thf(func_def_535,type,
sup_sup_real_o: ( real > $o ) > ( real > $o ) > real > $o ).
thf(func_def_536,type,
sup_sup_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).
thf(func_def_537,type,
sup_sup_int: int > int > int ).
thf(func_def_538,type,
sup_sup_nat: nat > nat > nat ).
thf(func_def_539,type,
sup_sup_rat: rat > rat > rat ).
thf(func_def_540,type,
sup_sup_real: real > real > real ).
thf(func_def_541,type,
sup_sup_set_o: set_o > set_o > set_o ).
thf(func_def_542,type,
sup_sup_set_complex: set_complex > set_complex > set_complex ).
thf(func_def_543,type,
sup_su4489774667511045786d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat ).
thf(func_def_544,type,
sup_sup_set_int: set_int > set_int > set_int ).
thf(func_def_545,type,
sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat ).
thf(func_def_546,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_547,type,
sup_sup_set_num: set_num > set_num > set_num ).
thf(func_def_548,type,
sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_549,type,
sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_550,type,
sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_551,type,
sup_sup_set_rat: set_rat > set_rat > set_rat ).
thf(func_def_552,type,
sup_sup_set_real: set_real > set_real > set_real ).
thf(func_def_553,type,
sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat ).
thf(func_def_554,type,
lattic921264341876707157d_enat: set_Extended_enat > extended_enat ).
thf(func_def_555,type,
lattic8265883725875713057ax_nat: set_nat > nat ).
thf(func_def_556,type,
lattic8556559851467007577_o_num: ( $o > num ) > set_o > $o ).
thf(func_def_557,type,
lattic2140725968369957399_o_rat: ( $o > rat ) > set_o > $o ).
thf(func_def_558,type,
lattic8697145971487455083o_real: ( $o > real ) > set_o > $o ).
thf(func_def_559,type,
lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).
thf(func_def_560,type,
lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).
thf(func_def_561,type,
lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).
thf(func_def_562,type,
lattic402713867396545063at_num: ( extended_enat > num ) > set_Extended_enat > extended_enat ).
thf(func_def_563,type,
lattic3210252021154270693at_rat: ( extended_enat > rat ) > set_Extended_enat > extended_enat ).
thf(func_def_564,type,
lattic1189837152898106425t_real: ( extended_enat > real ) > set_Extended_enat > extended_enat ).
thf(func_def_565,type,
lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).
thf(func_def_566,type,
lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
thf(func_def_567,type,
lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
thf(func_def_568,type,
lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
thf(func_def_569,type,
lattic1613168225601753569al_num: ( real > num ) > set_real > real ).
thf(func_def_570,type,
lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).
thf(func_def_571,type,
lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
thf(func_def_572,type,
lattic7826324295020591184_F_nat: ( nat > nat > nat ) > nat > set_nat > nat ).
thf(func_def_573,type,
bfun_nat_real: ( nat > real ) > filter_nat > $o ).
thf(func_def_574,type,
at_infinity_real: filter_real ).
thf(func_def_575,type,
append_int: list_int > list_int > list_int ).
thf(func_def_576,type,
append_nat: list_nat > list_nat > list_nat ).
thf(func_def_577,type,
concat_o: list_list_o > list_o ).
thf(func_def_578,type,
concat_nat: list_list_nat > list_nat ).
thf(func_def_579,type,
concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_580,type,
enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).
thf(func_def_581,type,
enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
thf(func_def_582,type,
enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
thf(func_def_583,type,
enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
thf(func_def_584,type,
find_o: ( $o > $o ) > list_o > option_o ).
thf(func_def_585,type,
find_int: ( int > $o ) > list_int > option_int ).
thf(func_def_586,type,
find_nat: ( nat > $o ) > list_nat > option_nat ).
thf(func_def_587,type,
find_num: ( num > $o ) > list_num > option_num ).
thf(func_def_588,type,
find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).
thf(func_def_589,type,
find_real: ( real > $o ) > list_real > option_real ).
thf(func_def_590,type,
find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).
thf(func_def_591,type,
find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).
thf(func_def_592,type,
fold_nat_nat: ( nat > nat > nat ) > list_nat > nat > nat ).
thf(func_def_593,type,
last_nat: list_nat > nat ).
thf(func_def_594,type,
linord2614967742042102400et_nat: set_nat > list_nat ).
thf(func_def_595,type,
cons_o: $o > list_o > list_o ).
thf(func_def_596,type,
cons_int: int > list_int > list_int ).
thf(func_def_597,type,
cons_nat: nat > list_nat > list_nat ).
thf(func_def_598,type,
cons_num: num > list_num > list_num ).
thf(func_def_599,type,
cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(func_def_600,type,
cons_real: real > list_real > list_real ).
thf(func_def_601,type,
cons_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(func_def_602,type,
cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_603,type,
nil_int: list_int ).
thf(func_def_604,type,
nil_nat: list_nat ).
thf(func_def_605,type,
hd_nat: list_nat > nat ).
thf(func_def_606,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(func_def_607,type,
set_o2: list_o > set_o ).
thf(func_def_608,type,
set_complex2: list_complex > set_complex ).
thf(func_def_609,type,
set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
thf(func_def_610,type,
set_int2: list_int > set_int ).
thf(func_def_611,type,
set_list_o2: list_list_o > set_list_o ).
thf(func_def_612,type,
set_list_nat2: list_list_nat > set_list_nat ).
thf(func_def_613,type,
set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).
thf(func_def_614,type,
set_nat2: list_nat > set_nat ).
thf(func_def_615,type,
set_num2: list_num > set_num ).
thf(func_def_616,type,
set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_617,type,
set_real2: list_real > set_real ).
thf(func_def_618,type,
set_set_nat2: list_set_nat > set_set_nat ).
thf(func_def_619,type,
set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_620,type,
size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
thf(func_def_621,type,
list_update_o: list_o > nat > $o > list_o ).
thf(func_def_622,type,
list_update_int: list_int > nat > int > list_int ).
thf(func_def_623,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(func_def_624,type,
list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
thf(func_def_625,type,
list_update_real: list_real > nat > real > list_real ).
thf(func_def_626,type,
list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
thf(func_def_627,type,
list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_628,type,
nth_o: list_o > nat > $o ).
thf(func_def_629,type,
nth_int: list_int > nat > int ).
thf(func_def_630,type,
nth_nat: list_nat > nat > nat ).
thf(func_def_631,type,
nth_num: list_num > nat > num ).
thf(func_def_632,type,
nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
thf(func_def_633,type,
nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
thf(func_def_634,type,
nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).
thf(func_def_635,type,
nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).
thf(func_def_636,type,
nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).
thf(func_def_637,type,
nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).
thf(func_def_638,type,
nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
thf(func_def_639,type,
nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
thf(func_def_640,type,
nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
thf(func_def_641,type,
nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
thf(func_def_642,type,
nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
thf(func_def_643,type,
nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
thf(func_def_644,type,
nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
thf(func_def_645,type,
nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
thf(func_def_646,type,
nth_real: list_real > nat > real ).
thf(func_def_647,type,
nth_set_nat: list_set_nat > nat > set_nat ).
thf(func_def_648,type,
nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_649,type,
product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_650,type,
product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_651,type,
product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
thf(func_def_652,type,
product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(func_def_653,type,
product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
thf(func_def_654,type,
produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
thf(func_def_655,type,
product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_656,type,
produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_657,type,
produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(func_def_658,type,
produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_659,type,
removeAll_o: $o > list_o > list_o ).
thf(func_def_660,type,
removeAll_int: int > list_int > list_int ).
thf(func_def_661,type,
removeAll_nat: nat > list_nat > list_nat ).
thf(func_def_662,type,
remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
thf(func_def_663,type,
removeAll_real: real > list_real > list_real ).
thf(func_def_664,type,
removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
thf(func_def_665,type,
removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_666,type,
replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_667,type,
rotate1_o: list_o > list_o ).
thf(func_def_668,type,
rotate1_int: list_int > list_int ).
thf(func_def_669,type,
rotate1_nat: list_nat > list_nat ).
thf(func_def_670,type,
rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_671,type,
sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
thf(func_def_672,type,
sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
thf(func_def_673,type,
take_nat: nat > list_nat > list_nat ).
thf(func_def_674,type,
upt: nat > nat > list_nat ).
thf(func_def_675,type,
upto: int > int > list_int ).
thf(func_def_676,type,
upto_aux: int > int > list_int > list_int ).
thf(func_def_677,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_678,type,
zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
thf(func_def_679,type,
zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
thf(func_def_680,type,
zip_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
thf(func_def_681,type,
zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
thf(func_def_682,type,
zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
thf(func_def_683,type,
zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
thf(func_def_684,type,
zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
thf(func_def_685,type,
zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
thf(func_def_686,type,
zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
thf(func_def_687,type,
zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
thf(func_def_688,type,
suc: nat > nat ).
thf(func_def_689,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_690,type,
case_nat_o: $o > ( nat > $o ) > nat > $o ).
thf(func_def_691,type,
case_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(func_def_692,type,
pred: nat > nat ).
thf(func_def_693,type,
semiri4939895301339042750nteger: nat > code_integer ).
thf(func_def_694,type,
semiri8010041392384452111omplex: nat > complex ).
thf(func_def_695,type,
semiri4216267220026989637d_enat: nat > extended_enat ).
thf(func_def_696,type,
semiri1314217659103216013at_int: nat > int ).
thf(func_def_697,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(func_def_698,type,
semiri681578069525770553at_rat: nat > rat ).
thf(func_def_699,type,
semiri5074537144036343181t_real: nat > real ).
thf(func_def_700,type,
semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
thf(func_def_701,type,
semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
thf(func_def_702,type,
semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
thf(func_def_703,type,
semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
thf(func_def_704,type,
size_size_list_o: list_o > nat ).
thf(func_def_705,type,
size_s3451745648224563538omplex: list_complex > nat ).
thf(func_def_706,type,
size_s3941691890525107288d_enat: list_Extended_enat > nat ).
thf(func_def_707,type,
size_size_list_int: list_int > nat ).
thf(func_def_708,type,
size_s2710708370519433104list_o: list_list_o > nat ).
thf(func_def_709,type,
size_s3023201423986296836st_nat: list_list_nat > nat ).
thf(func_def_710,type,
size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat ).
thf(func_def_711,type,
size_size_list_nat: list_nat > nat ).
thf(func_def_712,type,
size_size_list_num: list_num > nat ).
thf(func_def_713,type,
size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
thf(func_def_714,type,
size_size_list_real: list_real > nat ).
thf(func_def_715,type,
size_s3254054031482475050et_nat: list_set_nat > nat ).
thf(func_def_716,type,
size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
thf(func_def_717,type,
size_size_num: num > nat ).
thf(func_def_718,type,
size_size_option_nat: option_nat > nat ).
thf(func_def_719,type,
size_size_option_num: option_num > nat ).
thf(func_def_720,type,
size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
thf(func_def_721,type,
size_size_VEBT_VEBT: vEBT_VEBT > nat ).
thf(func_def_722,type,
nat_list_decode: nat > list_nat ).
thf(func_def_723,type,
nat_list_decode_rel: nat > nat > $o ).
thf(func_def_724,type,
nat_list_encode: list_nat > nat ).
thf(func_def_725,type,
nat_list_encode_rel: list_nat > list_nat > $o ).
thf(func_def_726,type,
nat_prod_decode: nat > product_prod_nat_nat ).
thf(func_def_727,type,
nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
thf(func_def_728,type,
nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_729,type,
nat_prod_encode: product_prod_nat_nat > nat ).
thf(func_def_730,type,
nat_set_decode: nat > set_nat ).
thf(func_def_731,type,
nat_set_encode: set_nat > nat ).
thf(func_def_732,type,
root: nat > real > real ).
thf(func_def_733,type,
sqrt: real > real ).
thf(func_def_734,type,
inc: num > num ).
thf(func_def_735,type,
neg_nu7757733837767384882nteger: code_integer > code_integer ).
thf(func_def_736,type,
neg_nu6511756317524482435omplex: complex > complex ).
thf(func_def_737,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(func_def_738,type,
neg_nu3179335615603231917ec_rat: rat > rat ).
thf(func_def_739,type,
neg_nu6075765906172075777c_real: real > real ).
thf(func_def_740,type,
neg_nu8557863876264182079omplex: complex > complex ).
thf(func_def_741,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(func_def_742,type,
neg_nu5219082963157363817nc_rat: rat > rat ).
thf(func_def_743,type,
neg_nu8295874005876285629c_real: real > real ).
thf(func_def_744,type,
bit0: num > num ).
thf(func_def_745,type,
bit1: num > num ).
thf(func_def_746,type,
one: num ).
thf(func_def_747,type,
size_num: num > nat ).
thf(func_def_748,type,
num_of_nat: nat > num ).
thf(func_def_749,type,
numera6620942414471956472nteger: num > code_integer ).
thf(func_def_750,type,
numera6690914467698888265omplex: num > complex ).
thf(func_def_751,type,
numera1916890842035813515d_enat: num > extended_enat ).
thf(func_def_752,type,
numeral_numeral_int: num > int ).
thf(func_def_753,type,
numeral_numeral_nat: num > nat ).
thf(func_def_754,type,
numeral_numeral_rat: num > rat ).
thf(func_def_755,type,
numeral_numeral_real: num > real ).
thf(func_def_756,type,
pred_numeral: num > nat ).
thf(func_def_757,type,
none_o: option_o ).
thf(func_def_758,type,
none_int: option_int ).
thf(func_def_759,type,
none_nat: option_nat ).
thf(func_def_760,type,
none_num: option_num ).
thf(func_def_761,type,
none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
thf(func_def_762,type,
none_real: option_real ).
thf(func_def_763,type,
none_set_nat: option_set_nat ).
thf(func_def_764,type,
none_VEBT_VEBT: option_VEBT_VEBT ).
thf(func_def_765,type,
some_o: $o > option_o ).
thf(func_def_766,type,
some_int: int > option_int ).
thf(func_def_767,type,
some_nat: nat > option_nat ).
thf(func_def_768,type,
some_num: num > option_num ).
thf(func_def_769,type,
some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
thf(func_def_770,type,
some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).
thf(func_def_771,type,
case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
thf(func_def_772,type,
size_option_nat: ( nat > nat ) > option_nat > nat ).
thf(func_def_773,type,
size_option_num: ( num > nat ) > option_num > nat ).
thf(func_def_774,type,
size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
thf(func_def_775,type,
the_nat: option_nat > nat ).
thf(func_def_776,type,
the_num: option_num > num ).
thf(func_def_777,type,
the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
thf(func_def_778,type,
order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
thf(func_def_779,type,
order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_780,type,
bot_bot_o_o: $o > $o ).
thf(func_def_781,type,
bot_bot_int_int_o: int > int > $o ).
thf(func_def_782,type,
bot_bot_int_o: int > $o ).
thf(func_def_783,type,
bot_bot_list_nat_o: list_nat > $o ).
thf(func_def_784,type,
bot_bot_nat_nat_o: nat > nat > $o ).
thf(func_def_785,type,
bot_bot_nat_o: nat > $o ).
thf(func_def_786,type,
bot_bo5043116465536727218_nat_o: option_nat > option_nat > $o ).
thf(func_def_787,type,
bot_bot_real_o: real > $o ).
thf(func_def_788,type,
bot_bot_set_nat_o: set_nat > $o ).
thf(func_def_789,type,
bot_bo394778441745866138_nat_o: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_790,type,
bot_bo3364206721330744218_nat_o: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(func_def_792,type,
bot_bo4199563552545308370d_enat: extended_enat ).
thf(func_def_793,type,
bot_bot_filter_nat: filter_nat ).
thf(func_def_794,type,
bot_bot_nat: nat ).
thf(func_def_795,type,
bot_bot_set_o: set_o ).
thf(func_def_796,type,
bot_bot_set_complex: set_complex ).
thf(func_def_797,type,
bot_bo7653980558646680370d_enat: set_Extended_enat ).
thf(func_def_798,type,
bot_bot_set_int: set_int ).
thf(func_def_799,type,
bot_bot_set_list_nat: set_list_nat ).
thf(func_def_800,type,
bot_bot_set_nat: set_nat ).
thf(func_def_801,type,
bot_bot_set_num: set_num ).
thf(func_def_802,type,
bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).
thf(func_def_803,type,
bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_804,type,
bot_bo232370072503712749on_nat: set_Pr6588086440996610945on_nat ).
thf(func_def_805,type,
bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_nat ).
thf(func_def_806,type,
bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).
thf(func_def_807,type,
bot_bo4948859079157340979at_nat: set_Pr7459493094073627847at_nat ).
thf(func_def_808,type,
bot_bot_set_rat: set_rat ).
thf(func_def_809,type,
bot_bot_set_real: set_real ).
thf(func_def_810,type,
bot_bot_set_set_int: set_set_int ).
thf(func_def_811,type,
bot_bot_set_set_nat: set_set_nat ).
thf(func_def_812,type,
bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).
thf(func_def_813,type,
ord_Le1955565732374568822d_enat: ( extended_enat > $o ) > extended_enat ).
thf(func_def_814,type,
ord_Least_nat: ( nat > $o ) > nat ).
thf(func_def_815,type,
ord_Least_real: ( real > $o ) > real ).
thf(func_def_816,type,
ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(func_def_817,type,
ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_818,type,
ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_819,type,
ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_820,type,
ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(func_def_821,type,
ord_less_o: $o > $o > $o ).
thf(func_def_822,type,
ord_le6747313008572928689nteger: code_integer > code_integer > $o ).
thf(func_def_823,type,
ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).
thf(func_def_824,type,
ord_less_int: int > int > $o ).
thf(func_def_825,type,
ord_less_nat: nat > nat > $o ).
thf(func_def_826,type,
ord_less_num: num > num > $o ).
thf(func_def_827,type,
ord_less_rat: rat > rat > $o ).
thf(func_def_828,type,
ord_less_real: real > real > $o ).
thf(func_def_829,type,
ord_less_set_o: set_o > set_o > $o ).
thf(func_def_830,type,
ord_less_set_complex: set_complex > set_complex > $o ).
thf(func_def_831,type,
ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o ).
thf(func_def_832,type,
ord_less_set_int: set_int > set_int > $o ).
thf(func_def_833,type,
ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o ).
thf(func_def_834,type,
ord_less_set_nat: set_nat > set_nat > $o ).
thf(func_def_835,type,
ord_less_set_num: set_num > set_num > $o ).
thf(func_def_836,type,
ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_837,type,
ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_838,type,
ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(func_def_839,type,
ord_less_set_rat: set_rat > set_rat > $o ).
thf(func_def_840,type,
ord_less_set_real: set_real > set_real > $o ).
thf(func_def_841,type,
ord_less_set_set_int: set_set_int > set_set_int > $o ).
thf(func_def_842,type,
ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_843,type,
ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).
thf(func_def_844,type,
ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).
thf(func_def_845,type,
ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).
thf(func_def_846,type,
ord_le6558929396352911974_nat_o: ( list_nat > list_nat > $o ) > ( list_nat > list_nat > $o ) > $o ).
thf(func_def_847,type,
ord_le1520216061033275535_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > $o ).
thf(func_def_848,type,
ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).
thf(func_def_849,type,
ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).
thf(func_def_850,type,
ord_le8905833333647802342_nat_o: ( option_nat > option_nat > $o ) > ( option_nat > option_nat > $o ) > $o ).
thf(func_def_851,type,
ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).
thf(func_def_852,type,
ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $o ).
thf(func_def_853,type,
ord_le5604493270027003598_nat_o: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > $o ).
thf(func_def_854,type,
ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).
thf(func_def_855,type,
ord_le7862513914298786254_nat_o: ( produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ) > ( produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ) > $o ).
thf(func_def_856,type,
ord_le8126618931240741628_nat_o: ( produc4953844613479565601on_nat > $o ) > ( produc4953844613479565601on_nat > $o ) > $o ).
thf(func_def_857,type,
ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).
thf(func_def_858,type,
ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).
thf(func_def_859,type,
ord_le3935385432712749774_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > $o ).
thf(func_def_860,type,
ord_le3072208448688395470_nat_o: ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > ( set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ) > $o ).
thf(func_def_861,type,
ord_less_eq_o: $o > $o > $o ).
thf(func_def_862,type,
ord_le3102999989581377725nteger: code_integer > code_integer > $o ).
thf(func_def_863,type,
ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).
thf(func_def_864,type,
ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).
thf(func_def_865,type,
ord_le4104064031414453916r_real: filter_real > filter_real > $o ).
thf(func_def_866,type,
ord_less_eq_int: int > int > $o ).
thf(func_def_867,type,
ord_less_eq_nat: nat > nat > $o ).
thf(func_def_868,type,
ord_less_eq_num: num > num > $o ).
thf(func_def_869,type,
ord_less_eq_rat: rat > rat > $o ).
thf(func_def_870,type,
ord_less_eq_real: real > real > $o ).
thf(func_def_871,type,
ord_less_eq_set_o: set_o > set_o > $o ).
thf(func_def_872,type,
ord_le211207098394363844omplex: set_complex > set_complex > $o ).
thf(func_def_873,type,
ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o ).
thf(func_def_874,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(func_def_875,type,
ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).
thf(func_def_876,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(func_def_877,type,
ord_less_eq_set_num: set_num > set_num > $o ).
thf(func_def_878,type,
ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_879,type,
ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_880,type,
ord_le6406482658798684961on_nat: set_Pr6588086440996610945on_nat > set_Pr6588086440996610945on_nat > $o ).
thf(func_def_881,type,
ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_882,type,
ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(func_def_883,type,
ord_le5997549366648089703at_nat: set_Pr7459493094073627847at_nat > set_Pr7459493094073627847at_nat > $o ).
thf(func_def_884,type,
ord_less_eq_set_rat: set_rat > set_rat > $o ).
thf(func_def_885,type,
ord_less_eq_set_real: set_real > set_real > $o ).
thf(func_def_886,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(func_def_887,type,
ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).
thf(func_def_888,type,
ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_889,type,
ord_max_Code_integer: code_integer > code_integer > code_integer ).
thf(func_def_890,type,
ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat ).
thf(func_def_891,type,
ord_max_int: int > int > int ).
thf(func_def_892,type,
ord_max_nat: nat > nat > nat ).
thf(func_def_893,type,
ord_max_num: num > num > num ).
thf(func_def_894,type,
ord_max_rat: rat > rat > rat ).
thf(func_def_895,type,
ord_max_real: real > real > real ).
thf(func_def_896,type,
ord_max_set_o: set_o > set_o > set_o ).
thf(func_def_897,type,
ord_max_set_int: set_int > set_int > set_int ).
thf(func_def_898,type,
ord_max_set_nat: set_nat > set_nat > set_nat ).
thf(func_def_899,type,
ord_max_set_real: set_real > set_real > set_real ).
thf(func_def_900,type,
ord_min_nat: nat > nat > nat ).
thf(func_def_901,type,
order_Greatest_nat: ( nat > $o ) > nat ).
thf(func_def_902,type,
order_9091379641038594480t_real: ( nat > real ) > $o ).
thf(func_def_903,type,
order_mono_nat_nat: ( nat > nat ) > $o ).
thf(func_def_904,type,
order_mono_nat_real: ( nat > real ) > $o ).
thf(func_def_905,type,
order_5726023648592871131at_nat: ( nat > nat ) > $o ).
thf(func_def_906,type,
order_7092887310737990675l_real: ( real > real ) > $o ).
thf(func_def_907,type,
ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).
thf(func_def_908,type,
top_top_set_o: set_o ).
thf(func_def_909,type,
top_top_set_nat: set_nat ).
thf(func_def_910,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(func_def_911,type,
top_top_set_real: set_real ).
thf(func_def_912,type,
top_top_set_char: set_char ).
thf(func_def_913,type,
power_8256067586552552935nteger: code_integer > nat > code_integer ).
thf(func_def_914,type,
power_power_complex: complex > nat > complex ).
thf(func_def_915,type,
power_power_int: int > nat > int ).
thf(func_def_916,type,
power_power_nat: nat > nat > nat ).
thf(func_def_917,type,
power_power_rat: rat > nat > rat ).
thf(func_def_918,type,
power_power_real: real > nat > real ).
thf(func_def_919,type,
produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).
thf(func_def_920,type,
produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).
thf(func_def_921,type,
produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).
thf(func_def_922,type,
produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).
thf(func_def_923,type,
produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).
thf(func_def_924,type,
produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
thf(func_def_925,type,
produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).
thf(func_def_926,type,
product_Pair_o_int: $o > int > product_prod_o_int ).
thf(func_def_927,type,
produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
thf(func_def_928,type,
produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).
thf(func_def_929,type,
produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).
thf(func_def_930,type,
product_Pair_int_o: int > $o > product_prod_int_o ).
thf(func_def_931,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(func_def_932,type,
product_Pair_int_nat: int > nat > product_prod_int_nat ).
thf(func_def_933,type,
produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).
thf(func_def_934,type,
product_Pair_nat_o: nat > $o > product_prod_nat_o ).
thf(func_def_935,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(func_def_936,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(func_def_937,type,
produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).
thf(func_def_938,type,
produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).
thf(func_def_939,type,
produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
thf(func_def_940,type,
produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).
thf(func_def_941,type,
produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).
thf(func_def_942,type,
produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).
thf(func_def_943,type,
produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).
thf(func_def_944,type,
produc6639060556116774935d_enat: set_Extended_enat > set_Extended_enat > produc1621487020699730983d_enat ).
thf(func_def_945,type,
produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
thf(func_def_946,type,
produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
thf(func_def_947,type,
produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
thf(func_def_948,type,
produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).
thf(func_def_949,type,
produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
thf(func_def_950,type,
produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
thf(func_def_951,type,
produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
thf(func_def_952,type,
produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
thf(func_def_953,type,
produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_954,type,
produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_955,type,
produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
thf(func_def_956,type,
produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
thf(func_def_957,type,
produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
thf(func_def_958,type,
produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
thf(func_def_959,type,
produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
thf(func_def_960,type,
produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
thf(func_def_961,type,
produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
thf(func_def_962,type,
produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
thf(func_def_963,type,
produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
thf(func_def_964,type,
produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_965,type,
produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
thf(func_def_966,type,
produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
thf(func_def_967,type,
produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
thf(func_def_968,type,
product_fst_int_int: product_prod_int_int > int ).
thf(func_def_969,type,
product_fst_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_970,type,
product_snd_int_int: product_prod_int_int > int ).
thf(func_def_971,type,
product_snd_nat_nat: product_prod_nat_nat > nat ).
thf(func_def_972,type,
product_Abs_unit: $o > product_unit ).
thf(func_def_973,type,
product_Rep_unit: product_unit > $o ).
thf(func_def_974,type,
abs_Rat: product_prod_int_int > rat ).
thf(func_def_975,type,
fract: int > int > rat ).
thf(func_def_976,type,
rep_Rat: rat > product_prod_int_int ).
thf(func_def_977,type,
field_5140801741446780682s_real: set_real ).
thf(func_def_978,type,
field_7254667332652039916t_real: rat > real ).
thf(func_def_979,type,
normalize: product_prod_int_int > product_prod_int_int ).
thf(func_def_980,type,
pcr_rat: product_prod_int_int > rat > $o ).
thf(func_def_981,type,
positive: rat > $o ).
thf(func_def_982,type,
quotient_of: rat > product_prod_int_int ).
thf(func_def_983,type,
ratrel: product_prod_int_int > product_prod_int_int > $o ).
thf(func_def_984,type,
real2: ( nat > rat ) > real ).
thf(func_def_985,type,
cauchy: ( nat > rat ) > $o ).
thf(func_def_986,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(func_def_987,type,
positive2: real > $o ).
thf(func_def_988,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(func_def_989,type,
rep_real: real > nat > rat ).
thf(func_def_990,type,
vanishes: ( nat > rat ) > $o ).
thf(func_def_991,type,
real_V3694042436643373181omplex: complex > complex > real ).
thf(func_def_992,type,
real_V975177566351809787t_real: real > real > real ).
thf(func_def_993,type,
real_V1022390504157884413omplex: complex > real ).
thf(func_def_994,type,
real_V7735802525324610683m_real: real > real ).
thf(func_def_995,type,
real_V4546457046886955230omplex: real > complex ).
thf(func_def_996,type,
field_nat: set_Pr1261947904930325089at_nat > set_nat ).
thf(func_def_997,type,
id_Pro2258643101195443293at_nat: set_Pr8693737435421807431at_nat ).
thf(func_def_998,type,
total_3592101749530773125at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_999,type,
trans_4347625901269045472at_nat: set_Pr8693737435421807431at_nat > $o ).
thf(func_def_1000,type,
algebr932160517623751201me_int: int > int > $o ).
thf(func_def_1001,type,
algebr934650988132801477me_nat: nat > nat > $o ).
thf(func_def_1002,type,
divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
thf(func_def_1003,type,
divide1717551699836669952omplex: complex > complex > complex ).
thf(func_def_1004,type,
divide_divide_int: int > int > int ).
thf(func_def_1005,type,
divide_divide_nat: nat > nat > nat ).
thf(func_def_1006,type,
divide_divide_rat: rat > rat > rat ).
thf(func_def_1007,type,
divide_divide_real: real > real > real ).
thf(func_def_1008,type,
dvd_dvd_int: int > int > $o ).
thf(func_def_1009,type,
dvd_dvd_nat: nat > nat > $o ).
thf(func_def_1010,type,
modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
thf(func_def_1011,type,
modulo_modulo_int: int > int > int ).
thf(func_def_1012,type,
modulo_modulo_nat: nat > nat > nat ).
thf(func_def_1013,type,
unit_f2748546683901255202or_nat: nat > nat ).
thf(func_def_1014,type,
zero_n2684676970156552555ol_int: $o > int ).
thf(func_def_1015,type,
zero_n2687167440665602831ol_nat: $o > nat ).
thf(func_def_1016,type,
suminf_complex: ( nat > complex ) > complex ).
thf(func_def_1017,type,
suminf_int: ( nat > int ) > int ).
thf(func_def_1018,type,
suminf_nat: ( nat > nat ) > nat ).
thf(func_def_1019,type,
suminf_real: ( nat > real ) > real ).
thf(func_def_1020,type,
summable_complex: ( nat > complex ) > $o ).
thf(func_def_1021,type,
summable_int: ( nat > int ) > $o ).
thf(func_def_1022,type,
summable_nat: ( nat > nat ) > $o ).
thf(func_def_1023,type,
summable_real: ( nat > real ) > $o ).
thf(func_def_1024,type,
sums_complex: ( nat > complex ) > complex > $o ).
thf(func_def_1025,type,
sums_int: ( nat > int ) > int > $o ).
thf(func_def_1026,type,
sums_nat: ( nat > nat ) > nat > $o ).
thf(func_def_1027,type,
sums_real: ( nat > real ) > real > $o ).
thf(func_def_1028,type,
collect_o: ( $o > $o ) > set_o ).
thf(func_def_1029,type,
collect_complex: ( complex > $o ) > set_complex ).
thf(func_def_1030,type,
collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
thf(func_def_1031,type,
collect_int: ( int > $o ) > set_int ).
thf(func_def_1032,type,
collect_list_o: ( list_o > $o ) > set_list_o ).
thf(func_def_1033,type,
collect_list_complex: ( list_complex > $o ) > set_list_complex ).
thf(func_def_1034,type,
collec8433460942617342167d_enat: ( list_Extended_enat > $o ) > set_li5464603477888414924d_enat ).
thf(func_def_1035,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(func_def_1036,type,
collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).
thf(func_def_1037,type,
collect_list_nat: ( list_nat > $o ) > set_list_nat ).
thf(func_def_1038,type,
collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).
thf(func_def_1039,type,
collect_list_set_nat: ( list_set_nat > $o ) > set_list_set_nat ).
thf(func_def_1040,type,
collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).
thf(func_def_1041,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(func_def_1042,type,
collect_num: ( num > $o ) > set_num ).
thf(func_def_1043,type,
collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).
thf(func_def_1044,type,
collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).
thf(func_def_1045,type,
collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1046,type,
collec7088162979684241874at_nat: ( produc859450856879609959at_nat > $o ) > set_Pr8693737435421807431at_nat ).
thf(func_def_1047,type,
collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).
thf(func_def_1048,type,
collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).
thf(func_def_1049,type,
collect_rat: ( rat > $o ) > set_rat ).
thf(func_def_1050,type,
collect_real: ( real > $o ) > set_real ).
thf(func_def_1051,type,
collect_set_complex: ( set_complex > $o ) > set_set_complex ).
thf(func_def_1052,type,
collec2260605976452661553d_enat: ( set_Extended_enat > $o ) > set_se7270636423289371942d_enat ).
thf(func_def_1053,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(func_def_1054,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(func_def_1055,type,
collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).
thf(func_def_1056,type,
pow_nat: set_nat > set_set_nat ).
thf(func_def_1057,type,
image_80655429650038917d_enat: ( extended_enat > extended_enat ) > set_Extended_enat > set_Extended_enat ).
thf(func_def_1058,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(func_def_1059,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(func_def_1060,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(func_def_1061,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_1062,type,
image_nat_real: ( nat > real ) > set_nat > set_real ).
thf(func_def_1063,type,
image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).
thf(func_def_1064,type,
image_nat_char: ( nat > char ) > set_nat > set_char ).
thf(func_def_1065,type,
image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).
thf(func_def_1066,type,
image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).
thf(func_def_1067,type,
image_real_real: ( real > real ) > set_real > set_real ).
thf(func_def_1068,type,
image_char_nat: ( char > nat ) > set_char > set_nat ).
thf(func_def_1069,type,
insert_o: $o > set_o > set_o ).
thf(func_def_1070,type,
insert_complex: complex > set_complex > set_complex ).
thf(func_def_1071,type,
insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat ).
thf(func_def_1072,type,
insert_int: int > set_int > set_int ).
thf(func_def_1073,type,
insert_list_nat: list_nat > set_list_nat > set_list_nat ).
thf(func_def_1074,type,
insert_nat: nat > set_nat > set_nat ).
thf(func_def_1075,type,
insert_num: num > set_num > set_num ).
thf(func_def_1076,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1077,type,
insert5050368324300391991at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_1078,type,
insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_1079,type,
insert_rat: rat > set_rat > set_rat ).
thf(func_def_1080,type,
insert_real: real > set_real > set_real ).
thf(func_def_1081,type,
insert_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(func_def_1082,type,
insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).
thf(func_def_1083,type,
is_empty_o: set_o > $o ).
thf(func_def_1084,type,
is_empty_int: set_int > $o ).
thf(func_def_1085,type,
is_empty_nat: set_nat > $o ).
thf(func_def_1086,type,
is_empty_real: set_real > $o ).
thf(func_def_1087,type,
is_singleton_o: set_o > $o ).
thf(func_def_1088,type,
is_singleton_complex: set_complex > $o ).
thf(func_def_1089,type,
is_singleton_int: set_int > $o ).
thf(func_def_1090,type,
is_sin2641923865335537900st_nat: set_list_nat > $o ).
thf(func_def_1091,type,
is_singleton_nat: set_nat > $o ).
thf(func_def_1092,type,
is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1093,type,
is_singleton_real: set_real > $o ).
thf(func_def_1094,type,
is_singleton_set_nat: set_set_nat > $o ).
thf(func_def_1095,type,
remove_o: $o > set_o > set_o ).
thf(func_def_1096,type,
remove_int: int > set_int > set_int ).
thf(func_def_1097,type,
remove_nat: nat > set_nat > set_nat ).
thf(func_def_1098,type,
remove6466555014256735590at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1099,type,
remove_real: real > set_real > set_real ).
thf(func_def_1100,type,
remove_set_nat: set_nat > set_set_nat > set_set_nat ).
thf(func_def_1101,type,
the_elem_o: set_o > $o ).
thf(func_def_1102,type,
the_elem_int: set_int > int ).
thf(func_def_1103,type,
the_elem_nat: set_nat > nat ).
thf(func_def_1104,type,
the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).
thf(func_def_1105,type,
the_elem_real: set_real > real ).
thf(func_def_1106,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(func_def_1107,type,
set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).
thf(func_def_1108,type,
set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).
thf(func_def_1109,type,
set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).
thf(func_def_1110,type,
set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).
thf(func_def_1111,type,
set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ).
thf(func_def_1112,type,
set_or8904488021354931149Most_o: $o > $o > set_o ).
thf(func_def_1113,type,
set_or1266510415728281911st_int: int > int > set_int ).
thf(func_def_1114,type,
set_or1269000886237332187st_nat: nat > nat > set_nat ).
thf(func_def_1115,type,
set_or7049704709247886629st_num: num > num > set_num ).
thf(func_def_1116,type,
set_or633870826150836451st_rat: rat > rat > set_rat ).
thf(func_def_1117,type,
set_or1222579329274155063t_real: real > real > set_real ).
thf(func_def_1118,type,
set_or370866239135849197et_int: set_int > set_int > set_set_int ).
thf(func_def_1119,type,
set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).
thf(func_def_1120,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(func_def_1121,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(func_def_1122,type,
set_ord_atLeast_nat: nat > set_nat ).
thf(func_def_1123,type,
set_ord_atLeast_real: real > set_real ).
thf(func_def_1124,type,
set_ord_atMost_int: int > set_int ).
thf(func_def_1125,type,
set_ord_atMost_nat: nat > set_nat ).
thf(func_def_1126,type,
set_or6656581121297822940st_int: int > int > set_int ).
thf(func_def_1127,type,
set_or6659071591806873216st_nat: nat > nat > set_nat ).
thf(func_def_1128,type,
set_or5832277885323065728an_int: int > int > set_int ).
thf(func_def_1129,type,
set_or5834768355832116004an_nat: nat > nat > set_nat ).
thf(func_def_1130,type,
set_or1633881224788618240n_real: real > real > set_real ).
thf(func_def_1131,type,
set_or1210151606488870762an_nat: nat > set_nat ).
thf(func_def_1132,type,
set_or5849166863359141190n_real: real > set_real ).
thf(func_def_1133,type,
set_ord_lessThan_o: $o > set_o ).
thf(func_def_1134,type,
set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).
thf(func_def_1135,type,
set_ord_lessThan_int: int > set_int ).
thf(func_def_1136,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(func_def_1137,type,
set_ord_lessThan_num: num > set_num ).
thf(func_def_1138,type,
set_ord_lessThan_rat: rat > set_rat ).
thf(func_def_1139,type,
set_or5984915006950818249n_real: real > set_real ).
thf(func_def_1140,type,
set_or890127255671739683et_nat: set_nat > set_set_nat ).
thf(func_def_1141,type,
abort_real: literal > ( product_unit > real ) > real ).
thf(func_def_1142,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(func_def_1143,type,
comm_s629917340098488124ar_nat: char > nat ).
thf(func_def_1144,type,
unique3096191561947761185of_nat: nat > char ).
thf(func_def_1145,type,
topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
thf(func_def_1146,type,
topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
thf(func_def_1147,type,
topolo4899668324122417113eq_int: ( nat > int ) > $o ).
thf(func_def_1148,type,
topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
thf(func_def_1149,type,
topolo1459490580787246023eq_num: ( nat > num ) > $o ).
thf(func_def_1150,type,
topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
thf(func_def_1151,type,
topolo6980174941875973593q_real: ( nat > real ) > $o ).
thf(func_def_1152,type,
topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
thf(func_def_1153,type,
topolo2177554685111907308n_real: real > set_real > filter_real ).
thf(func_def_1154,type,
topolo7531315842566124627t_real: ( nat > real ) > $o ).
thf(func_def_1155,type,
topolo2815343760600316023s_real: real > filter_real ).
thf(func_def_1156,type,
topolo4055970368930404560y_real: ( nat > real ) > $o ).
thf(func_def_1157,type,
topolo896644834953643431omplex: filter6041513312241820739omplex ).
thf(func_def_1158,type,
topolo1511823702728130853y_real: filter2146258269922977983l_real ).
thf(func_def_1159,type,
arccos: real > real ).
thf(func_def_1160,type,
arcosh_real: real > real ).
thf(func_def_1161,type,
arcsin: real > real ).
thf(func_def_1162,type,
arctan: real > real ).
thf(func_def_1163,type,
arsinh_real: real > real ).
thf(func_def_1164,type,
artanh_real: real > real ).
thf(func_def_1165,type,
cos_real: real > real ).
thf(func_def_1166,type,
cos_coeff: nat > real ).
thf(func_def_1167,type,
cosh_real: real > real ).
thf(func_def_1168,type,
cot_real: real > real ).
thf(func_def_1169,type,
diffs_complex: ( nat > complex ) > nat > complex ).
thf(func_def_1170,type,
diffs_real: ( nat > real ) > nat > real ).
thf(func_def_1171,type,
exp_complex: complex > complex ).
thf(func_def_1172,type,
exp_real: real > real ).
thf(func_def_1173,type,
ln_ln_real: real > real ).
thf(func_def_1174,type,
log: real > real > real ).
thf(func_def_1175,type,
pi: real ).
thf(func_def_1176,type,
powr_real: real > real > real ).
thf(func_def_1177,type,
powr_real2: real > real > real ).
thf(func_def_1178,type,
sin_real: real > real ).
thf(func_def_1179,type,
sin_coeff: nat > real ).
thf(func_def_1180,type,
sinh_complex: complex > complex ).
thf(func_def_1181,type,
sinh_real: real > real ).
thf(func_def_1182,type,
tan_real: real > real ).
thf(func_def_1183,type,
tanh_real: real > real ).
thf(func_def_1184,type,
transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1185,type,
transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(func_def_1186,type,
type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
thf(func_def_1187,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(func_def_1188,type,
vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1189,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(func_def_1190,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(func_def_1191,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(func_def_1192,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(func_def_1193,type,
vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
thf(func_def_1194,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(func_def_1195,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(func_def_1196,type,
vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1197,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(func_def_1198,type,
vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1199,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(func_def_1200,type,
vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1201,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(func_def_1202,type,
vEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_1203,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(func_def_1204,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(func_def_1205,type,
vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1206,type,
vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1207,type,
vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1208,type,
vEBT_VEBT_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1209,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1210,type,
vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1211,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(func_def_1212,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(func_def_1213,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1214,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
thf(func_def_1215,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(func_def_1216,type,
vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1217,type,
vEBT_VEBT_add: option_nat > option_nat > option_nat ).
thf(func_def_1218,type,
vEBT_VEBT_greater: option_nat > option_nat > $o ).
thf(func_def_1219,type,
vEBT_V5711637165310795180er_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
thf(func_def_1220,type,
vEBT_VEBT_less: option_nat > option_nat > $o ).
thf(func_def_1221,type,
vEBT_VEBT_less_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
thf(func_def_1222,type,
vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
thf(func_def_1223,type,
vEBT_VEBT_lesseq_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
thf(func_def_1224,type,
vEBT_VEBT_max_in_set: set_nat > nat > $o ).
thf(func_def_1225,type,
vEBT_VEBT_min_in_set: set_nat > nat > $o ).
thf(func_def_1226,type,
vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
thf(func_def_1227,type,
vEBT_V2881884560877996034ft_nat: ( nat > nat > $o ) > option_nat > option_nat > $o ).
thf(func_def_1228,type,
vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
thf(func_def_1229,type,
vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
thf(func_def_1230,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_1231,type,
vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ).
thf(func_def_1232,type,
vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o ).
thf(func_def_1233,type,
vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ).
thf(func_def_1234,type,
vEBT_VEBT_power: option_nat > option_nat > option_nat ).
thf(func_def_1235,type,
vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
thf(func_def_1236,type,
vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1237,type,
vEBT_vebt_mint: vEBT_VEBT > option_nat ).
thf(func_def_1238,type,
vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1239,type,
vEBT_is_pred_in_set: set_nat > nat > nat > $o ).
thf(func_def_1240,type,
vEBT_vebt_pred: vEBT_VEBT > nat > option_nat ).
thf(func_def_1241,type,
vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1242,type,
vEBT_is_succ_in_set: set_nat > nat > nat > $o ).
thf(func_def_1243,type,
vEBT_vebt_succ: vEBT_VEBT > nat > option_nat ).
thf(func_def_1244,type,
vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
thf(func_def_1245,type,
accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
thf(func_def_1246,type,
accp_nat: ( nat > nat > $o ) > nat > $o ).
thf(func_def_1247,type,
accp_P6019419558468335806at_nat: ( produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o ) > produc4471711990508489141at_nat > $o ).
thf(func_def_1248,type,
accp_P5496254298877145759on_nat: ( produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o ) > produc8306885398267862888on_nat > $o ).
thf(func_def_1249,type,
accp_P7605991808943153877on_num: ( produc1193250871479095198on_num > produc1193250871479095198on_num > $o ) > produc1193250871479095198on_num > $o ).
thf(func_def_1250,type,
accp_P3267385326087170368at_nat: ( produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o ) > produc5542196010084753463at_nat > $o ).
thf(func_def_1251,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(func_def_1252,type,
accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).
thf(func_def_1253,type,
accp_P8646395344606611882on_nat: ( produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ) > produc4953844613479565601on_nat > $o ).
thf(func_def_1254,type,
accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).
thf(func_def_1255,type,
accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).
thf(func_def_1256,type,
finite8643634255014194347omplex: set_Pr6308028481084910985omplex ).
thf(func_def_1257,type,
finite4251489430341359785d_enat: set_Pr2112562347474612743d_enat ).
thf(func_def_1258,type,
finite_psubset_int: set_Pr2522554150109002629et_int ).
thf(func_def_1259,type,
finite_psubset_nat: set_Pr5488025237498180813et_nat ).
thf(func_def_1260,type,
finite469560695537375940at_nat: set_Pr4329608150637261639at_nat ).
thf(func_def_1261,type,
finite4695646753290404266at_nat: set_Pr7459493094073627847at_nat ).
thf(func_def_1262,type,
less_than: set_Pr1261947904930325089at_nat ).
thf(func_def_1263,type,
lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat ).
thf(func_def_1264,type,
max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_1265,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(func_def_1266,type,
measure_nat: ( nat > nat ) > set_Pr1261947904930325089at_nat ).
thf(func_def_1267,type,
measure_option_nat: ( option_nat > nat ) > set_Pr6588086440996610945on_nat ).
thf(func_def_1268,type,
measur1827424007717751593at_nat: ( set_Pr1261947904930325089at_nat > nat ) > set_Pr4329608150637261639at_nat ).
thf(func_def_1269,type,
measur4922264856574889999at_nat: ( set_Pr4329608150637261639at_nat > nat ) > set_Pr7459493094073627847at_nat ).
thf(func_def_1270,type,
min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).
thf(func_def_1271,type,
pred_nat: set_Pr1261947904930325089at_nat ).
thf(func_def_1272,type,
wf_nat: set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1273,type,
wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o ).
thf(func_def_1274,type,
fChoice_real: ( real > $o ) > real ).
thf(func_def_1275,type,
member_o: $o > set_o > $o ).
thf(func_def_1276,type,
member_Code_integer: code_integer > set_Code_integer > $o ).
thf(func_def_1277,type,
member_complex: complex > set_complex > $o ).
thf(func_def_1278,type,
member_Extended_enat: extended_enat > set_Extended_enat > $o ).
thf(func_def_1279,type,
member_int: int > set_int > $o ).
thf(func_def_1280,type,
member_list_o: list_o > set_list_o > $o ).
thf(func_def_1281,type,
member_list_nat: list_nat > set_list_nat > $o ).
thf(func_def_1282,type,
member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
thf(func_def_1283,type,
member_nat: nat > set_nat > $o ).
thf(func_def_1284,type,
member_num: num > set_num > $o ).
thf(func_def_1285,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(func_def_1286,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(func_def_1287,type,
member4117937158525611210on_nat: produc4953844613479565601on_nat > set_Pr6588086440996610945on_nat > $o ).
thf(func_def_1288,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(func_def_1289,type,
member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
thf(func_def_1290,type,
member4453595087596390480d_enat: produc1621487020699730983d_enat > set_Pr2112562347474612743d_enat > $o ).
thf(func_def_1291,type,
member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
thf(func_def_1292,type,
member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
thf(func_def_1293,type,
member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
thf(func_def_1294,type,
member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).
thf(func_def_1295,type,
member_rat: rat > set_rat > $o ).
thf(func_def_1296,type,
member_real: real > set_real > $o ).
thf(func_def_1297,type,
member_set_int: set_int > set_set_int > $o ).
thf(func_def_1298,type,
member_set_nat: set_nat > set_set_nat > $o ).
thf(func_def_1299,type,
member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
thf(func_def_1300,type,
a: nat ).
thf(func_def_1301,type,
b: nat ).
thf(func_def_1302,type,
deg: nat ).
thf(func_def_1303,type,
h: nat ).
thf(func_def_1304,type,
info: option4927543243414619207at_nat ).
thf(func_def_1305,type,
k: vEBT_VEBT ).
thf(func_def_1306,type,
m: nat ).
thf(func_def_1307,type,
ma: nat ).
thf(func_def_1308,type,
mi: nat ).
thf(func_def_1309,type,
na: nat ).
thf(func_def_1310,type,
sa: vEBT_VEBT ).
thf(func_def_1311,type,
summary: vEBT_VEBT ).
thf(func_def_1312,type,
summary2: vEBT_VEBT ).
thf(func_def_1313,type,
ta: vEBT_VEBT ).
thf(func_def_1314,type,
treeList: list_VEBT_VEBT ).
thf(func_def_1315,type,
treeList2: list_VEBT_VEBT ).
thf(func_def_1334,type,
sP0: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1335,type,
sP1: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1336,type,
sP2: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1337,type,
sP3: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1338,type,
sP4: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1339,type,
sP5: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1340,type,
sP6: ( option_nat > option_nat > $o ) > option_nat > option_nat > $o ).
thf(func_def_1341,type,
sP7: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1342,type,
sP8: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1343,type,
sP9: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1344,type,
sP10: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1345,type,
sP11: nat > nat > $o ).
thf(func_def_1346,type,
sP12: nat > nat > $o ).
thf(func_def_1349,type,
sP15: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1350,type,
sP16: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1351,type,
sP17: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1352,type,
sP18: option_nat > nat > vEBT_VEBT > $o ).
thf(func_def_1353,type,
sP19: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1354,type,
sP20: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1355,type,
sP21: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1356,type,
sP22: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1357,type,
sP23: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1358,type,
sP24: option_nat > vEBT_VEBT > nat > $o ).
thf(func_def_1359,type,
sP25: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1360,type,
sP26: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1361,type,
sP27: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1362,type,
sP28: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1363,type,
sP29: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1364,type,
sP30: vEBT_VEBT > option_nat > $o ).
thf(func_def_1365,type,
sP31: vEBT_VEBT > option_nat > $o ).
thf(func_def_1366,type,
sP32: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1367,type,
sP33: option_nat > nat > vEBT_VEBT > $o ).
thf(func_def_1368,type,
sP34: vEBT_VEBT > nat > option_nat > $o ).
thf(func_def_1369,type,
sP35: vEBT_VEBT > option_nat > $o ).
thf(func_def_1370,type,
sK36: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1371,type,
sK37: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1372,type,
sK38: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1373,type,
sK39: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1374,type,
sK40: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1375,type,
sK41: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1376,type,
sK42: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1377,type,
sK43: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1378,type,
sK44: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1379,type,
sK45: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1380,type,
sK46: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1381,type,
sK47: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1382,type,
sK48: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1383,type,
sK49: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1384,type,
sK50: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1385,type,
sK51: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1386,type,
sK52: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1387,type,
sK53: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1388,type,
sK54: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1389,type,
sK55: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1390,type,
sK56: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1391,type,
sK57: vEBT_VEBT > nat > vEBT_VEBT > nat ).
thf(func_def_1392,type,
sK58: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1393,type,
sK59: option_nat > nat ).
thf(func_def_1394,type,
sK60: ( option_nat > option_nat > $o ) > option_nat > option_nat > nat ).
thf(func_def_1395,type,
sK61: ( option_nat > option_nat > $o ) > option_nat > option_nat > nat ).
thf(func_def_1396,type,
sK62: ( nat > $o ) > nat ).
thf(func_def_1397,type,
sK63: nat > nat ).
thf(func_def_1398,type,
sK64: nat > nat ).
thf(func_def_1399,type,
sK65: vEBT_VEBT > $o ).
thf(func_def_1400,type,
sK66: vEBT_VEBT > $o ).
thf(func_def_1401,type,
sK67: vEBT_VEBT > nat > option_nat > nat ).
thf(func_def_1402,type,
sK68: vEBT_VEBT > nat > option_nat > nat ).
thf(func_def_1403,type,
sK69: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT ).
thf(func_def_1404,type,
sK70: vEBT_VEBT > nat > option_nat > vEBT_VEBT ).
thf(func_def_1405,type,
sK71: vEBT_VEBT > nat > option_nat > nat ).
thf(func_def_1406,type,
sK72: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1407,type,
sK73: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1408,type,
sK74: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1409,type,
sK75: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1410,type,
sK76: nat > nat ).
thf(func_def_1411,type,
sK77: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1412,type,
sK78: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1413,type,
sK79: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1414,type,
sK80: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1415,type,
sK81: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1416,type,
sK82: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1417,type,
sK83: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1418,type,
sK84: vEBT_VEBT > nat ).
thf(func_def_1419,type,
sK85: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1420,type,
sK86: ( nat > $o ) > nat ).
thf(func_def_1421,type,
sK87: nat > nat ).
thf(func_def_1422,type,
sK88: nat ).
thf(func_def_1423,type,
sK89: nat ).
thf(func_def_1424,type,
sK90: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT ).
thf(func_def_1425,type,
sK91: vEBT_VEBT > nat > option_nat > vEBT_VEBT ).
thf(func_def_1426,type,
sK92: vEBT_VEBT > nat > option_nat > product_prod_nat_nat ).
thf(func_def_1427,type,
sK93: vEBT_VEBT > option_nat > nat > product_prod_nat_nat ).
thf(func_def_1428,type,
sK94: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT ).
thf(func_def_1429,type,
sK95: vEBT_VEBT > option_nat > nat > vEBT_VEBT ).
thf(func_def_1430,type,
sK96: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1431,type,
sK97: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1432,type,
sK98: option_nat > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1433,type,
sK99: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1434,type,
sK100: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1435,type,
sK101: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1436,type,
sK102: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1437,type,
sK103: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1438,type,
sK104: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1439,type,
sK105: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1440,type,
sK106: vEBT_VEBT > option_nat > nat > $o ).
thf(func_def_1441,type,
sK107: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1442,type,
sK108: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1443,type,
sK109: option_nat > vEBT_VEBT > nat > nat ).
thf(func_def_1444,type,
sK110: option_nat > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1445,type,
sK111: ( nat > $o ) > nat ).
thf(func_def_1446,type,
sK112: nat > option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1447,type,
sK113: nat > option_nat > vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1448,type,
sK114: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1449,type,
sK115: option_nat > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1450,type,
sK116: option_nat > vEBT_VEBT > nat > product_prod_nat_nat ).
thf(func_def_1451,type,
sK117: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1452,type,
sK118: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT ).
thf(func_def_1453,type,
sK119: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1454,type,
sK120: vEBT_VEBT > option_nat > nat > vEBT_VEBT ).
thf(func_def_1455,type,
sK121: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1456,type,
sK122: vEBT_VEBT > option_nat > nat > nat ).
thf(func_def_1457,type,
sK123: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1458,type,
sK124: nat > option_nat > vEBT_VEBT > $o ).
thf(func_def_1459,type,
sK125: nat > option_nat > vEBT_VEBT > nat ).
thf(func_def_1460,type,
sK126: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1461,type,
sK127: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1462,type,
sK128: nat > vEBT_VEBT > option_nat > list_VEBT_VEBT ).
thf(func_def_1463,type,
sK129: nat > vEBT_VEBT > option_nat > vEBT_VEBT ).
thf(func_def_1464,type,
sK130: nat > vEBT_VEBT > option_nat > nat ).
thf(func_def_1465,type,
sK131: nat > nat ).
thf(func_def_1466,type,
sK132: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1467,type,
sK133: vEBT_VEBT > nat > vEBT_VEBT > $o ).
thf(func_def_1468,type,
sK134: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1469,type,
sK135: nat > vEBT_VEBT > vEBT_VEBT > $o ).
thf(func_def_1470,type,
sK136: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1471,type,
sK137: vEBT_VEBT > vEBT_VEBT > nat > $o ).
thf(func_def_1472,type,
sK138: nat > nat ).
thf(func_def_1473,type,
sK139: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1474,type,
sK140: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT ).
thf(func_def_1475,type,
sK141: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1476,type,
sK142: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT ).
thf(func_def_1477,type,
sK143: vEBT_VEBT > vEBT_VEBT > nat > nat ).
thf(func_def_1478,type,
sK144: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1479,type,
sK145: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1480,type,
sK146: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1481,type,
sK147: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1482,type,
sK148: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1483,type,
sK149: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1484,type,
sK150: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1485,type,
sK151: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1486,type,
sK152: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1487,type,
sK153: vEBT_VEBT > vEBT_VEBT > nat ).
thf(func_def_1488,type,
sK154: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1489,type,
sK155: nat > nat ).
thf(func_def_1490,type,
sK156: vEBT_VEBT > option_nat > nat ).
thf(func_def_1491,type,
sK157: vEBT_VEBT > option_nat > list_VEBT_VEBT ).
thf(func_def_1492,type,
sK158: vEBT_VEBT > option_nat > vEBT_VEBT ).
thf(func_def_1493,type,
sK159: vEBT_VEBT > option_nat > nat ).
thf(func_def_1494,type,
sK160: vEBT_VEBT > option_nat > nat ).
thf(func_def_1495,type,
sK161: vEBT_VEBT > option_nat > $o ).
thf(func_def_1496,type,
sK162: vEBT_VEBT > option_nat > $o ).
thf(func_def_1497,type,
sK163: option_nat > vEBT_VEBT > nat ).
thf(func_def_1498,type,
sK164: option_nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1499,type,
sK165: option_nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1500,type,
sK166: vEBT_VEBT > nat > nat ).
thf(func_def_1501,type,
sK167: ( nat > nat > $o ) > nat ).
thf(func_def_1502,type,
sK168: ( nat > nat > $o ) > nat ).
thf(func_def_1503,type,
sK169: ( nat > nat > $o ) > nat ).
thf(func_def_1504,type,
sK170: ( nat > nat > $o ) > nat ).
thf(func_def_1505,type,
sK171: ( nat > nat > $o ) > nat ).
thf(func_def_1506,type,
sK172: nat > nat ).
thf(func_def_1507,type,
sK173: nat > nat ).
thf(func_def_1508,type,
sK174: vEBT_VEBT > $o ).
thf(func_def_1509,type,
sK175: vEBT_VEBT > $o ).
thf(func_def_1510,type,
sK176: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1511,type,
sK177: option_nat > nat > vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1512,type,
sK178: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1513,type,
sK179: option_nat > nat > vEBT_VEBT > nat ).
thf(func_def_1514,type,
sK180: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1515,type,
sK181: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1516,type,
sK182: nat > vEBT_VEBT > option_nat > $o ).
thf(func_def_1517,type,
sK183: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1518,type,
sK184: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1519,type,
sK185: vEBT_VEBT > nat ).
thf(func_def_1520,type,
sK186: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1521,type,
sK187: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1522,type,
sK188: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1523,type,
sK189: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1524,type,
sK190: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1525,type,
sK191: vEBT_VEBT > product_prod_nat_nat ).
thf(func_def_1526,type,
sK192: nat > nat ).
thf(func_def_1527,type,
sK193: nat > nat ).
thf(func_def_1528,type,
sK194: nat > nat ).
thf(func_def_1529,type,
sK195: vEBT_VEBT > option_nat > $o ).
thf(func_def_1530,type,
sK196: vEBT_VEBT > option_nat > $o ).
thf(func_def_1531,type,
sK197: vEBT_VEBT > vEBT_VEBT ).
thf(func_def_1532,type,
sK198: vEBT_VEBT > nat ).
thf(func_def_1533,type,
sK199: vEBT_VEBT > list_VEBT_VEBT ).
thf(func_def_1534,type,
sK200: vEBT_VEBT > option_nat > nat ).
thf(func_def_1535,type,
sK201: vEBT_VEBT > option_nat > vEBT_VEBT ).
thf(func_def_1536,type,
sK202: vEBT_VEBT > option_nat > nat ).
thf(func_def_1537,type,
sK203: vEBT_VEBT > option_nat > list_VEBT_VEBT ).
thf(func_def_1538,type,
sK204: vEBT_VEBT > option_nat > nat ).
thf(func_def_1539,type,
sK205: ( nat > $o ) > nat ).
thf(func_def_1541,type,
sK207: option_nat > nat ).
thf(func_def_1542,type,
sK208: option_nat > nat ).
thf(func_def_1543,type,
ph209:
!>[X0: $tType] : X0 ).
thf(func_def_1544,type,
sK210: ( option_nat > $o ) > option_nat ).
thf(func_def_1545,type,
sK211: ( option_nat > $o ) > option_nat ).
thf(func_def_1546,type,
sK212: ( option_nat > $o ) > nat ).
thf(func_def_1547,type,
sK213: ( option_nat > $o ) > nat ).
thf(func_def_1548,type,
sK214: ( option_nat > $o ) > option_nat ).
thf(func_def_1549,type,
sK215: option_nat > $o ).
thf(func_def_1550,type,
sK216: nat ).
thf(func_def_1551,type,
sK217: option_nat ).
thf(func_def_1552,type,
sK218: ( option_nat > $o ) > option_nat ).
thf(func_def_1553,type,
sK219: ( option_nat > $o ) > option_nat ).
thf(func_def_1554,type,
sK220: ( option_nat > $o ) > option_nat ).
thf(func_def_1555,type,
sK221: ( option_nat > $o ) > nat ).
thf(f28214,plain,
$false,
inference(avatar_sat_refutation,[status(thm)],[f27114,f27147,f27301,f27478,f28213]) ).
thf(f28213,plain,
( ~ spl206_4
| ~ spl206_12
| ~ spl206_16 ),
inference(avatar_contradiction_clause,[status(thm)],[f28212]) ).
thf(f28212,plain,
( $false
| ~ spl206_4
| ~ spl206_12
| ~ spl206_16 ),
inference(subsumption_resolution,[status(thm)],[f28211,f27150]) ).
thf(f27150,plain,
( ( ( some_nat @ b )
!= ( some_nat @ a ) )
| ~ spl206_4
| ~ spl206_12 ),
inference(backward_demodulation,[status(thm)],[f27149,f27087]) ).
thf(f27087,plain,
( ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) )
| ~ spl206_4 ),
inference(avatar_component_clause,[status(thm)],[f27085]) ).
thf(f27085,definition,
( spl206_4
<=> ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) ) ),
introduced(definition,[new_symbols(naming,[spl206_4])],[avatar_definition]) ).
thf(f27149,plain,
( ( ( vEBT_vebt_mint @ ta )
!= ( some_nat @ b ) )
| ~ spl206_12 ),
inference(forward_demodulation,[status(thm)],[f26690,f27128]) ).
thf(f27128,plain,
( ( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) )
| ~ spl206_12 ),
inference(avatar_component_clause,[status(thm)],[f27127]) ).
thf(f27127,definition,
( spl206_12
<=> ( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) ) ),
introduced(definition,[new_symbols(naming,[spl206_12])],[avatar_definition]) ).
thf(f26690,plain,
( ( vEBT_vebt_mint @ ta )
!= ( vEBT_vebt_mint @ k ) ),
inference(cnf_transformation,[status(thm)],[f1]) ).
thf(f1,axiom,
( ( vEBT_vebt_mint @ ta )
!= ( vEBT_vebt_mint @ k ) ),
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062) ).
thf(f28211,plain,
( ( ( some_nat @ b )
= ( some_nat @ a ) )
| ~ spl206_12
| ~ spl206_16 ),
inference(trivial_inequality_removal,[status(thm)],[f28210]) ).
thf(f28210,plain,
( ( ( some_nat @ b )
= ( some_nat @ a ) )
| ( $true != $true )
| ~ spl206_12
| ~ spl206_16 ),
inference(superposition,[status(thm)],[f27300,f27737]) ).
thf(f27737,plain,
( ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ b )
= $true )
| ~ spl206_12 ),
inference(equality_resolution,[status(thm)],[f27306]) ).
thf(f27306,plain,
( ! [X0: nat] :
( ( ( some_nat @ b )
!= ( some_nat @ X0 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
= $true ) )
| ~ spl206_12 ),
inference(forward_demodulation,[status(thm)],[f27305,f27128]) ).
thf(f27305,plain,
! [X0: nat] :
( ( ( vEBT_vebt_mint @ k )
!= ( some_nat @ X0 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
= $true ) ),
inference(trivial_inequality_removal,[status(thm)],[f27303]) ).
thf(f27303,plain,
! [X0: nat] :
( ( ( vEBT_vebt_mint @ k )
!= ( some_nat @ X0 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
= $true )
| ( $true != $true ) ),
inference(superposition,[status(thm)],[f26895,f26937]) ).
thf(f26937,plain,
( ( vEBT_invar_vebt @ k @ h )
= $true ),
inference(cnf_transformation,[status(thm)],[f16969]) ).
thf(f16969,plain,
( ( vEBT_invar_vebt @ k @ h )
= $true ),
inference(fool_elimination,[status(thm)],[f16968]) ).
thf(f16968,plain,
vEBT_invar_vebt @ k @ h,
inference(rectify,[status(thm)],[f18]) ).
thf(f18,axiom,
vEBT_invar_vebt @ k @ h,
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_17_assms_I2_J) ).
thf(f26895,plain,
! [X2: vEBT_VEBT,X0: nat,X1: nat] :
( ( ( vEBT_invar_vebt @ X2 @ X1 )
!= $true )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X0 )
= $true )
| ( ( some_nat @ X0 )
!= ( vEBT_vebt_mint @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f26276]) ).
thf(f26276,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( some_nat @ X0 )
!= ( vEBT_vebt_mint @ X2 ) )
| ( ( vEBT_invar_vebt @ X2 @ X1 )
!= $true )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X0 )
= $true ) ),
inference(flattening,[status(thm)],[f26275]) ).
thf(f26275,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X0 )
= $true )
| ( ( some_nat @ X0 )
!= ( vEBT_vebt_mint @ X2 ) )
| ( ( vEBT_invar_vebt @ X2 @ X1 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f11441]) ).
thf(f11441,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( vEBT_invar_vebt @ X2 @ X1 )
= $true )
=> ( ( ( some_nat @ X0 )
= ( vEBT_vebt_mint @ X2 ) )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X0 )
= $true ) ) ),
inference(fool_elimination,[status(thm)],[f11440]) ).
thf(f11440,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ X2 @ X1 )
=> ( ( ( some_nat @ X0 )
= ( vEBT_vebt_mint @ X2 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X0 ) ) ),
inference(rectify,[status(thm)],[f31]) ).
thf(f31,axiom,
! [X4: nat,X11: nat,X5: vEBT_VEBT] :
( ( vEBT_invar_vebt @ X5 @ X11 )
=> ( ( ( vEBT_vebt_mint @ X5 )
= ( some_nat @ X4 ) )
=> ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ X4 ) ) ),
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_30_mint__corr) ).
thf(f27300,plain,
( ! [X0: nat] :
( ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
!= $true )
| ( ( some_nat @ a )
= ( some_nat @ X0 ) ) )
| ~ spl206_16 ),
inference(avatar_component_clause,[status(thm)],[f27299]) ).
thf(f27299,definition,
( spl206_16
<=> ! [X0: nat] :
( ( ( some_nat @ a )
= ( some_nat @ X0 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
!= $true ) ) ),
introduced(definition,[new_symbols(naming,[spl206_16])],[avatar_definition]) ).
thf(f27478,plain,
~ spl206_15,
inference(avatar_contradiction_clause,[status(thm)],[f27477]) ).
thf(f27477,plain,
( $false
| ~ spl206_15 ),
inference(trivial_inequality_removal,[status(thm)],[f27476]) ).
thf(f27476,plain,
( ( $true != $true )
| ~ spl206_15 ),
inference(superposition,[status(thm)],[f27297,f26740]) ).
thf(f26740,plain,
( ( vEBT_invar_vebt @ ta @ h )
= $true ),
inference(cnf_transformation,[status(thm)],[f23888]) ).
thf(f23888,plain,
( ( vEBT_invar_vebt @ ta @ h )
= $true ),
inference(fool_elimination,[status(thm)],[f23887]) ).
thf(f23887,plain,
vEBT_invar_vebt @ ta @ h,
inference(rectify,[status(thm)],[f19]) ).
thf(f19,axiom,
vEBT_invar_vebt @ ta @ h,
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_18_assms_I1_J) ).
thf(f27297,plain,
( ! [X1: nat] :
( ( vEBT_invar_vebt @ ta @ X1 )
!= $true )
| ~ spl206_15 ),
inference(avatar_component_clause,[status(thm)],[f27296]) ).
thf(f27296,definition,
( spl206_15
<=> ! [X1: nat] :
( ( vEBT_invar_vebt @ ta @ X1 )
!= $true ) ),
introduced(definition,[new_symbols(naming,[spl206_15])],[avatar_definition]) ).
thf(f27301,plain,
( spl206_15
| spl206_16
| ~ spl206_4 ),
inference(avatar_split_clause,[status(thm)],[f27294,f27085,f27299,f27296]) ).
thf(f27294,plain,
( ! [X0: nat,X1: nat] :
( ( ( some_nat @ a )
= ( some_nat @ X0 ) )
| ( ( vEBT_invar_vebt @ ta @ X1 )
!= $true )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
!= $true ) )
| ~ spl206_4 ),
inference(forward_demodulation,[status(thm)],[f27293,f27087]) ).
thf(f27293,plain,
! [X0: nat,X1: nat] :
( ( ( vEBT_invar_vebt @ ta @ X1 )
!= $true )
| ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ X0 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ k ) @ X0 )
!= $true ) ),
inference(superposition,[status(thm)],[f26700,f26683]) ).
thf(f26683,plain,
( ( vEBT_VEBT_set_vebt @ ta )
= ( vEBT_VEBT_set_vebt @ k ) ),
inference(cnf_transformation,[status(thm)],[f3]) ).
thf(f3,axiom,
( ( vEBT_VEBT_set_vebt @ ta )
= ( vEBT_VEBT_set_vebt @ k ) ),
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_2_assms_I3_J) ).
thf(f26700,plain,
! [X2: vEBT_VEBT,X0: nat,X1: nat] :
( ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X1 )
!= $true )
| ( ( vEBT_invar_vebt @ X2 @ X0 )
!= $true )
| ( ( some_nat @ X1 )
= ( vEBT_vebt_mint @ X2 ) ) ),
inference(cnf_transformation,[status(thm)],[f26374]) ).
thf(f26374,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( some_nat @ X1 )
= ( vEBT_vebt_mint @ X2 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X1 )
!= $true )
| ( ( vEBT_invar_vebt @ X2 @ X0 )
!= $true ) ),
inference(flattening,[status(thm)],[f26373]) ).
thf(f26373,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( some_nat @ X1 )
= ( vEBT_vebt_mint @ X2 ) )
| ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X1 )
!= $true )
| ( ( vEBT_invar_vebt @ X2 @ X0 )
!= $true ) ),
inference(ennf_transformation,[status(thm)],[f25750]) ).
thf(f25750,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( ( vEBT_invar_vebt @ X2 @ X0 )
= $true )
=> ( ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X1 )
= $true )
=> ( ( some_nat @ X1 )
= ( vEBT_vebt_mint @ X2 ) ) ) ),
inference(fool_elimination,[status(thm)],[f25749]) ).
thf(f25749,plain,
! [X0: nat,X1: nat,X2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ X2 @ X0 )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X2 ) @ X1 )
=> ( ( some_nat @ X1 )
= ( vEBT_vebt_mint @ X2 ) ) ) ),
inference(rectify,[status(thm)],[f30]) ).
thf(f30,axiom,
! [X11: nat,X4: nat,X5: vEBT_VEBT] :
( ( vEBT_invar_vebt @ X5 @ X11 )
=> ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ X5 ) @ X4 )
=> ( ( vEBT_vebt_mint @ X5 )
= ( some_nat @ X4 ) ) ) ),
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_29_mint__sound) ).
thf(f27147,plain,
spl206_12,
inference(avatar_split_clause,[status(thm)],[f26695,f27127]) ).
thf(f26695,plain,
( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) ),
inference(cnf_transformation,[status(thm)],[f11385]) ).
thf(f11385,plain,
( ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) )
& ( ( ord_less_nat @ a @ b )
= $true )
& ( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) ) ),
inference(fool_elimination,[status(thm)],[f11384]) ).
thf(f11384,plain,
( ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) )
& ( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) )
& ( ord_less_nat @ a @ b ) ),
inference(rectify,[status(thm)],[f10164]) ).
thf(f10164,axiom,
( ( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) )
& ( ( vEBT_vebt_mint @ k )
= ( some_nat @ b ) )
& ( ord_less_nat @ a @ b ) ),
file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',conj_2) ).
thf(f27114,plain,
spl206_4,
inference(avatar_split_clause,[status(thm)],[f26697,f27085]) ).
thf(f26697,plain,
( ( vEBT_vebt_mint @ ta )
= ( some_nat @ a ) ),
inference(cnf_transformation,[status(thm)],[f11385]) ).
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : ITP277^3 : 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
% 0.17/0.34 % Computer : n019.cluster.edu
% 0.17/0.34 % Model : x86_64 x86_64
% 0.17/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.17/0.34 % Memory : 8042.1875MB
% 0.17/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.17/0.34 % CPULimit : 300
% 0.17/0.34 % WCLimit : 300
% 0.17/0.34 % DateTime : Fri Feb 27 14:01:54 EST 2026
% 0.17/0.35 % CPUTime :
% 0.17/0.35 Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.34/0.49 ---- Original DTF file ---
% 0.34/0.49 thf(spec,logic,$$dhol).
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % File : ITP277^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.34/0.49 % Domain : Interactive Theorem Proving
% 0.34/0.49 % Problem : Sledgehammer problem VEBT_Uniqueness 00371_024092
% 0.34/0.49 % Version : [Des22] axioms.
% 0.34/0.49 % English :
% 0.34/0.49
% 0.34/0.49 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.34/0.49 % : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.34/0.49 % Source : [Des22]
% 0.34/0.49 % Names : 0075_VEBT_Uniqueness_00371_024092 [Des22]
% 0.34/0.49
% 0.34/0.49 % Status : Theorem
% 0.34/0.49 % Rating : ? v9.2.1
% 0.34/0.49 % Syntax : Number of formulae : 11481 (4686 unt;1316 typ; 0 def)
% 0.34/0.49 % Number of atoms : 32221 (11947 equ; 0 cnn)
% 0.34/0.49 % Maximal formula atoms : 71 ( 3 avg)
% 0.34/0.49 % Number of connectives : 118110 (3292 ~; 543 |;2555 &;98472 @)
% 0.34/0.49 % ( 0 <=>;13248 =>; 0 <=; 0 <~>)
% 0.34/0.49 % Maximal formula depth : 39 ( 6 avg)
% 0.34/0.49 % Number of types : 128 ( 127 usr)
% 0.34/0.49 % Number of type conns : 4875 (4875 >; 0 *; 0 +; 0 <<)
% 0.34/0.49 % Number of symbols : 1192 (1189 usr; 104 con; 0-8 aty)
% 0.34/0.49 % Number of variables : 27245 (2185 ^;24068 !; 992 ?;27245 :)
% 0.34/0.49 % SPC : TH0_THM_EQU_NAR
% 0.34/0.49
% 0.34/0.49 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.34/0.49 % from the van Emde Boas Trees session in the Archive of Formal
% 0.34/0.49 % proofs -
% 0.34/0.49 % www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.34/0.49 % 2022-02-18 16:12:27.698
% 0.34/0.49 % Bugfixes : Added ()s around negated sides of equations.
% 0.34/0.49 %------------------------------------------------------------------------------
% 0.34/0.49 % Could-be-implicit typings (127)
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% 0.34/0.49 set_Pr7459493094073627847at_nat: $tType ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
% 0.34/0.50 removeAll_int: int > list_int > list_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
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% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.50 removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
% 0.34/0.50 removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 0.34/0.50 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Orotate1_001_Eo,type,
% 0.34/0.50 rotate1_o: list_o > list_o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Orotate1_001t__Int__Oint,type,
% 0.34/0.50 rotate1_int: list_int > list_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
% 0.34/0.50 rotate1_nat: list_nat > list_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
% 0.34/0.50 rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 0.34/0.50 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 0.34/0.50 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 0.34/0.50 take_nat: nat > list_nat > list_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Oupt,type,
% 0.34/0.50 upt: nat > nat > list_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Oupto,type,
% 0.34/0.50 upto: int > int > list_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Oupto__aux,type,
% 0.34/0.50 upto_aux: int > int > list_int > list_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Oupto__rel,type,
% 0.34/0.50 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
% 0.34/0.50 zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
% 0.34/0.50 zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
% 0.34/0.50 zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
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% 0.34/0.50 thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_OSuc,type,
% 0.34/0.50 suc: nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 0.34/0.50 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 0.34/0.50 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Onat_Opred,type,
% 0.34/0.50 pred: nat > nat ).
% 0.34/0.50
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% 0.34/0.50 semiri4939895301339042750nteger: nat > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 0.34/0.50 semiri8010041392384452111omplex: nat > complex ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 0.34/0.50 size_size_list_o: list_o > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 0.34/0.50 size_s3451745648224563538omplex: list_complex > nat ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
% 0.34/0.50 size_s2710708370519433104list_o: list_list_o > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 0.34/0.50 size_s3023201423986296836st_nat: list_list_nat > nat ).
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% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 0.34/0.50 size_size_list_nat: list_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 0.34/0.50 size_size_list_num: list_num > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 0.34/0.50 size_size_list_real: list_real > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 0.34/0.50 size_s3254054031482475050et_nat: list_set_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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% 0.34/0.50
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% 0.34/0.50 size_size_num: num > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 0.34/0.50 size_size_option_nat: option_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Olist__decode,type,
% 0.34/0.50 nat_list_decode: nat > list_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
% 0.34/0.50 nat_list_decode_rel: nat > nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Olist__encode,type,
% 0.34/0.50 nat_list_encode: list_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 0.34/0.50 nat_list_encode_rel: list_nat > list_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oprod__decode,type,
% 0.34/0.50 nat_prod_decode: nat > product_prod_nat_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 0.34/0.50 nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 0.34/0.50 nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 0.34/0.50 nat_prod_encode: product_prod_nat_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oset__decode,type,
% 0.34/0.50 nat_set_decode: nat > set_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Nat__Bijection_Oset__encode,type,
% 0.34/0.50 nat_set_encode: set_nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_NthRoot_Oroot,type,
% 0.34/0.50 root: nat > real > real ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_NthRoot_Osqrt,type,
% 0.34/0.50 sqrt: real > real ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oinc,type,
% 0.34/0.50 inc: num > num ).
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% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 0.34/0.50 neg_nu6511756317524482435omplex: complex > complex ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 0.34/0.50 neg_nu3811975205180677377ec_int: int > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 0.34/0.50 neg_nu3179335615603231917ec_rat: rat > rat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 0.34/0.50 neg_nu6075765906172075777c_real: real > real ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex,type,
% 0.34/0.50 neg_nu8557863876264182079omplex: complex > complex ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
% 0.34/0.50 neg_nu5851722552734809277nc_int: int > int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Rat__Orat,type,
% 0.34/0.50 neg_nu5219082963157363817nc_rat: rat > rat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Real__Oreal,type,
% 0.34/0.50 neg_nu8295874005876285629c_real: real > real ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onum_OBit0,type,
% 0.34/0.50 bit0: num > num ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onum_OBit1,type,
% 0.34/0.50 bit1: num > num ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onum_OOne,type,
% 0.34/0.50 one: num ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onum_Osize__num,type,
% 0.34/0.50 size_num: num > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onum__of__nat,type,
% 0.34/0.50 num_of_nat: nat > num ).
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% 0.34/0.50 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 numera6620942414471956472nteger: num > code_integer ).
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% 0.34/0.50 numera6690914467698888265omplex: num > complex ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 0.34/0.50 numeral_numeral_nat: num > nat ).
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% 0.34/0.50 thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
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% 0.34/0.50 pred_numeral: num > nat ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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% 0.34/0.50 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,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 0.34/0.50 produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.50 produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.50 produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
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% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 0.34/0.50 produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 0.34/0.50 produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
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% 0.34/0.50 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,
% 0.34/0.50 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 0.34/0.50
% 0.34/0.50 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,
% 0.34/0.50 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
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% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
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% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Product__Type_Ounit_ORep__unit,type,
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% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Rat_OFract,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Rat_ORep__Rat,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
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% 0.34/0.50 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
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% 0.34/0.50 thf(sy_c_Rat_Onormalize,type,
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% 0.34/0.50 thf(sy_c_Rat_Opcr__rat,type,
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% 0.34/0.50 thf(sy_c_Rat_Opositive,type,
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% 0.34/0.50 thf(sy_c_Rat_Oquotient__of,type,
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% 0.34/0.50 thf(sy_c_Rat_Oratrel,type,
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% 0.34/0.50 thf(sy_c_Real_Opcr__real,type,
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% 0.34/0.50
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% 0.34/0.50
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% 0.34/0.50
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
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% 0.34/0.50
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% 0.34/0.50 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 0.34/0.50 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 0.34/0.50 vEBT_VEBT_low: nat > nat > nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 0.34/0.50 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
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% 0.34/0.50
% 0.34/0.50 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
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% 0.34/0.50 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
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% 0.34/0.50 thf(sy_c_VEBT__Delete_Ovebt__delete,type,
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% 0.34/0.50 thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
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% 0.34/0.50 thf(sy_c_member_001_Eo,type,
% 0.34/0.50 member_o: $o > set_o > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Code____Numeral__Ointeger,type,
% 0.34/0.50 member_Code_integer: code_integer > set_Code_integer > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Complex__Ocomplex,type,
% 0.34/0.50 member_complex: complex > set_complex > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Extended____Nat__Oenat,type,
% 0.34/0.50 member_Extended_enat: extended_enat > set_Extended_enat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Int__Oint,type,
% 0.34/0.50 member_int: int > set_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
% 0.34/0.50 member_list_o: list_o > set_list_o > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
% 0.34/0.50 member_list_nat: list_nat > set_list_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 0.34/0.50 member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Nat__Onat,type,
% 0.34/0.50 member_nat: nat > set_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Num__Onum,type,
% 0.34/0.50 member_num: num > set_num > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.34/0.50 member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.34/0.50 member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
% 0.34/0.50 member4117937158525611210on_nat: produc4953844613479565601on_nat > set_Pr6588086440996610945on_nat > $o ).
% 0.34/0.50
% 0.34/0.50 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,
% 0.34/0.50 member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Complex__Ocomplex_J_Mt__Set__Oset_It__Complex__Ocomplex_J_J,type,
% 0.34/0.50 member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Extended____Nat__Oenat_J_Mt__Set__Oset_It__Extended____Nat__Oenat_J_J,type,
% 0.34/0.50 member4453595087596390480d_enat: produc1621487020699730983d_enat > set_Pr2112562347474612743d_enat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Int__Oint_J_Mt__Set__Oset_It__Int__Oint_J_J,type,
% 0.34/0.50 member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.34/0.50 member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
% 0.34/0.50 member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J,type,
% 0.34/0.50 member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Rat__Orat,type,
% 0.34/0.50 member_rat: rat > set_rat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Real__Oreal,type,
% 0.34/0.50 member_real: real > set_real > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.34/0.50 member_set_int: set_int > set_set_int > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.34/0.50 member_set_nat: set_nat > set_set_nat > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.34/0.50 member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_a____,type,
% 0.34/0.50 a: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_b____,type,
% 0.34/0.50 b: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_deg____,type,
% 0.34/0.50 deg: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_h____,type,
% 0.34/0.50 h: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_info____,type,
% 0.34/0.50 info: option4927543243414619207at_nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_k____,type,
% 0.34/0.50 k: vEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_m____,type,
% 0.34/0.50 m: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_ma____,type,
% 0.34/0.50 ma: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_mi____,type,
% 0.34/0.50 mi: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_na____,type,
% 0.34/0.50 na: nat ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_sa____,type,
% 0.34/0.50 sa: vEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_summary_H____,type,
% 0.34/0.50 summary: vEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_summary____,type,
% 0.34/0.50 summary2: vEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_ta____,type,
% 0.34/0.50 ta: vEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_treeList_H____,type,
% 0.34/0.50 treeList: list_VEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 thf(sy_v_treeList____,type,
% 0.34/0.50 treeList2: list_VEBT_VEBT ).
% 0.34/0.50
% 0.34/0.50 % Relevant facts (10123)
% 0.34/0.50 thf(fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062,axiom,
% 0.34/0.50 ( ( vEBT_vebt_mint @ ta )
% 0.34/0.50 != ( vEBT_vebt_mint @ k ) ) ).
% 0.34/0.50
% 0.34/0.50 % \<open>vebt_mint t \<noteq> vebt_mint k\<close>
% 0.34/0.50 thf(fact_1_abdef,axiom,
% 0.34/0.50 ( ( ( ( vEBT_vebt_mint @ ta )
% 0.34/0.50 = none_nat )
% 0.34/0.50 & ( ( vEBT_vebt_mint @ k )
% 0.34/0.50 = ( some_nat @ b ) ) )
% 0.34/0.50 | ( ( ( vEBT_vebt_mint @ ta )
% 0.34/0.50 = ( some_nat @ a ) )
% 0.34/0.50 & ( ( vEBT_vebt_mint @ k )
% 0.34/0.50 = none_nat ) )
% 0.34/0.50 | ( ( ord_less_nat @ a @ b )
% 0.34/0.50 & ( ( some_nat @ a )
% 0.34/0.50 = ( vEBT_vebt_mint @ ta ) )
% 0.34/0.50 & ( ( some_nat @ b )
% 0.34/0.50 = ( vEBT_vebt_mint @ k ) ) )
% 0.34/0.50 | ( ( ord_less_nat @ b @ a )
% 0.34/0.50 & ( ( some_nat @ a )
% 0.34/0.50 = ( vEBT_vebt_mint @ ta ) )
% 0.34/0.50 & ( ( some_nat @ b )
% 0.34/0.50 = ( vEBT_vebt_mint @ k ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % abdef
% 0.34/0.50 thf(fact_2_assms_I3_J,axiom,
% 0.34/0.50 ( ( vEBT_VEBT_set_vebt @ ta )
% 0.34/0.50 = ( vEBT_VEBT_set_vebt @ k ) ) ).
% 0.34/0.50
% 0.34/0.50 % assms(3)
% 0.34/0.50 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_Ab_O_Avebt__mint_At_A_061_ANone_A_092_060and_062_Avebt__mint_Ak_A_061_ASome_Ab_A_092_060or_062_Avebt__mint_At_A_061_ASome_Aa_A_092_060and_062_Avebt__mint_Ak_A_061_ANone_A_092_060or_062_Aa_A_060_Ab_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060or_062_Ab_A_060_Aa_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.34/0.50 ~ ! [A: nat,B: nat] :
% 0.34/0.50 ~ ( ( ( ( vEBT_vebt_mint @ ta )
% 0.34/0.50 = none_nat )
% 0.34/0.50 & ( ( vEBT_vebt_mint @ k )
% 0.34/0.50 = ( some_nat @ B ) ) )
% 0.34/0.50 | ( ( ( vEBT_vebt_mint @ ta )
% 0.34/0.50 = ( some_nat @ A ) )
% 0.34/0.50 & ( ( vEBT_vebt_mint @ k )
% 0.34/0.50 = none_nat ) )
% 0.34/0.50 | ( ( ord_less_nat @ A @ B )
% 0.34/0.50 & ( ( some_nat @ A )
% 0.34/0.50 = ( vEBT_vebt_mint @ ta ) )
% 0.34/0.50 & ( ( some_nat @ B )
% 0.34/0.50 = ( vEBT_vebt_mint @ k ) ) )
% 0.34/0.50 | ( ( ord_less_nat @ B @ A )
% 0.34/0.50 & ( ( some_nat @ A )
% 0.34/0.50 = ( vEBT_vebt_mint @ ta ) )
% 0.34/0.50 & ( ( some_nat @ B )
% 0.34/0.50 = ( vEBT_vebt_mint @ k ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % \<open>\<And>thesis. (\<And>a b. vebt_mint t = None \<and> vebt_mint k = Some b \<or> vebt_mint t = Some a \<and> vebt_mint k = None \<or> a < b \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<or> b < a \<and> Some a = vebt_mint t \<and> Some b = vebt_mint k \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.34/0.50 thf(fact_4_greater__shift,axiom,
% 0.34/0.50 ( ord_less_nat
% 0.34/0.50 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % greater_shift
% 0.34/0.50 thf(fact_5_less__shift,axiom,
% 0.34/0.50 ( ord_less_nat
% 0.34/0.50 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % less_shift
% 0.34/0.50 thf(fact_6_not__None__eq,axiom,
% 0.34/0.50 ! [X2: option_nat] :
% 0.34/0.50 ( ( X2 != none_nat )
% 0.34/0.50 = ( ? [Y: nat] :
% 0.34/0.50 ( X2
% 0.34/0.50 = ( some_nat @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_None_eq
% 0.34/0.50 thf(fact_7_not__None__eq,axiom,
% 0.34/0.50 ! [X2: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( X2 != none_P5556105721700978146at_nat )
% 0.34/0.50 = ( ? [Y: product_prod_nat_nat] :
% 0.34/0.50 ( X2
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_None_eq
% 0.34/0.50 thf(fact_8_not__None__eq,axiom,
% 0.34/0.50 ! [X2: option_num] :
% 0.34/0.50 ( ( X2 != none_num )
% 0.34/0.50 = ( ? [Y: num] :
% 0.34/0.50 ( X2
% 0.34/0.50 = ( some_num @ Y ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_None_eq
% 0.34/0.50 thf(fact_9_not__Some__eq,axiom,
% 0.34/0.50 ! [X2: option_nat] :
% 0.34/0.50 ( ( ! [Y: nat] :
% 0.34/0.50 ( X2
% 0.34/0.50 != ( some_nat @ Y ) ) )
% 0.34/0.50 = ( X2 = none_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_Some_eq
% 0.34/0.50 thf(fact_10_not__Some__eq,axiom,
% 0.34/0.50 ! [X2: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( ! [Y: product_prod_nat_nat] :
% 0.34/0.50 ( X2
% 0.34/0.50 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 0.34/0.50 = ( X2 = none_P5556105721700978146at_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_Some_eq
% 0.34/0.50 thf(fact_11_not__Some__eq,axiom,
% 0.34/0.50 ! [X2: option_num] :
% 0.34/0.50 ( ( ! [Y: num] :
% 0.34/0.50 ( X2
% 0.34/0.50 != ( some_num @ Y ) ) )
% 0.34/0.50 = ( X2 = none_num ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_Some_eq
% 0.34/0.50 thf(fact_12_minNullmin,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT] :
% 0.34/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.34/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = none_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % minNullmin
% 0.34/0.50 thf(fact_13_minminNull,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT] :
% 0.34/0.50 ( ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = none_nat )
% 0.34/0.50 => ( vEBT_VEBT_minNull @ T ) ) ).
% 0.34/0.50
% 0.34/0.50 % minminNull
% 0.34/0.50 thf(fact_14_option_Oinject,axiom,
% 0.34/0.50 ! [X22: nat,Y2: nat] :
% 0.34/0.50 ( ( ( some_nat @ X22 )
% 0.34/0.50 = ( some_nat @ Y2 ) )
% 0.34/0.50 = ( X22 = Y2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.inject
% 0.34/0.50 thf(fact_15_option_Oinject,axiom,
% 0.34/0.50 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 0.34/0.50 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 0.34/0.50 = ( X22 = Y2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.inject
% 0.34/0.50 thf(fact_16_option_Oinject,axiom,
% 0.34/0.50 ! [X22: num,Y2: num] :
% 0.34/0.50 ( ( ( some_num @ X22 )
% 0.34/0.50 = ( some_num @ Y2 ) )
% 0.34/0.50 = ( X22 = Y2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.inject
% 0.34/0.50 thf(fact_17_assms_I2_J,axiom,
% 0.34/0.50 vEBT_invar_vebt @ k @ h ).
% 0.34/0.50
% 0.34/0.50 % assms(2)
% 0.34/0.50 thf(fact_18_assms_I1_J,axiom,
% 0.34/0.50 vEBT_invar_vebt @ ta @ h ).
% 0.34/0.50
% 0.34/0.50 % assms(1)
% 0.34/0.50 thf(fact_19_option_Odistinct_I1_J,axiom,
% 0.34/0.50 ! [X22: nat] :
% 0.34/0.50 ( none_nat
% 0.34/0.50 != ( some_nat @ X22 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.distinct(1)
% 0.34/0.50 thf(fact_20_option_Odistinct_I1_J,axiom,
% 0.34/0.50 ! [X22: product_prod_nat_nat] :
% 0.34/0.50 ( none_P5556105721700978146at_nat
% 0.34/0.50 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.distinct(1)
% 0.34/0.50 thf(fact_21_option_Odistinct_I1_J,axiom,
% 0.34/0.50 ! [X22: num] :
% 0.34/0.50 ( none_num
% 0.34/0.50 != ( some_num @ X22 ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.distinct(1)
% 0.34/0.50 thf(fact_22_option_OdiscI,axiom,
% 0.34/0.50 ! [Option: option_nat,X22: nat] :
% 0.34/0.50 ( ( Option
% 0.34/0.50 = ( some_nat @ X22 ) )
% 0.34/0.50 => ( Option != none_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.discI
% 0.34/0.50 thf(fact_23_option_OdiscI,axiom,
% 0.34/0.50 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.34/0.50 ( ( Option
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.34/0.50 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.discI
% 0.34/0.50 thf(fact_24_option_OdiscI,axiom,
% 0.34/0.50 ! [Option: option_num,X22: num] :
% 0.34/0.50 ( ( Option
% 0.34/0.50 = ( some_num @ X22 ) )
% 0.34/0.50 => ( Option != none_num ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.discI
% 0.34/0.50 thf(fact_25_option_Oexhaust,axiom,
% 0.34/0.50 ! [Y3: option_nat] :
% 0.34/0.50 ( ( Y3 != none_nat )
% 0.34/0.50 => ~ ! [X23: nat] :
% 0.34/0.50 ( Y3
% 0.34/0.50 != ( some_nat @ X23 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.exhaust
% 0.34/0.50 thf(fact_26_option_Oexhaust,axiom,
% 0.34/0.50 ! [Y3: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( Y3 != none_P5556105721700978146at_nat )
% 0.34/0.50 => ~ ! [X23: product_prod_nat_nat] :
% 0.34/0.50 ( Y3
% 0.34/0.50 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.exhaust
% 0.34/0.50 thf(fact_27_option_Oexhaust,axiom,
% 0.34/0.50 ! [Y3: option_num] :
% 0.34/0.50 ( ( Y3 != none_num )
% 0.34/0.50 => ~ ! [X23: num] :
% 0.34/0.50 ( Y3
% 0.34/0.50 != ( some_num @ X23 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % option.exhaust
% 0.34/0.50 thf(fact_28_insert_H__pres__valid,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % insert'_pres_valid
% 0.34/0.50 thf(fact_29_mint__sound,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 0.34/0.50 => ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = ( some_nat @ X2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % mint_sound
% 0.34/0.50 thf(fact_30_mint__corr,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = ( some_nat @ X2 ) )
% 0.34/0.50 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % mint_corr
% 0.34/0.50 thf(fact_31_set__vebt__set__vebt_H__valid,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( vEBT_set_vebt @ T )
% 0.34/0.50 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % set_vebt_set_vebt'_valid
% 0.34/0.50 thf(fact_32_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 0.34/0.50 ( ( ( X2 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: nat,B: nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_33_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( ( X2 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: nat,B: product_prod_nat_nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_34_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 0.34/0.50 ( ( ( X2 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: nat,B: num] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_num @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_35_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 0.34/0.50 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: product_prod_nat_nat,B: nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_36_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_37_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 0.34/0.50 ( ( ( X2 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: product_prod_nat_nat,B: num] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_num @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_38_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 0.34/0.50 ( ( ( X2 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: num,B: nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_num @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_39_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.34/0.50 ( ( ( X2 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: num,B: product_prod_nat_nat] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_num @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_P7363390416028606310at_nat @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_40_combine__options__cases,axiom,
% 0.34/0.50 ! [X2: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 0.34/0.50 ( ( ( X2 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ( ( Y3 = none_num )
% 0.34/0.50 => ( P @ X2 @ Y3 ) )
% 0.34/0.50 => ( ! [A: num,B: num] :
% 0.34/0.50 ( ( X2
% 0.34/0.50 = ( some_num @ A ) )
% 0.34/0.50 => ( ( Y3
% 0.34/0.50 = ( some_num @ B ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) )
% 0.34/0.50 => ( P @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % combine_options_cases
% 0.34/0.50 thf(fact_41_split__option__all,axiom,
% 0.34/0.50 ( ( ^ [P2: option_nat > $o] :
% 0.34/0.50 ! [X3: option_nat] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option_nat > $o] :
% 0.34/0.50 ( ( P3 @ none_nat )
% 0.34/0.50 & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_all
% 0.34/0.50 thf(fact_42_split__option__all,axiom,
% 0.34/0.50 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.34/0.50 ! [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.34/0.50 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.34/0.50 & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_all
% 0.34/0.50 thf(fact_43_split__option__all,axiom,
% 0.34/0.50 ( ( ^ [P2: option_num > $o] :
% 0.34/0.50 ! [X3: option_num] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option_num > $o] :
% 0.34/0.50 ( ( P3 @ none_num )
% 0.34/0.50 & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_all
% 0.34/0.50 thf(fact_44_split__option__ex,axiom,
% 0.34/0.50 ( ( ^ [P2: option_nat > $o] :
% 0.34/0.50 ? [X3: option_nat] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option_nat > $o] :
% 0.34/0.50 ( ( P3 @ none_nat )
% 0.34/0.50 | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_ex
% 0.34/0.50 thf(fact_45_split__option__ex,axiom,
% 0.34/0.50 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.34/0.50 ? [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.34/0.50 ( ( P3 @ none_P5556105721700978146at_nat )
% 0.34/0.50 | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_ex
% 0.34/0.50 thf(fact_46_split__option__ex,axiom,
% 0.34/0.50 ( ( ^ [P2: option_num > $o] :
% 0.34/0.50 ? [X3: option_num] : ( P2 @ X3 ) )
% 0.34/0.50 = ( ^ [P3: option_num > $o] :
% 0.34/0.50 ( ( P3 @ none_num )
% 0.34/0.50 | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % split_option_ex
% 0.34/0.50 thf(fact_47_mint__corr__help__empty,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = none_nat )
% 0.34/0.50 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.34/0.50 = bot_bot_set_nat ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % mint_corr_help_empty
% 0.34/0.50 thf(fact_48_mint__member,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = ( some_nat @ Maxi ) )
% 0.34/0.50 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % mint_member
% 0.34/0.50 thf(fact_49_valid__eq,axiom,
% 0.34/0.50 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.34/0.50
% 0.34/0.50 % valid_eq
% 0.34/0.50 thf(fact_50_valid__eq1,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,D: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ D )
% 0.34/0.50 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.34/0.50
% 0.34/0.50 % valid_eq1
% 0.34/0.50 thf(fact_51_valid__eq2,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,D: nat] :
% 0.34/0.50 ( ( vEBT_VEBT_valid @ T @ D )
% 0.34/0.50 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.34/0.50
% 0.34/0.50 % valid_eq2
% 0.34/0.50 thf(fact_52_deg__not__0,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % deg_not_0
% 0.34/0.50 thf(fact_53_set__vebt__finite,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % set_vebt_finite
% 0.34/0.50 thf(fact_54_case4_I12_J,axiom,
% 0.34/0.50 vEBT_invar_vebt @ sa @ deg ).
% 0.34/0.50
% 0.34/0.50 % case4(12)
% 0.34/0.50 thf(fact_55_succ__corr,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.34/0.50 = ( some_nat @ Sx ) )
% 0.34/0.50 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % succ_corr
% 0.34/0.50 thf(fact_56_pred__corr,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.34/0.50 = ( some_nat @ Px ) )
% 0.34/0.50 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pred_corr
% 0.34/0.50 thf(fact_57_valid__tree__deg__neq__0,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT] :
% 0.34/0.50 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.34/0.50
% 0.34/0.50 % valid_tree_deg_neq_0
% 0.34/0.50 thf(fact_58_valid__0__not,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT] :
% 0.34/0.50 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.34/0.50
% 0.34/0.50 % valid_0_not
% 0.34/0.50 thf(fact_59_min__Null__member,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,X2: nat] :
% 0.34/0.50 ( ( vEBT_VEBT_minNull @ T )
% 0.34/0.50 => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % min_Null_member
% 0.34/0.50 thf(fact_60_pred__none__empty,axiom,
% 0.34/0.50 ! [Xs: set_nat,A2: nat] :
% 0.34/0.50 ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A2 @ X_1 )
% 0.34/0.50 => ( ( finite_finite_nat @ Xs )
% 0.34/0.50 => ~ ? [X4: nat] :
% 0.34/0.50 ( ( member_nat @ X4 @ Xs )
% 0.34/0.50 & ( ord_less_nat @ X4 @ A2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pred_none_empty
% 0.34/0.50 thf(fact_61_succ__none__empty,axiom,
% 0.34/0.50 ! [Xs: set_nat,A2: nat] :
% 0.34/0.50 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A2 @ X_1 )
% 0.34/0.50 => ( ( finite_finite_nat @ Xs )
% 0.34/0.50 => ~ ? [X4: nat] :
% 0.34/0.50 ( ( member_nat @ X4 @ Xs )
% 0.34/0.50 & ( ord_less_nat @ A2 @ X4 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % succ_none_empty
% 0.34/0.50 thf(fact_62_member__correct,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.34/0.50 = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % member_correct
% 0.34/0.50 thf(fact_63_obtain__set__pred,axiom,
% 0.34/0.50 ! [Z: nat,X2: nat,A3: set_nat] :
% 0.34/0.50 ( ( ord_less_nat @ Z @ X2 )
% 0.34/0.50 => ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
% 0.34/0.50 => ( ( finite_finite_nat @ A3 )
% 0.34/0.50 => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X2 @ X_1 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % obtain_set_pred
% 0.34/0.50 thf(fact_64_pred__correct,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.34/0.50 = ( some_nat @ Sx ) )
% 0.34/0.50 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pred_correct
% 0.34/0.50 thf(fact_65_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: $o,P: $o > $o] :
% 0.34/0.50 ( ( member_o @ A2 @ ( collect_o @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_66_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: real,P: real > $o] :
% 0.34/0.50 ( ( member_real @ A2 @ ( collect_real @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_67_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: list_nat,P: list_nat > $o] :
% 0.34/0.50 ( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_68_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: set_nat,P: set_nat > $o] :
% 0.34/0.50 ( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_69_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: nat,P: nat > $o] :
% 0.34/0.50 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_70_mem__Collect__eq,axiom,
% 0.34/0.50 ! [A2: int,P: int > $o] :
% 0.34/0.50 ( ( member_int @ A2 @ ( collect_int @ P ) )
% 0.34/0.50 = ( P @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % mem_Collect_eq
% 0.34/0.50 thf(fact_71_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_o] :
% 0.34/0.50 ( ( collect_o
% 0.34/0.50 @ ^ [X: $o] : ( member_o @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_72_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_real] :
% 0.34/0.50 ( ( collect_real
% 0.34/0.50 @ ^ [X: real] : ( member_real @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_73_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_list_nat] :
% 0.34/0.50 ( ( collect_list_nat
% 0.34/0.50 @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_74_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_set_nat] :
% 0.34/0.50 ( ( collect_set_nat
% 0.34/0.50 @ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_75_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_nat] :
% 0.34/0.50 ( ( collect_nat
% 0.34/0.50 @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_76_Collect__mem__eq,axiom,
% 0.34/0.50 ! [A3: set_int] :
% 0.34/0.50 ( ( collect_int
% 0.34/0.50 @ ^ [X: int] : ( member_int @ X @ A3 ) )
% 0.34/0.50 = A3 ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_mem_eq
% 0.34/0.50 thf(fact_77_Collect__cong,axiom,
% 0.34/0.50 ! [P: real > $o,Q: real > $o] :
% 0.34/0.50 ( ! [X5: real] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect_real @ P )
% 0.34/0.50 = ( collect_real @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_78_Collect__cong,axiom,
% 0.34/0.50 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.34/0.50 ( ! [X5: list_nat] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect_list_nat @ P )
% 0.34/0.50 = ( collect_list_nat @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_79_Collect__cong,axiom,
% 0.34/0.50 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.34/0.50 ( ! [X5: set_nat] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect_set_nat @ P )
% 0.34/0.50 = ( collect_set_nat @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_80_Collect__cong,axiom,
% 0.34/0.50 ! [P: nat > $o,Q: nat > $o] :
% 0.34/0.50 ( ! [X5: nat] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect_nat @ P )
% 0.34/0.50 = ( collect_nat @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_81_Collect__cong,axiom,
% 0.34/0.50 ! [P: int > $o,Q: int > $o] :
% 0.34/0.50 ( ! [X5: int] :
% 0.34/0.50 ( ( P @ X5 )
% 0.34/0.50 = ( Q @ X5 ) )
% 0.34/0.50 => ( ( collect_int @ P )
% 0.34/0.50 = ( collect_int @ Q ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % Collect_cong
% 0.34/0.50 thf(fact_82_succ__correct,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.34/0.50 = ( some_nat @ Sx ) )
% 0.34/0.50 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % succ_correct
% 0.34/0.50 thf(fact_83_obtain__set__succ,axiom,
% 0.34/0.50 ! [X2: nat,Z: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.50 ( ( ord_less_nat @ X2 @ Z )
% 0.34/0.50 => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
% 0.34/0.50 => ( ( finite_finite_nat @ B2 )
% 0.34/0.50 => ( ( A3 = B2 )
% 0.34/0.50 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X2 @ X_1 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % obtain_set_succ
% 0.34/0.50 thf(fact_84_buildup__gives__valid,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.50 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % buildup_gives_valid
% 0.34/0.50 thf(fact_85_buildup__gives__empty,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 0.34/0.50 = bot_bot_set_nat ) ).
% 0.34/0.50
% 0.34/0.50 % buildup_gives_empty
% 0.34/0.50 thf(fact_86_bot__nat__0_Onot__eq__extremum,axiom,
% 0.34/0.50 ! [A2: nat] :
% 0.34/0.50 ( ( A2 != zero_zero_nat )
% 0.34/0.50 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 0.34/0.50
% 0.34/0.50 % bot_nat_0.not_eq_extremum
% 0.34/0.50 thf(fact_87_neq0__conv,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( ( N != zero_zero_nat )
% 0.34/0.50 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.50
% 0.34/0.50 % neq0_conv
% 0.34/0.50 thf(fact_88_less__nat__zero__code,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.34/0.50
% 0.34/0.50 % less_nat_zero_code
% 0.34/0.50 thf(fact_89_not__gr__zero,axiom,
% 0.34/0.50 ! [N: nat] :
% 0.34/0.50 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.34/0.50 = ( N = zero_zero_nat ) ) ).
% 0.34/0.50
% 0.34/0.50 % not_gr_zero
% 0.34/0.50 thf(fact_90_succ__member,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 0.34/0.50 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 0.34/0.50 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.34/0.50 & ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.50 & ! [Z2: nat] :
% 0.34/0.50 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 0.34/0.50 & ( ord_less_nat @ X2 @ Z2 ) )
% 0.34/0.50 => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % succ_member
% 0.34/0.50 thf(fact_91_pred__member,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 0.34/0.50 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 0.34/0.50 = ( ( vEBT_vebt_member @ T @ Y3 )
% 0.34/0.50 & ( ord_less_nat @ Y3 @ X2 )
% 0.34/0.50 & ! [Z2: nat] :
% 0.34/0.50 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 0.34/0.50 & ( ord_less_nat @ Z2 @ X2 ) )
% 0.34/0.50 => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % pred_member
% 0.34/0.50 thf(fact_92_mint__corr__help,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_mint @ T )
% 0.34/0.50 = ( some_nat @ Mini ) )
% 0.34/0.50 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.34/0.50 => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % mint_corr_help
% 0.34/0.50 thf(fact_93_maxt__corr__help__empty,axiom,
% 0.34/0.50 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.50 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.50 => ( ( ( vEBT_vebt_maxt @ T )
% 0.34/0.50 = none_nat )
% 0.34/0.50 => ( ( vEBT_VEBT_set_vebt @ T )
% 0.34/0.50 = bot_bot_set_nat ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % maxt_corr_help_empty
% 0.34/0.50 thf(fact_94_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_Extended_enat] :
% 0.34/0.50 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.50 => ( ( S != bot_bo7653980558646680370d_enat )
% 0.34/0.50 => ? [X5: extended_enat] :
% 0.34/0.50 ( ( member_Extended_enat @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: extended_enat] :
% 0.34/0.50 ( ( member_Extended_enat @ Xa @ S )
% 0.34/0.50 & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_95_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_o] :
% 0.34/0.50 ( ( finite_finite_o @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_o )
% 0.34/0.50 => ? [X5: $o] :
% 0.34/0.50 ( ( member_o @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: $o] :
% 0.34/0.50 ( ( member_o @ Xa @ S )
% 0.34/0.50 & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_96_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_real] :
% 0.34/0.50 ( ( finite_finite_real @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_real )
% 0.34/0.50 => ? [X5: real] :
% 0.34/0.50 ( ( member_real @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: real] :
% 0.34/0.50 ( ( member_real @ Xa @ S )
% 0.34/0.50 & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_97_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_rat] :
% 0.34/0.50 ( ( finite_finite_rat @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_rat )
% 0.34/0.50 => ? [X5: rat] :
% 0.34/0.50 ( ( member_rat @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: rat] :
% 0.34/0.50 ( ( member_rat @ Xa @ S )
% 0.34/0.50 & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_98_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_num] :
% 0.34/0.50 ( ( finite_finite_num @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_num )
% 0.34/0.50 => ? [X5: num] :
% 0.34/0.50 ( ( member_num @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: num] :
% 0.34/0.50 ( ( member_num @ Xa @ S )
% 0.34/0.50 & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_99_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_nat] :
% 0.34/0.50 ( ( finite_finite_nat @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_nat )
% 0.34/0.50 => ? [X5: nat] :
% 0.34/0.50 ( ( member_nat @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: nat] :
% 0.34/0.50 ( ( member_nat @ Xa @ S )
% 0.34/0.50 & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_100_ex__min__if__finite,axiom,
% 0.34/0.50 ! [S: set_int] :
% 0.34/0.50 ( ( finite_finite_int @ S )
% 0.34/0.50 => ( ( S != bot_bot_set_int )
% 0.34/0.50 => ? [X5: int] :
% 0.34/0.50 ( ( member_int @ X5 @ S )
% 0.34/0.50 & ~ ? [Xa: int] :
% 0.34/0.50 ( ( member_int @ Xa @ S )
% 0.34/0.50 & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % ex_min_if_finite
% 0.34/0.50 thf(fact_101_infinite__growing,axiom,
% 0.34/0.50 ! [X6: set_Extended_enat] :
% 0.34/0.50 ( ( X6 != bot_bo7653980558646680370d_enat )
% 0.34/0.50 => ( ! [X5: extended_enat] :
% 0.34/0.50 ( ( member_Extended_enat @ X5 @ X6 )
% 0.34/0.50 => ? [Xa: extended_enat] :
% 0.34/0.50 ( ( member_Extended_enat @ Xa @ X6 )
% 0.34/0.50 & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
% 0.34/0.50 => ~ ( finite4001608067531595151d_enat @ X6 ) ) ) ).
% 0.34/0.50
% 0.34/0.50 % infinite_growing
% 0.34/0.50 thf(fact_102_infinite__growing,axiom,
% 0.34/0.50 ! [X6: set_o] :
% 0.34/0.50 ( ( X6 != bot_bot_set_o )
% 0.34/0.50 => ( ! [X5: $o] :
% 0.34/0.50 ( ( member_o @ X5 @ X6 )
% 0.34/0.50 => ? [Xa: $o] :
% 0.34/0.50 ( ( member_o @ Xa @ X6 )
% 0.34/0.50 & ( ord_less_o @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_o @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_103_infinite__growing,axiom,
% 0.34/0.51 ! [X6: set_real] :
% 0.34/0.51 ( ( X6 != bot_bot_set_real )
% 0.34/0.51 => ( ! [X5: real] :
% 0.34/0.51 ( ( member_real @ X5 @ X6 )
% 0.34/0.51 => ? [Xa: real] :
% 0.34/0.51 ( ( member_real @ Xa @ X6 )
% 0.34/0.51 & ( ord_less_real @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_real @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_104_infinite__growing,axiom,
% 0.34/0.51 ! [X6: set_rat] :
% 0.34/0.51 ( ( X6 != bot_bot_set_rat )
% 0.34/0.51 => ( ! [X5: rat] :
% 0.34/0.51 ( ( member_rat @ X5 @ X6 )
% 0.34/0.51 => ? [Xa: rat] :
% 0.34/0.51 ( ( member_rat @ Xa @ X6 )
% 0.34/0.51 & ( ord_less_rat @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_rat @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_105_infinite__growing,axiom,
% 0.34/0.51 ! [X6: set_num] :
% 0.34/0.51 ( ( X6 != bot_bot_set_num )
% 0.34/0.51 => ( ! [X5: num] :
% 0.34/0.51 ( ( member_num @ X5 @ X6 )
% 0.34/0.51 => ? [Xa: num] :
% 0.34/0.51 ( ( member_num @ Xa @ X6 )
% 0.34/0.51 & ( ord_less_num @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_num @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_106_infinite__growing,axiom,
% 0.34/0.51 ! [X6: set_nat] :
% 0.34/0.51 ( ( X6 != bot_bot_set_nat )
% 0.34/0.51 => ( ! [X5: nat] :
% 0.34/0.51 ( ( member_nat @ X5 @ X6 )
% 0.34/0.51 => ? [Xa: nat] :
% 0.34/0.51 ( ( member_nat @ Xa @ X6 )
% 0.34/0.51 & ( ord_less_nat @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_nat @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_107_infinite__growing,axiom,
% 0.34/0.51 ! [X6: set_int] :
% 0.34/0.51 ( ( X6 != bot_bot_set_int )
% 0.34/0.51 => ( ! [X5: int] :
% 0.34/0.51 ( ( member_int @ X5 @ X6 )
% 0.34/0.51 => ? [Xa: int] :
% 0.34/0.51 ( ( member_int @ Xa @ X6 )
% 0.34/0.51 & ( ord_less_int @ X5 @ Xa ) ) )
% 0.34/0.51 => ~ ( finite_finite_int @ X6 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_growing
% 0.34/0.51 thf(fact_108_max__in__set__def,axiom,
% 0.34/0.51 ( vEBT_VEBT_max_in_set
% 0.34/0.51 = ( ^ [Xs2: set_nat,X: nat] :
% 0.34/0.51 ( ( member_nat @ X @ Xs2 )
% 0.34/0.51 & ! [Y: nat] :
% 0.34/0.51 ( ( member_nat @ Y @ Xs2 )
% 0.34/0.51 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % max_in_set_def
% 0.34/0.51 thf(fact_109_min__in__set__def,axiom,
% 0.34/0.51 ( vEBT_VEBT_min_in_set
% 0.34/0.51 = ( ^ [Xs2: set_nat,X: nat] :
% 0.34/0.51 ( ( member_nat @ X @ Xs2 )
% 0.34/0.51 & ! [Y: nat] :
% 0.34/0.51 ( ( member_nat @ Y @ Xs2 )
% 0.34/0.51 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % min_in_set_def
% 0.34/0.51 thf(fact_110_maxt__member,axiom,
% 0.34/0.51 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 0.34/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.51 => ( ( ( vEBT_vebt_maxt @ T )
% 0.34/0.51 = ( some_nat @ Maxi ) )
% 0.34/0.51 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % maxt_member
% 0.34/0.51 thf(fact_111_maxt__corr__help,axiom,
% 0.34/0.51 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
% 0.34/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.51 => ( ( ( vEBT_vebt_maxt @ T )
% 0.34/0.51 = ( some_nat @ Maxi ) )
% 0.34/0.51 => ( ( vEBT_vebt_member @ T @ X2 )
% 0.34/0.51 => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % maxt_corr_help
% 0.34/0.51 thf(fact_112_maxt__corr,axiom,
% 0.34/0.51 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.51 => ( ( ( vEBT_vebt_maxt @ T )
% 0.34/0.51 = ( some_nat @ X2 ) )
% 0.34/0.51 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % maxt_corr
% 0.34/0.51 thf(fact_113_maxt__sound,axiom,
% 0.34/0.51 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.51 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.51 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 0.34/0.51 => ( ( vEBT_vebt_maxt @ T )
% 0.34/0.51 = ( some_nat @ X2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % maxt_sound
% 0.34/0.51 thf(fact_114_le__zero__eq,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.34/0.51 = ( N = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_zero_eq
% 0.34/0.51 thf(fact_115_bot__nat__0_Oextremum,axiom,
% 0.34/0.51 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % bot_nat_0.extremum
% 0.34/0.51 thf(fact_116_le0,axiom,
% 0.34/0.51 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.34/0.51
% 0.34/0.51 % le0
% 0.34/0.51 thf(fact_117_lesseq__shift,axiom,
% 0.34/0.51 ( ord_less_eq_nat
% 0.34/0.51 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % lesseq_shift
% 0.34/0.51 thf(fact_118_le__refl,axiom,
% 0.34/0.51 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 0.34/0.51
% 0.34/0.51 % le_refl
% 0.34/0.51 thf(fact_119_le__trans,axiom,
% 0.34/0.51 ! [I: nat,J: nat,K: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.51 => ( ( ord_less_eq_nat @ J @ K )
% 0.34/0.51 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_trans
% 0.34/0.51 thf(fact_120_eq__imp__le,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( M = N )
% 0.34/0.51 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % eq_imp_le
% 0.34/0.51 thf(fact_121_le__antisym,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.51 => ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.51 => ( M = N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_antisym
% 0.34/0.51 thf(fact_122_nat__le__linear,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.51 | ( ord_less_eq_nat @ N @ M ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_le_linear
% 0.34/0.51 thf(fact_123_Nat_Oex__has__greatest__nat,axiom,
% 0.34/0.51 ! [P: nat > $o,K: nat,B3: nat] :
% 0.34/0.51 ( ( P @ K )
% 0.34/0.51 => ( ! [Y4: nat] :
% 0.34/0.51 ( ( P @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 0.34/0.51 => ? [X5: nat] :
% 0.34/0.51 ( ( P @ X5 )
% 0.34/0.51 & ! [Y5: nat] :
% 0.34/0.51 ( ( P @ Y5 )
% 0.34/0.51 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Nat.ex_has_greatest_nat
% 0.34/0.51 thf(fact_124_zero__le,axiom,
% 0.34/0.51 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % zero_le
% 0.34/0.51 thf(fact_125_less__eq__nat_Osimps_I1_J,axiom,
% 0.34/0.51 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 0.34/0.51
% 0.34/0.51 % less_eq_nat.simps(1)
% 0.34/0.51 thf(fact_126_bot__nat__0_Oextremum__unique,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.51 = ( A2 = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_nat_0.extremum_unique
% 0.34/0.51 thf(fact_127_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.34/0.51 => ( A2 = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_nat_0.extremum_uniqueI
% 0.34/0.51 thf(fact_128_le__0__eq,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 0.34/0.51 = ( N = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_0_eq
% 0.34/0.51 thf(fact_129_less__mono__imp__le__mono,axiom,
% 0.34/0.51 ! [F: nat > nat,I: nat,J: nat] :
% 0.34/0.51 ( ! [I2: nat,J2: nat] :
% 0.34/0.51 ( ( ord_less_nat @ I2 @ J2 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 0.34/0.51 => ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_mono_imp_le_mono
% 0.34/0.51 thf(fact_130_le__neq__implies__less,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.51 => ( ( M != N )
% 0.34/0.51 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_neq_implies_less
% 0.34/0.51 thf(fact_131_less__or__eq__imp__le,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ( ord_less_nat @ M @ N )
% 0.34/0.51 | ( M = N ) )
% 0.34/0.51 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_or_eq_imp_le
% 0.34/0.51 thf(fact_132_le__eq__less__or__eq,axiom,
% 0.34/0.51 ( ord_less_eq_nat
% 0.34/0.51 = ( ^ [M2: nat,N2: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M2 @ N2 )
% 0.34/0.51 | ( M2 = N2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_eq_less_or_eq
% 0.34/0.51 thf(fact_133_less__imp__le__nat,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M @ N )
% 0.34/0.51 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_le_nat
% 0.34/0.51 thf(fact_134_nat__less__le,axiom,
% 0.34/0.51 ( ord_less_nat
% 0.34/0.51 = ( ^ [M2: nat,N2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.34/0.51 & ( M2 != N2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_less_le
% 0.34/0.51 thf(fact_135_ex__least__nat__le,axiom,
% 0.34/0.51 ! [P: nat > $o,N: nat] :
% 0.34/0.51 ( ( P @ N )
% 0.34/0.51 => ( ~ ( P @ zero_zero_nat )
% 0.34/0.51 => ? [K2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ K2 @ N )
% 0.34/0.51 & ! [I3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ I3 @ K2 )
% 0.34/0.51 => ~ ( P @ I3 ) )
% 0.34/0.51 & ( P @ K2 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_least_nat_le
% 0.34/0.51 thf(fact_136_zero__reorient,axiom,
% 0.34/0.51 ! [X2: literal] :
% 0.34/0.51 ( ( zero_zero_literal = X2 )
% 0.34/0.51 = ( X2 = zero_zero_literal ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_reorient
% 0.34/0.51 thf(fact_137_zero__reorient,axiom,
% 0.34/0.51 ! [X2: real] :
% 0.34/0.51 ( ( zero_zero_real = X2 )
% 0.34/0.51 = ( X2 = zero_zero_real ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_reorient
% 0.34/0.51 thf(fact_138_zero__reorient,axiom,
% 0.34/0.51 ! [X2: rat] :
% 0.34/0.51 ( ( zero_zero_rat = X2 )
% 0.34/0.51 = ( X2 = zero_zero_rat ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_reorient
% 0.34/0.51 thf(fact_139_zero__reorient,axiom,
% 0.34/0.51 ! [X2: nat] :
% 0.34/0.51 ( ( zero_zero_nat = X2 )
% 0.34/0.51 = ( X2 = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_reorient
% 0.34/0.51 thf(fact_140_zero__reorient,axiom,
% 0.34/0.51 ! [X2: int] :
% 0.34/0.51 ( ( zero_zero_int = X2 )
% 0.34/0.51 = ( X2 = zero_zero_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_reorient
% 0.34/0.51 thf(fact_141_linorder__neqE__nat,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE_nat
% 0.34/0.51 thf(fact_142_infinite__descent,axiom,
% 0.34/0.51 ! [P: nat > $o,N: nat] :
% 0.34/0.51 ( ! [N3: nat] :
% 0.34/0.51 ( ~ ( P @ N3 )
% 0.34/0.51 => ? [M3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M3 @ N3 )
% 0.34/0.51 & ~ ( P @ M3 ) ) )
% 0.34/0.51 => ( P @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_descent
% 0.34/0.51 thf(fact_143_nat__less__induct,axiom,
% 0.34/0.51 ! [P: nat > $o,N: nat] :
% 0.34/0.51 ( ! [N3: nat] :
% 0.34/0.51 ( ! [M3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M3 @ N3 )
% 0.34/0.51 => ( P @ M3 ) )
% 0.34/0.51 => ( P @ N3 ) )
% 0.34/0.51 => ( P @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_less_induct
% 0.34/0.51 thf(fact_144_less__irrefl__nat,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ N @ N ) ).
% 0.34/0.51
% 0.34/0.51 % less_irrefl_nat
% 0.34/0.51 thf(fact_145_less__not__refl3,axiom,
% 0.34/0.51 ! [S2: nat,T: nat] :
% 0.34/0.51 ( ( ord_less_nat @ S2 @ T )
% 0.34/0.51 => ( S2 != T ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_not_refl3
% 0.34/0.51 thf(fact_146_less__not__refl2,axiom,
% 0.34/0.51 ! [N: nat,M: nat] :
% 0.34/0.51 ( ( ord_less_nat @ N @ M )
% 0.34/0.51 => ( M != N ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_not_refl2
% 0.34/0.51 thf(fact_147_less__not__refl,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ N @ N ) ).
% 0.34/0.51
% 0.34/0.51 % less_not_refl
% 0.34/0.51 thf(fact_148_nat__neq__iff,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( M != N )
% 0.34/0.51 = ( ( ord_less_nat @ M @ N )
% 0.34/0.51 | ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nat_neq_iff
% 0.34/0.51 thf(fact_149_zero__less__iff__neq__zero,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.51 = ( N != zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % zero_less_iff_neq_zero
% 0.34/0.51 thf(fact_150_gr__implies__not__zero,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M @ N )
% 0.34/0.51 => ( N != zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % gr_implies_not_zero
% 0.34/0.51 thf(fact_151_not__less__zero,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_zero
% 0.34/0.51 thf(fact_152_gr__zeroI,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( N != zero_zero_nat )
% 0.34/0.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % gr_zeroI
% 0.34/0.51 thf(fact_153_infinite__descent0,axiom,
% 0.34/0.51 ! [P: nat > $o,N: nat] :
% 0.34/0.51 ( ( P @ zero_zero_nat )
% 0.34/0.51 => ( ! [N3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.34/0.51 => ( ~ ( P @ N3 )
% 0.34/0.51 => ? [M3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M3 @ N3 )
% 0.34/0.51 & ~ ( P @ M3 ) ) ) )
% 0.34/0.51 => ( P @ N ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_descent0
% 0.34/0.51 thf(fact_154_gr__implies__not0,axiom,
% 0.34/0.51 ! [M: nat,N: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M @ N )
% 0.34/0.51 => ( N != zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % gr_implies_not0
% 0.34/0.51 thf(fact_155_less__zeroE,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.34/0.51
% 0.34/0.51 % less_zeroE
% 0.34/0.51 thf(fact_156_not__less0,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 0.34/0.51
% 0.34/0.51 % not_less0
% 0.34/0.51 thf(fact_157_not__gr0,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 0.34/0.51 = ( N = zero_zero_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_gr0
% 0.34/0.51 thf(fact_158_gr0I,axiom,
% 0.34/0.51 ! [N: nat] :
% 0.34/0.51 ( ( N != zero_zero_nat )
% 0.34/0.51 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.51
% 0.34/0.51 % gr0I
% 0.34/0.51 thf(fact_159_bot__nat__0_Oextremum__strict,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% 0.34/0.51
% 0.34/0.51 % bot_nat_0.extremum_strict
% 0.34/0.51 thf(fact_160_buildup__nothing__in__leaf,axiom,
% 0.34/0.51 ! [N: nat,X2: nat] :
% 0.34/0.51 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % buildup_nothing_in_leaf
% 0.34/0.51 thf(fact_161_is__succ__in__set__def,axiom,
% 0.34/0.51 ( vEBT_is_succ_in_set
% 0.34/0.51 = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
% 0.34/0.51 ( ( member_nat @ Y @ Xs2 )
% 0.34/0.51 & ( ord_less_nat @ X @ Y )
% 0.34/0.51 & ! [Z2: nat] :
% 0.34/0.51 ( ( member_nat @ Z2 @ Xs2 )
% 0.34/0.51 => ( ( ord_less_nat @ X @ Z2 )
% 0.34/0.51 => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_succ_in_set_def
% 0.34/0.51 thf(fact_162_is__pred__in__set__def,axiom,
% 0.34/0.51 ( vEBT_is_pred_in_set
% 0.34/0.51 = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
% 0.34/0.51 ( ( member_nat @ Y @ Xs2 )
% 0.34/0.51 & ( ord_less_nat @ Y @ X )
% 0.34/0.51 & ! [Z2: nat] :
% 0.34/0.51 ( ( member_nat @ Z2 @ Xs2 )
% 0.34/0.51 => ( ( ord_less_nat @ Z2 @ X )
% 0.34/0.51 => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % is_pred_in_set_def
% 0.34/0.51 thf(fact_163_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_Extended_enat] :
% 0.34/0.51 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.34/0.51 => ? [X5: extended_enat] :
% 0.34/0.51 ( ( member_Extended_enat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: extended_enat] :
% 0.34/0.51 ( ( member_Extended_enat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_164_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ( finite_finite_real @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_real )
% 0.34/0.51 => ? [X5: real] :
% 0.34/0.51 ( ( member_real @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: real] :
% 0.34/0.51 ( ( member_real @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_165_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ( finite_finite_o @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_o )
% 0.34/0.51 => ? [X5: $o] :
% 0.34/0.51 ( ( member_o @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: $o] :
% 0.34/0.51 ( ( member_o @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_o @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_166_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_set_int] :
% 0.34/0.51 ( ( finite6197958912794628473et_int @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_set_int )
% 0.34/0.51 => ? [X5: set_int] :
% 0.34/0.51 ( ( member_set_int @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: set_int] :
% 0.34/0.51 ( ( member_set_int @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_167_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_rat] :
% 0.34/0.51 ( ( finite_finite_rat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_rat )
% 0.34/0.51 => ? [X5: rat] :
% 0.34/0.51 ( ( member_rat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: rat] :
% 0.34/0.51 ( ( member_rat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_168_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_num] :
% 0.34/0.51 ( ( finite_finite_num @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_num )
% 0.34/0.51 => ? [X5: num] :
% 0.34/0.51 ( ( member_num @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: num] :
% 0.34/0.51 ( ( member_num @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_169_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ( finite_finite_nat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_nat )
% 0.34/0.51 => ? [X5: nat] :
% 0.34/0.51 ( ( member_nat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: nat] :
% 0.34/0.51 ( ( member_nat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_170_finite__has__maximal,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ( finite_finite_int @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_int )
% 0.34/0.51 => ? [X5: int] :
% 0.34/0.51 ( ( member_int @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: int] :
% 0.34/0.51 ( ( member_int @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_maximal
% 0.34/0.51 thf(fact_171_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_Extended_enat] :
% 0.34/0.51 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.34/0.51 => ? [X5: extended_enat] :
% 0.34/0.51 ( ( member_Extended_enat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: extended_enat] :
% 0.34/0.51 ( ( member_Extended_enat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_172_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ( finite_finite_real @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_real )
% 0.34/0.51 => ? [X5: real] :
% 0.34/0.51 ( ( member_real @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: real] :
% 0.34/0.51 ( ( member_real @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_173_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ( finite_finite_o @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_o )
% 0.34/0.51 => ? [X5: $o] :
% 0.34/0.51 ( ( member_o @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: $o] :
% 0.34/0.51 ( ( member_o @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_o @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_174_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_set_int] :
% 0.34/0.51 ( ( finite6197958912794628473et_int @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_set_int )
% 0.34/0.51 => ? [X5: set_int] :
% 0.34/0.51 ( ( member_set_int @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: set_int] :
% 0.34/0.51 ( ( member_set_int @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_175_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_rat] :
% 0.34/0.51 ( ( finite_finite_rat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_rat )
% 0.34/0.51 => ? [X5: rat] :
% 0.34/0.51 ( ( member_rat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: rat] :
% 0.34/0.51 ( ( member_rat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_176_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_num] :
% 0.34/0.51 ( ( finite_finite_num @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_num )
% 0.34/0.51 => ? [X5: num] :
% 0.34/0.51 ( ( member_num @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: num] :
% 0.34/0.51 ( ( member_num @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_177_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ( finite_finite_nat @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_nat )
% 0.34/0.51 => ? [X5: nat] :
% 0.34/0.51 ( ( member_nat @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: nat] :
% 0.34/0.51 ( ( member_nat @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_178_finite__has__minimal,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ( finite_finite_int @ A3 )
% 0.34/0.51 => ( ( A3 != bot_bot_set_int )
% 0.34/0.51 => ? [X5: int] :
% 0.34/0.51 ( ( member_int @ X5 @ A3 )
% 0.34/0.51 & ! [Xa: int] :
% 0.34/0.51 ( ( member_int @ Xa @ A3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.34/0.51 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_has_minimal
% 0.34/0.51 thf(fact_179_empty__iff,axiom,
% 0.34/0.51 ! [C: set_nat] :
% 0.34/0.51 ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_180_empty__iff,axiom,
% 0.34/0.51 ! [C: real] :
% 0.34/0.51 ~ ( member_real @ C @ bot_bot_set_real ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_181_empty__iff,axiom,
% 0.34/0.51 ! [C: $o] :
% 0.34/0.51 ~ ( member_o @ C @ bot_bot_set_o ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_182_empty__iff,axiom,
% 0.34/0.51 ! [C: nat] :
% 0.34/0.51 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_183_empty__iff,axiom,
% 0.34/0.51 ! [C: int] :
% 0.34/0.51 ~ ( member_int @ C @ bot_bot_set_int ) ).
% 0.34/0.51
% 0.34/0.51 % empty_iff
% 0.34/0.51 thf(fact_184_all__not__in__conv,axiom,
% 0.34/0.51 ! [A3: set_set_nat] :
% 0.34/0.51 ( ( ! [X: set_nat] :
% 0.34/0.51 ~ ( member_set_nat @ X @ A3 ) )
% 0.34/0.51 = ( A3 = bot_bot_set_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_185_all__not__in__conv,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ( ! [X: real] :
% 0.34/0.51 ~ ( member_real @ X @ A3 ) )
% 0.34/0.51 = ( A3 = bot_bot_set_real ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_186_all__not__in__conv,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ( ! [X: $o] :
% 0.34/0.51 ~ ( member_o @ X @ A3 ) )
% 0.34/0.51 = ( A3 = bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_187_all__not__in__conv,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ( ! [X: nat] :
% 0.34/0.51 ~ ( member_nat @ X @ A3 ) )
% 0.34/0.51 = ( A3 = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_188_all__not__in__conv,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ( ! [X: int] :
% 0.34/0.51 ~ ( member_int @ X @ A3 ) )
% 0.34/0.51 = ( A3 = bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % all_not_in_conv
% 0.34/0.51 thf(fact_189_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: list_nat > $o] :
% 0.34/0.51 ( ( ( collect_list_nat @ P )
% 0.34/0.51 = bot_bot_set_list_nat )
% 0.34/0.51 = ( ! [X: list_nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_190_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: set_nat > $o] :
% 0.34/0.51 ( ( ( collect_set_nat @ P )
% 0.34/0.51 = bot_bot_set_set_nat )
% 0.34/0.51 = ( ! [X: set_nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_191_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: real > $o] :
% 0.34/0.51 ( ( ( collect_real @ P )
% 0.34/0.51 = bot_bot_set_real )
% 0.34/0.51 = ( ! [X: real] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_192_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: $o > $o] :
% 0.34/0.51 ( ( ( collect_o @ P )
% 0.34/0.51 = bot_bot_set_o )
% 0.34/0.51 = ( ! [X: $o] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_193_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: nat > $o] :
% 0.34/0.51 ( ( ( collect_nat @ P )
% 0.34/0.51 = bot_bot_set_nat )
% 0.34/0.51 = ( ! [X: nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_194_Collect__empty__eq,axiom,
% 0.34/0.51 ! [P: int > $o] :
% 0.34/0.51 ( ( ( collect_int @ P )
% 0.34/0.51 = bot_bot_set_int )
% 0.34/0.51 = ( ! [X: int] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Collect_empty_eq
% 0.34/0.51 thf(fact_195_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: list_nat > $o] :
% 0.34/0.51 ( ( bot_bot_set_list_nat
% 0.34/0.51 = ( collect_list_nat @ P ) )
% 0.34/0.51 = ( ! [X: list_nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_196_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: set_nat > $o] :
% 0.34/0.51 ( ( bot_bot_set_set_nat
% 0.34/0.51 = ( collect_set_nat @ P ) )
% 0.34/0.51 = ( ! [X: set_nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_197_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: real > $o] :
% 0.34/0.51 ( ( bot_bot_set_real
% 0.34/0.51 = ( collect_real @ P ) )
% 0.34/0.51 = ( ! [X: real] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_198_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: $o > $o] :
% 0.34/0.51 ( ( bot_bot_set_o
% 0.34/0.51 = ( collect_o @ P ) )
% 0.34/0.51 = ( ! [X: $o] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_199_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: nat > $o] :
% 0.34/0.51 ( ( bot_bot_set_nat
% 0.34/0.51 = ( collect_nat @ P ) )
% 0.34/0.51 = ( ! [X: nat] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_200_empty__Collect__eq,axiom,
% 0.34/0.51 ! [P: int > $o] :
% 0.34/0.51 ( ( bot_bot_set_int
% 0.34/0.51 = ( collect_int @ P ) )
% 0.34/0.51 = ( ! [X: int] :
% 0.34/0.51 ~ ( P @ X ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % empty_Collect_eq
% 0.34/0.51 thf(fact_201_order__refl,axiom,
% 0.34/0.51 ! [X2: set_int] : ( ord_less_eq_set_int @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_refl
% 0.34/0.51 thf(fact_202_order__refl,axiom,
% 0.34/0.51 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_refl
% 0.34/0.51 thf(fact_203_order__refl,axiom,
% 0.34/0.51 ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_refl
% 0.34/0.51 thf(fact_204_order__refl,axiom,
% 0.34/0.51 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_refl
% 0.34/0.51 thf(fact_205_order__refl,axiom,
% 0.34/0.51 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_refl
% 0.34/0.51 thf(fact_206_dual__order_Orefl,axiom,
% 0.34/0.51 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.refl
% 0.34/0.51 thf(fact_207_dual__order_Orefl,axiom,
% 0.34/0.51 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.refl
% 0.34/0.51 thf(fact_208_dual__order_Orefl,axiom,
% 0.34/0.51 ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.refl
% 0.34/0.51 thf(fact_209_dual__order_Orefl,axiom,
% 0.34/0.51 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.refl
% 0.34/0.51 thf(fact_210_dual__order_Orefl,axiom,
% 0.34/0.51 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.refl
% 0.34/0.51 thf(fact_211_empty__subsetI,axiom,
% 0.34/0.51 ! [A3: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A3 ) ).
% 0.34/0.51
% 0.34/0.51 % empty_subsetI
% 0.34/0.51 thf(fact_212_empty__subsetI,axiom,
% 0.34/0.51 ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).
% 0.34/0.51
% 0.34/0.51 % empty_subsetI
% 0.34/0.51 thf(fact_213_empty__subsetI,axiom,
% 0.34/0.51 ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% 0.34/0.51
% 0.34/0.51 % empty_subsetI
% 0.34/0.51 thf(fact_214_empty__subsetI,axiom,
% 0.34/0.51 ! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).
% 0.34/0.51
% 0.34/0.51 % empty_subsetI
% 0.34/0.51 thf(fact_215_subset__empty,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
% 0.34/0.51 = ( A3 = bot_bot_set_real ) ) ).
% 0.34/0.51
% 0.34/0.51 % subset_empty
% 0.34/0.51 thf(fact_216_subset__empty,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
% 0.34/0.51 = ( A3 = bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % subset_empty
% 0.34/0.51 thf(fact_217_subset__empty,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
% 0.34/0.51 = ( A3 = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % subset_empty
% 0.34/0.51 thf(fact_218_subset__empty,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
% 0.34/0.51 = ( A3 = bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % subset_empty
% 0.34/0.51 thf(fact_219_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_list_nat
% 0.34/0.51 = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_220_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_set_nat
% 0.34/0.51 = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_221_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_real
% 0.34/0.51 = ( collect_real @ bot_bot_real_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_222_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_o
% 0.34/0.51 = ( collect_o @ bot_bot_o_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_223_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_nat
% 0.34/0.51 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_224_bot__set__def,axiom,
% 0.34/0.51 ( bot_bot_set_int
% 0.34/0.51 = ( collect_int @ bot_bot_int_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % bot_set_def
% 0.34/0.51 thf(fact_225_rev__finite__subset,axiom,
% 0.34/0.51 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.51 ( ( finite_finite_nat @ B2 )
% 0.34/0.51 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.51 => ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % rev_finite_subset
% 0.34/0.51 thf(fact_226_rev__finite__subset,axiom,
% 0.34/0.51 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.51 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.51 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.51 => ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % rev_finite_subset
% 0.34/0.51 thf(fact_227_rev__finite__subset,axiom,
% 0.34/0.51 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.51 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.51 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.51 => ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % rev_finite_subset
% 0.34/0.51 thf(fact_228_rev__finite__subset,axiom,
% 0.34/0.51 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.51 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.51 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.51 => ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % rev_finite_subset
% 0.34/0.51 thf(fact_229_rev__finite__subset,axiom,
% 0.34/0.51 ! [B2: set_int,A3: set_int] :
% 0.34/0.51 ( ( finite_finite_int @ B2 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.51 => ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % rev_finite_subset
% 0.34/0.51 thf(fact_230_infinite__super,axiom,
% 0.34/0.51 ! [S: set_nat,T2: set_nat] :
% 0.34/0.51 ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.34/0.51 => ( ~ ( finite_finite_nat @ S )
% 0.34/0.51 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_super
% 0.34/0.51 thf(fact_231_infinite__super,axiom,
% 0.34/0.51 ! [S: set_complex,T2: set_complex] :
% 0.34/0.51 ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.34/0.51 => ( ~ ( finite3207457112153483333omplex @ S )
% 0.34/0.51 => ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_super
% 0.34/0.51 thf(fact_232_infinite__super,axiom,
% 0.34/0.51 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.34/0.51 ( ( ord_le3146513528884898305at_nat @ S @ T2 )
% 0.34/0.51 => ( ~ ( finite6177210948735845034at_nat @ S )
% 0.34/0.51 => ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_super
% 0.34/0.51 thf(fact_233_infinite__super,axiom,
% 0.34/0.51 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 0.34/0.51 ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.34/0.51 => ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.51 => ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_super
% 0.34/0.51 thf(fact_234_infinite__super,axiom,
% 0.34/0.51 ! [S: set_int,T2: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ S @ T2 )
% 0.34/0.51 => ( ~ ( finite_finite_int @ S )
% 0.34/0.51 => ~ ( finite_finite_int @ T2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % infinite_super
% 0.34/0.51 thf(fact_235_finite__subset,axiom,
% 0.34/0.51 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.51 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.51 => ( ( finite_finite_nat @ B2 )
% 0.34/0.51 => ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_subset
% 0.34/0.51 thf(fact_236_finite__subset,axiom,
% 0.34/0.51 ! [A3: set_complex,B2: set_complex] :
% 0.34/0.51 ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.51 => ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.51 => ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_subset
% 0.34/0.51 thf(fact_237_finite__subset,axiom,
% 0.34/0.51 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.51 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.51 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.51 => ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_subset
% 0.34/0.51 thf(fact_238_finite__subset,axiom,
% 0.34/0.51 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.34/0.51 ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.51 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.51 => ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_subset
% 0.34/0.51 thf(fact_239_finite__subset,axiom,
% 0.34/0.51 ! [A3: set_int,B2: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.51 => ( ( finite_finite_int @ B2 )
% 0.34/0.51 => ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % finite_subset
% 0.34/0.51 thf(fact_240_bot__nat__def,axiom,
% 0.34/0.51 bot_bot_nat = zero_zero_nat ).
% 0.34/0.51
% 0.34/0.51 % bot_nat_def
% 0.34/0.51 thf(fact_241_order__antisym__conv,axiom,
% 0.34/0.51 ! [Y3: set_int,X2: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym_conv
% 0.34/0.51 thf(fact_242_order__antisym__conv,axiom,
% 0.34/0.51 ! [Y3: rat,X2: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym_conv
% 0.34/0.51 thf(fact_243_order__antisym__conv,axiom,
% 0.34/0.51 ! [Y3: num,X2: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.34/0.51 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym_conv
% 0.34/0.51 thf(fact_244_order__antisym__conv,axiom,
% 0.34/0.51 ! [Y3: nat,X2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym_conv
% 0.34/0.51 thf(fact_245_order__antisym__conv,axiom,
% 0.34/0.51 ! [Y3: int,X2: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.34/0.51 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym_conv
% 0.34/0.51 thf(fact_246_linorder__le__cases,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ~ ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_le_cases
% 0.34/0.51 thf(fact_247_linorder__le__cases,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ~ ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_le_cases
% 0.34/0.51 thf(fact_248_linorder__le__cases,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_le_cases
% 0.34/0.51 thf(fact_249_linorder__le__cases,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ~ ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_le_cases
% 0.34/0.51 thf(fact_250_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_251_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_252_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_253_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_254_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_255_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_256_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_257_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_258_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_259_ord__le__eq__subst,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,F: nat > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_subst
% 0.34/0.51 thf(fact_260_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_261_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_262_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_263_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_264_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_265_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: num,F: num > num,B3: num,C: num] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_266_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_267_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: int,F: num > int,B3: num,C: num] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_268_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_269_ord__eq__le__subst,axiom,
% 0.34/0.51 ! [A2: num,F: nat > num,B3: nat,C: nat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_subst
% 0.34/0.51 thf(fact_270_linorder__linear,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_linear
% 0.34/0.51 thf(fact_271_linorder__linear,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_linear
% 0.34/0.51 thf(fact_272_linorder__linear,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_linear
% 0.34/0.51 thf(fact_273_linorder__linear,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_linear
% 0.34/0.51 thf(fact_274_order__eq__refl,axiom,
% 0.34/0.51 ! [X2: set_int,Y3: set_int] :
% 0.34/0.51 ( ( X2 = Y3 )
% 0.34/0.51 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_eq_refl
% 0.34/0.51 thf(fact_275_order__eq__refl,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( X2 = Y3 )
% 0.34/0.51 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_eq_refl
% 0.34/0.51 thf(fact_276_order__eq__refl,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( X2 = Y3 )
% 0.34/0.51 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_eq_refl
% 0.34/0.51 thf(fact_277_order__eq__refl,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( X2 = Y3 )
% 0.34/0.51 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_eq_refl
% 0.34/0.51 thf(fact_278_order__eq__refl,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( X2 = Y3 )
% 0.34/0.51 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_eq_refl
% 0.34/0.51 thf(fact_279_order__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_280_order__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_281_order__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_282_order__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_283_order__subst2,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_284_order__subst2,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_285_order__subst2,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_286_order__subst2,axiom,
% 0.34/0.51 ! [A2: num,B3: num,F: num > int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_287_order__subst2,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_288_order__subst2,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,F: nat > num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst2
% 0.34/0.51 thf(fact_289_order__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_290_order__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_291_order__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_292_order__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_int @ B3 @ C )
% 0.34/0.51 => ( ! [X5: int,Y4: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_293_order__subst1,axiom,
% 0.34/0.51 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_294_order__subst1,axiom,
% 0.34/0.51 ! [A2: num,F: num > num,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_295_order__subst1,axiom,
% 0.34/0.51 ! [A2: num,F: nat > num,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_296_order__subst1,axiom,
% 0.34/0.51 ! [A2: num,F: int > num,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_int @ B3 @ C )
% 0.34/0.51 => ( ! [X5: int,Y4: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_297_order__subst1,axiom,
% 0.34/0.51 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_298_order__subst1,axiom,
% 0.34/0.51 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_subst1
% 0.34/0.51 thf(fact_299_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.34/0.51 & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Orderings.order_eq_iff
% 0.34/0.51 thf(fact_300_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.34/0.51 & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Orderings.order_eq_iff
% 0.34/0.51 thf(fact_301_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: num,B4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A4 @ B4 )
% 0.34/0.51 & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Orderings.order_eq_iff
% 0.34/0.51 thf(fact_302_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: nat,B4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.34/0.51 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Orderings.order_eq_iff
% 0.34/0.51 thf(fact_303_Orderings_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: int,B4: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ A4 @ B4 )
% 0.34/0.51 & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % Orderings.order_eq_iff
% 0.34/0.51 thf(fact_304_antisym,axiom,
% 0.34/0.51 ! [A2: set_int,B3: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym
% 0.34/0.51 thf(fact_305_antisym,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym
% 0.34/0.51 thf(fact_306_antisym,axiom,
% 0.34/0.51 ! [A2: num,B3: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ A2 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym
% 0.34/0.51 thf(fact_307_antisym,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym
% 0.34/0.51 thf(fact_308_antisym,axiom,
% 0.34/0.51 ! [A2: int,B3: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym
% 0.34/0.51 thf(fact_309_dual__order_Otrans,axiom,
% 0.34/0.51 ! [B3: set_int,A2: set_int,C: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ C @ B3 )
% 0.34/0.51 => ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.trans
% 0.34/0.51 thf(fact_310_dual__order_Otrans,axiom,
% 0.34/0.51 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ C @ B3 )
% 0.34/0.51 => ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.trans
% 0.34/0.51 thf(fact_311_dual__order_Otrans,axiom,
% 0.34/0.51 ! [B3: num,A2: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_num @ C @ B3 )
% 0.34/0.51 => ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.trans
% 0.34/0.51 thf(fact_312_dual__order_Otrans,axiom,
% 0.34/0.51 ! [B3: nat,A2: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.34/0.51 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.trans
% 0.34/0.51 thf(fact_313_dual__order_Otrans,axiom,
% 0.34/0.51 ! [B3: int,A2: int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_int @ C @ B3 )
% 0.34/0.51 => ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.trans
% 0.34/0.51 thf(fact_314_dual__order_Oantisym,axiom,
% 0.34/0.51 ! [B3: set_int,A2: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.antisym
% 0.34/0.51 thf(fact_315_dual__order_Oantisym,axiom,
% 0.34/0.51 ! [B3: rat,A2: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.antisym
% 0.34/0.51 thf(fact_316_dual__order_Oantisym,axiom,
% 0.34/0.51 ! [B3: num,A2: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.antisym
% 0.34/0.51 thf(fact_317_dual__order_Oantisym,axiom,
% 0.34/0.51 ! [B3: nat,A2: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.antisym
% 0.34/0.51 thf(fact_318_dual__order_Oantisym,axiom,
% 0.34/0.51 ! [B3: int,A2: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.51 => ( A2 = B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.antisym
% 0.34/0.51 thf(fact_319_dual__order_Oeq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: set_int,B4: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.34/0.51 & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.eq_iff
% 0.34/0.51 thf(fact_320_dual__order_Oeq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.34/0.51 & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.eq_iff
% 0.34/0.51 thf(fact_321_dual__order_Oeq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: num,B4: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ B4 @ A4 )
% 0.34/0.51 & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.eq_iff
% 0.34/0.51 thf(fact_322_dual__order_Oeq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: nat,B4: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.34/0.51 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.eq_iff
% 0.34/0.51 thf(fact_323_dual__order_Oeq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [A4: int,B4: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ B4 @ A4 )
% 0.34/0.51 & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.eq_iff
% 0.34/0.51 thf(fact_324_linorder__wlog,axiom,
% 0.34/0.51 ! [P: rat > rat > $o,A2: rat,B3: rat] :
% 0.34/0.51 ( ! [A: rat,B: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: rat,B: rat] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_wlog
% 0.34/0.51 thf(fact_325_linorder__wlog,axiom,
% 0.34/0.51 ! [P: num > num > $o,A2: num,B3: num] :
% 0.34/0.51 ( ! [A: num,B: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: num,B: num] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_wlog
% 0.34/0.51 thf(fact_326_linorder__wlog,axiom,
% 0.34/0.51 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.34/0.51 ( ! [A: nat,B: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: nat,B: nat] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_wlog
% 0.34/0.51 thf(fact_327_linorder__wlog,axiom,
% 0.34/0.51 ! [P: int > int > $o,A2: int,B3: int] :
% 0.34/0.51 ( ! [A: int,B: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: int,B: int] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_wlog
% 0.34/0.51 thf(fact_328_order__trans,axiom,
% 0.34/0.51 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_trans
% 0.34/0.51 thf(fact_329_order__trans,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_trans
% 0.34/0.51 thf(fact_330_order__trans,axiom,
% 0.34/0.51 ! [X2: num,Y3: num,Z: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_trans
% 0.34/0.51 thf(fact_331_order__trans,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat,Z: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_trans
% 0.34/0.51 thf(fact_332_order__trans,axiom,
% 0.34/0.51 ! [X2: int,Y3: int,Z: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_trans
% 0.34/0.51 thf(fact_333_order_Otrans,axiom,
% 0.34/0.51 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.trans
% 0.34/0.51 thf(fact_334_order_Otrans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.trans
% 0.34/0.51 thf(fact_335_order_Otrans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.trans
% 0.34/0.51 thf(fact_336_order_Otrans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.trans
% 0.34/0.51 thf(fact_337_order_Otrans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.trans
% 0.34/0.51 thf(fact_338_order__antisym,axiom,
% 0.34/0.51 ! [X2: set_int,Y3: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym
% 0.34/0.51 thf(fact_339_order__antisym,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym
% 0.34/0.51 thf(fact_340_order__antisym,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym
% 0.34/0.51 thf(fact_341_order__antisym,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym
% 0.34/0.51 thf(fact_342_order__antisym,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.34/0.51 => ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_antisym
% 0.34/0.51 thf(fact_343_ord__le__eq__trans,axiom,
% 0.34/0.51 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_trans
% 0.34/0.51 thf(fact_344_ord__le__eq__trans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_trans
% 0.34/0.51 thf(fact_345_ord__le__eq__trans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_trans
% 0.34/0.51 thf(fact_346_ord__le__eq__trans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_trans
% 0.34/0.51 thf(fact_347_ord__le__eq__trans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_le_eq_trans
% 0.34/0.51 thf(fact_348_ord__eq__le__trans,axiom,
% 0.34/0.51 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_eq_set_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_trans
% 0.34/0.51 thf(fact_349_ord__eq__le__trans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_trans
% 0.34/0.51 thf(fact_350_ord__eq__le__trans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_trans
% 0.34/0.51 thf(fact_351_ord__eq__le__trans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_trans
% 0.34/0.51 thf(fact_352_ord__eq__le__trans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_eq_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_le_trans
% 0.34/0.51 thf(fact_353_order__class_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [X: set_int,Y: set_int] :
% 0.34/0.51 ( ( ord_less_eq_set_int @ X @ Y )
% 0.34/0.51 & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_class.order_eq_iff
% 0.34/0.51 thf(fact_354_order__class_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [X: rat,Y: rat] :
% 0.34/0.51 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.51 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_class.order_eq_iff
% 0.34/0.51 thf(fact_355_order__class_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [X: num,Y: num] :
% 0.34/0.51 ( ( ord_less_eq_num @ X @ Y )
% 0.34/0.51 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_class.order_eq_iff
% 0.34/0.51 thf(fact_356_order__class_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [X: nat,Y: nat] :
% 0.34/0.51 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.51 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_class.order_eq_iff
% 0.34/0.51 thf(fact_357_order__class_Oorder__eq__iff,axiom,
% 0.34/0.51 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 0.34/0.51 = ( ^ [X: int,Y: int] :
% 0.34/0.51 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.51 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_class.order_eq_iff
% 0.34/0.51 thf(fact_358_le__cases3,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.51 ( ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ X2 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_rat @ X2 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ Y3 @ X2 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ Z @ X2 ) )
% 0.34/0.51 => ~ ( ( ord_less_eq_rat @ Z @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_cases3
% 0.34/0.51 thf(fact_359_le__cases3,axiom,
% 0.34/0.51 ! [X2: num,Y3: num,Z: num] :
% 0.34/0.51 ( ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_num @ X2 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_num @ X2 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_num @ Y3 @ X2 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_num @ Z @ X2 ) )
% 0.34/0.51 => ~ ( ( ord_less_eq_num @ Z @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_num @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_cases3
% 0.34/0.51 thf(fact_360_le__cases3,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat,Z: nat] :
% 0.34/0.51 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 0.34/0.51 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_cases3
% 0.34/0.51 thf(fact_361_le__cases3,axiom,
% 0.34/0.51 ! [X2: int,Y3: int,Z: int] :
% 0.34/0.51 ( ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_int @ X2 @ Z ) )
% 0.34/0.51 => ( ( ( ord_less_eq_int @ X2 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 0.34/0.51 => ~ ( ord_less_eq_int @ Y3 @ X2 ) )
% 0.34/0.51 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 0.34/0.51 => ~ ( ord_less_eq_int @ Z @ X2 ) )
% 0.34/0.51 => ~ ( ( ord_less_eq_int @ Z @ X2 )
% 0.34/0.51 => ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % le_cases3
% 0.34/0.51 thf(fact_362_nle__le,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat] :
% 0.34/0.51 ( ( ~ ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.34/0.51 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.51 & ( B3 != A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nle_le
% 0.34/0.51 thf(fact_363_nle__le,axiom,
% 0.34/0.51 ! [A2: num,B3: num] :
% 0.34/0.51 ( ( ~ ( ord_less_eq_num @ A2 @ B3 ) )
% 0.34/0.51 = ( ( ord_less_eq_num @ B3 @ A2 )
% 0.34/0.51 & ( B3 != A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nle_le
% 0.34/0.51 thf(fact_364_nle__le,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat] :
% 0.34/0.51 ( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
% 0.34/0.51 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.51 & ( B3 != A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nle_le
% 0.34/0.51 thf(fact_365_nle__le,axiom,
% 0.34/0.51 ! [A2: int,B3: int] :
% 0.34/0.51 ( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
% 0.34/0.51 = ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.51 & ( B3 != A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % nle_le
% 0.34/0.51 thf(fact_366_lt__ex,axiom,
% 0.34/0.51 ! [X2: real] :
% 0.34/0.51 ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % lt_ex
% 0.34/0.51 thf(fact_367_lt__ex,axiom,
% 0.34/0.51 ! [X2: rat] :
% 0.34/0.51 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % lt_ex
% 0.34/0.51 thf(fact_368_lt__ex,axiom,
% 0.34/0.51 ! [X2: int] :
% 0.34/0.51 ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % lt_ex
% 0.34/0.51 thf(fact_369_gt__ex,axiom,
% 0.34/0.51 ! [X2: real] :
% 0.34/0.51 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.34/0.51
% 0.34/0.51 % gt_ex
% 0.34/0.51 thf(fact_370_gt__ex,axiom,
% 0.34/0.51 ! [X2: rat] :
% 0.34/0.51 ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 0.34/0.51
% 0.34/0.51 % gt_ex
% 0.34/0.51 thf(fact_371_gt__ex,axiom,
% 0.34/0.51 ! [X2: nat] :
% 0.34/0.51 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 0.34/0.51
% 0.34/0.51 % gt_ex
% 0.34/0.51 thf(fact_372_gt__ex,axiom,
% 0.34/0.51 ! [X2: int] :
% 0.34/0.51 ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% 0.34/0.51
% 0.34/0.51 % gt_ex
% 0.34/0.51 thf(fact_373_dense,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ? [Z4: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Z4 )
% 0.34/0.51 & ( ord_less_real @ Z4 @ Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dense
% 0.34/0.51 thf(fact_374_dense,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ? [Z4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Z4 )
% 0.34/0.51 & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dense
% 0.34/0.51 thf(fact_375_less__imp__neq,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_neq
% 0.34/0.51 thf(fact_376_less__imp__neq,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_neq
% 0.34/0.51 thf(fact_377_less__imp__neq,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_neq
% 0.34/0.51 thf(fact_378_less__imp__neq,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_neq
% 0.34/0.51 thf(fact_379_less__imp__neq,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_imp_neq
% 0.34/0.51 thf(fact_380_order_Oasym,axiom,
% 0.34/0.51 ! [A2: real,B3: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_real @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.asym
% 0.34/0.51 thf(fact_381_order_Oasym,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.asym
% 0.34/0.51 thf(fact_382_order_Oasym,axiom,
% 0.34/0.51 ! [A2: num,B3: num] :
% 0.34/0.51 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_num @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.asym
% 0.34/0.51 thf(fact_383_order_Oasym,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.asym
% 0.34/0.51 thf(fact_384_order_Oasym,axiom,
% 0.34/0.51 ! [A2: int,B3: int] :
% 0.34/0.51 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_int @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.asym
% 0.34/0.51 thf(fact_385_ord__eq__less__trans,axiom,
% 0.34/0.51 ! [A2: real,B3: real,C: real] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ord_less_real @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_trans
% 0.34/0.51 thf(fact_386_ord__eq__less__trans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_trans
% 0.34/0.51 thf(fact_387_ord__eq__less__trans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.51 => ( ord_less_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_trans
% 0.34/0.51 thf(fact_388_ord__eq__less__trans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.51 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_trans
% 0.34/0.51 thf(fact_389_ord__eq__less__trans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( A2 = B3 )
% 0.34/0.51 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_trans
% 0.34/0.51 thf(fact_390_ord__less__eq__trans,axiom,
% 0.34/0.51 ! [A2: real,B3: real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_real @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_trans
% 0.34/0.51 thf(fact_391_ord__less__eq__trans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_trans
% 0.34/0.51 thf(fact_392_ord__less__eq__trans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_trans
% 0.34/0.51 thf(fact_393_ord__less__eq__trans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_trans
% 0.34/0.51 thf(fact_394_ord__less__eq__trans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.51 => ( ( B3 = C )
% 0.34/0.51 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_trans
% 0.34/0.51 thf(fact_395_less__induct,axiom,
% 0.34/0.51 ! [P: nat > $o,A2: nat] :
% 0.34/0.51 ( ! [X5: nat] :
% 0.34/0.51 ( ! [Y5: nat] :
% 0.34/0.51 ( ( ord_less_nat @ Y5 @ X5 )
% 0.34/0.51 => ( P @ Y5 ) )
% 0.34/0.51 => ( P @ X5 ) )
% 0.34/0.51 => ( P @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % less_induct
% 0.34/0.51 thf(fact_396_antisym__conv3,axiom,
% 0.34/0.51 ! [Y3: real,X2: real] :
% 0.34/0.51 ( ~ ( ord_less_real @ Y3 @ X2 )
% 0.34/0.51 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym_conv3
% 0.34/0.51 thf(fact_397_antisym__conv3,axiom,
% 0.34/0.51 ! [Y3: rat,X2: rat] :
% 0.34/0.51 ( ~ ( ord_less_rat @ Y3 @ X2 )
% 0.34/0.51 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym_conv3
% 0.34/0.51 thf(fact_398_antisym__conv3,axiom,
% 0.34/0.51 ! [Y3: num,X2: num] :
% 0.34/0.51 ( ~ ( ord_less_num @ Y3 @ X2 )
% 0.34/0.51 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym_conv3
% 0.34/0.51 thf(fact_399_antisym__conv3,axiom,
% 0.34/0.51 ! [Y3: nat,X2: nat] :
% 0.34/0.51 ( ~ ( ord_less_nat @ Y3 @ X2 )
% 0.34/0.51 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym_conv3
% 0.34/0.51 thf(fact_400_antisym__conv3,axiom,
% 0.34/0.51 ! [Y3: int,X2: int] :
% 0.34/0.51 ( ~ ( ord_less_int @ Y3 @ X2 )
% 0.34/0.51 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.34/0.51 = ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % antisym_conv3
% 0.34/0.51 thf(fact_401_linorder__cases,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( ( X2 != Y3 )
% 0.34/0.51 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_cases
% 0.34/0.51 thf(fact_402_linorder__cases,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ( X2 != Y3 )
% 0.34/0.51 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_cases
% 0.34/0.51 thf(fact_403_linorder__cases,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( ( X2 != Y3 )
% 0.34/0.51 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_cases
% 0.34/0.51 thf(fact_404_linorder__cases,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ( X2 != Y3 )
% 0.34/0.51 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_cases
% 0.34/0.51 thf(fact_405_linorder__cases,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( X2 != Y3 )
% 0.34/0.51 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_cases
% 0.34/0.51 thf(fact_406_dual__order_Oasym,axiom,
% 0.34/0.51 ! [B3: real,A2: real] :
% 0.34/0.51 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.51 => ~ ( ord_less_real @ A2 @ B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.asym
% 0.34/0.51 thf(fact_407_dual__order_Oasym,axiom,
% 0.34/0.51 ! [B3: rat,A2: rat] :
% 0.34/0.51 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.51 => ~ ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.asym
% 0.34/0.51 thf(fact_408_dual__order_Oasym,axiom,
% 0.34/0.51 ! [B3: num,A2: num] :
% 0.34/0.51 ( ( ord_less_num @ B3 @ A2 )
% 0.34/0.51 => ~ ( ord_less_num @ A2 @ B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.asym
% 0.34/0.51 thf(fact_409_dual__order_Oasym,axiom,
% 0.34/0.51 ! [B3: nat,A2: nat] :
% 0.34/0.51 ( ( ord_less_nat @ B3 @ A2 )
% 0.34/0.51 => ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.asym
% 0.34/0.51 thf(fact_410_dual__order_Oasym,axiom,
% 0.34/0.51 ! [B3: int,A2: int] :
% 0.34/0.51 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.51 => ~ ( ord_less_int @ A2 @ B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.asym
% 0.34/0.51 thf(fact_411_dual__order_Oirrefl,axiom,
% 0.34/0.51 ! [A2: real] :
% 0.34/0.51 ~ ( ord_less_real @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.irrefl
% 0.34/0.51 thf(fact_412_dual__order_Oirrefl,axiom,
% 0.34/0.51 ! [A2: rat] :
% 0.34/0.51 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.irrefl
% 0.34/0.51 thf(fact_413_dual__order_Oirrefl,axiom,
% 0.34/0.51 ! [A2: num] :
% 0.34/0.51 ~ ( ord_less_num @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.irrefl
% 0.34/0.51 thf(fact_414_dual__order_Oirrefl,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.irrefl
% 0.34/0.51 thf(fact_415_dual__order_Oirrefl,axiom,
% 0.34/0.51 ! [A2: int] :
% 0.34/0.51 ~ ( ord_less_int @ A2 @ A2 ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.irrefl
% 0.34/0.51 thf(fact_416_exists__least__iff,axiom,
% 0.34/0.51 ( ( ^ [P2: nat > $o] :
% 0.34/0.51 ? [X3: nat] : ( P2 @ X3 ) )
% 0.34/0.51 = ( ^ [P3: nat > $o] :
% 0.34/0.51 ? [N2: nat] :
% 0.34/0.51 ( ( P3 @ N2 )
% 0.34/0.51 & ! [M2: nat] :
% 0.34/0.51 ( ( ord_less_nat @ M2 @ N2 )
% 0.34/0.51 => ~ ( P3 @ M2 ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % exists_least_iff
% 0.34/0.51 thf(fact_417_linorder__less__wlog,axiom,
% 0.34/0.51 ! [P: real > real > $o,A2: real,B3: real] :
% 0.34/0.51 ( ! [A: real,B: real] :
% 0.34/0.51 ( ( ord_less_real @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: real] : ( P @ A @ A )
% 0.34/0.51 => ( ! [A: real,B: real] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_wlog
% 0.34/0.51 thf(fact_418_linorder__less__wlog,axiom,
% 0.34/0.51 ! [P: rat > rat > $o,A2: rat,B3: rat] :
% 0.34/0.51 ( ! [A: rat,B: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: rat] : ( P @ A @ A )
% 0.34/0.51 => ( ! [A: rat,B: rat] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_wlog
% 0.34/0.51 thf(fact_419_linorder__less__wlog,axiom,
% 0.34/0.51 ! [P: num > num > $o,A2: num,B3: num] :
% 0.34/0.51 ( ! [A: num,B: num] :
% 0.34/0.51 ( ( ord_less_num @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: num] : ( P @ A @ A )
% 0.34/0.51 => ( ! [A: num,B: num] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_wlog
% 0.34/0.51 thf(fact_420_linorder__less__wlog,axiom,
% 0.34/0.51 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.34/0.51 ( ! [A: nat,B: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: nat] : ( P @ A @ A )
% 0.34/0.51 => ( ! [A: nat,B: nat] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_wlog
% 0.34/0.51 thf(fact_421_linorder__less__wlog,axiom,
% 0.34/0.51 ! [P: int > int > $o,A2: int,B3: int] :
% 0.34/0.51 ( ! [A: int,B: int] :
% 0.34/0.51 ( ( ord_less_int @ A @ B )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( ! [A: int] : ( P @ A @ A )
% 0.34/0.51 => ( ! [A: int,B: int] :
% 0.34/0.51 ( ( P @ B @ A )
% 0.34/0.51 => ( P @ A @ B ) )
% 0.34/0.51 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_wlog
% 0.34/0.51 thf(fact_422_order_Ostrict__trans,axiom,
% 0.34/0.51 ! [A2: real,B3: real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ord_less_real @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_trans
% 0.34/0.51 thf(fact_423_order_Ostrict__trans,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_trans
% 0.34/0.51 thf(fact_424_order_Ostrict__trans,axiom,
% 0.34/0.51 ! [A2: num,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.51 => ( ord_less_num @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_trans
% 0.34/0.51 thf(fact_425_order_Ostrict__trans,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.51 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_trans
% 0.34/0.51 thf(fact_426_order_Ostrict__trans,axiom,
% 0.34/0.51 ! [A2: int,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.51 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_trans
% 0.34/0.51 thf(fact_427_not__less__iff__gr__or__eq,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.34/0.51 = ( ( ord_less_real @ Y3 @ X2 )
% 0.34/0.51 | ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_iff_gr_or_eq
% 0.34/0.51 thf(fact_428_not__less__iff__gr__or__eq,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.34/0.51 = ( ( ord_less_rat @ Y3 @ X2 )
% 0.34/0.51 | ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_iff_gr_or_eq
% 0.34/0.51 thf(fact_429_not__less__iff__gr__or__eq,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.34/0.51 = ( ( ord_less_num @ Y3 @ X2 )
% 0.34/0.51 | ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_iff_gr_or_eq
% 0.34/0.51 thf(fact_430_not__less__iff__gr__or__eq,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.34/0.51 = ( ( ord_less_nat @ Y3 @ X2 )
% 0.34/0.51 | ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_iff_gr_or_eq
% 0.34/0.51 thf(fact_431_not__less__iff__gr__or__eq,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.34/0.51 = ( ( ord_less_int @ Y3 @ X2 )
% 0.34/0.51 | ( X2 = Y3 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % not_less_iff_gr_or_eq
% 0.34/0.51 thf(fact_432_dual__order_Ostrict__trans,axiom,
% 0.34/0.51 ! [B3: real,A2: real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_real @ C @ B3 )
% 0.34/0.51 => ( ord_less_real @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_trans
% 0.34/0.51 thf(fact_433_dual__order_Ostrict__trans,axiom,
% 0.34/0.51 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_rat @ C @ B3 )
% 0.34/0.51 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_trans
% 0.34/0.51 thf(fact_434_dual__order_Ostrict__trans,axiom,
% 0.34/0.51 ! [B3: num,A2: num,C: num] :
% 0.34/0.51 ( ( ord_less_num @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_num @ C @ B3 )
% 0.34/0.51 => ( ord_less_num @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_trans
% 0.34/0.51 thf(fact_435_dual__order_Ostrict__trans,axiom,
% 0.34/0.51 ! [B3: nat,A2: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_nat @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_nat @ C @ B3 )
% 0.34/0.51 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_trans
% 0.34/0.51 thf(fact_436_dual__order_Ostrict__trans,axiom,
% 0.34/0.51 ! [B3: int,A2: int,C: int] :
% 0.34/0.51 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.51 => ( ( ord_less_int @ C @ B3 )
% 0.34/0.51 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_trans
% 0.34/0.51 thf(fact_437_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [A2: real,B3: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_implies_not_eq
% 0.34/0.51 thf(fact_438_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_implies_not_eq
% 0.34/0.51 thf(fact_439_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [A2: num,B3: num] :
% 0.34/0.51 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_implies_not_eq
% 0.34/0.51 thf(fact_440_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_implies_not_eq
% 0.34/0.51 thf(fact_441_order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [A2: int,B3: int] :
% 0.34/0.51 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order.strict_implies_not_eq
% 0.34/0.51 thf(fact_442_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [B3: real,A2: real] :
% 0.34/0.51 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_implies_not_eq
% 0.34/0.51 thf(fact_443_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [B3: rat,A2: rat] :
% 0.34/0.51 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_implies_not_eq
% 0.34/0.51 thf(fact_444_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [B3: num,A2: num] :
% 0.34/0.51 ( ( ord_less_num @ B3 @ A2 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_implies_not_eq
% 0.34/0.51 thf(fact_445_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [B3: nat,A2: nat] :
% 0.34/0.51 ( ( ord_less_nat @ B3 @ A2 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_implies_not_eq
% 0.34/0.51 thf(fact_446_dual__order_Ostrict__implies__not__eq,axiom,
% 0.34/0.51 ! [B3: int,A2: int] :
% 0.34/0.51 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.51 => ( A2 != B3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % dual_order.strict_implies_not_eq
% 0.34/0.51 thf(fact_447_linorder__neqE,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE
% 0.34/0.51 thf(fact_448_linorder__neqE,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE
% 0.34/0.51 thf(fact_449_linorder__neqE,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE
% 0.34/0.51 thf(fact_450_linorder__neqE,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE
% 0.34/0.51 thf(fact_451_linorder__neqE,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neqE
% 0.34/0.51 thf(fact_452_order__less__asym,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym
% 0.34/0.51 thf(fact_453_order__less__asym,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym
% 0.34/0.51 thf(fact_454_order__less__asym,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym
% 0.34/0.51 thf(fact_455_order__less__asym,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym
% 0.34/0.51 thf(fact_456_order__less__asym,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym
% 0.34/0.51 thf(fact_457_linorder__neq__iff,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 = ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neq_iff
% 0.34/0.51 thf(fact_458_linorder__neq__iff,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 = ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neq_iff
% 0.34/0.51 thf(fact_459_linorder__neq__iff,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 = ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neq_iff
% 0.34/0.51 thf(fact_460_linorder__neq__iff,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 = ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neq_iff
% 0.34/0.51 thf(fact_461_linorder__neq__iff,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( X2 != Y3 )
% 0.34/0.51 = ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 | ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_neq_iff
% 0.34/0.51 thf(fact_462_order__less__asym_H,axiom,
% 0.34/0.51 ! [A2: real,B3: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_real @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym'
% 0.34/0.51 thf(fact_463_order__less__asym_H,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym'
% 0.34/0.51 thf(fact_464_order__less__asym_H,axiom,
% 0.34/0.51 ! [A2: num,B3: num] :
% 0.34/0.51 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_num @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym'
% 0.34/0.51 thf(fact_465_order__less__asym_H,axiom,
% 0.34/0.51 ! [A2: nat,B3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym'
% 0.34/0.51 thf(fact_466_order__less__asym_H,axiom,
% 0.34/0.51 ! [A2: int,B3: int] :
% 0.34/0.51 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.51 => ~ ( ord_less_int @ B3 @ A2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_asym'
% 0.34/0.51 thf(fact_467_order__less__trans,axiom,
% 0.34/0.51 ! [X2: real,Y3: real,Z: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_real @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_trans
% 0.34/0.51 thf(fact_468_order__less__trans,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_rat @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_trans
% 0.34/0.51 thf(fact_469_order__less__trans,axiom,
% 0.34/0.51 ! [X2: num,Y3: num,Z: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_num @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_trans
% 0.34/0.51 thf(fact_470_order__less__trans,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat,Z: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_nat @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_trans
% 0.34/0.51 thf(fact_471_order__less__trans,axiom,
% 0.34/0.51 ! [X2: int,Y3: int,Z: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_int @ Y3 @ Z )
% 0.34/0.51 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_trans
% 0.34/0.51 thf(fact_472_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: real,F: real > real,B3: real,C: real] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_473_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_474_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: num,F: real > num,B3: real,C: real] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_475_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: nat,F: real > nat,B3: real,C: real] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_476_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: int,F: real > int,B3: real,C: real] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_477_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_478_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_479_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_480_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_481_ord__eq__less__subst,axiom,
% 0.34/0.51 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 0.34/0.51 ( ( A2
% 0.34/0.51 = ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_eq_less_subst
% 0.34/0.51 thf(fact_482_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_483_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_484_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > num,C: num] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_485_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_486_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > int,C: int] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_487_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_488_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_489_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_490_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_491_ord__less__eq__subst,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ( F @ B3 )
% 0.34/0.51 = C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % ord_less_eq_subst
% 0.34/0.51 thf(fact_492_order__less__irrefl,axiom,
% 0.34/0.51 ! [X2: real] :
% 0.34/0.51 ~ ( ord_less_real @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_irrefl
% 0.34/0.51 thf(fact_493_order__less__irrefl,axiom,
% 0.34/0.51 ! [X2: rat] :
% 0.34/0.51 ~ ( ord_less_rat @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_irrefl
% 0.34/0.51 thf(fact_494_order__less__irrefl,axiom,
% 0.34/0.51 ! [X2: num] :
% 0.34/0.51 ~ ( ord_less_num @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_irrefl
% 0.34/0.51 thf(fact_495_order__less__irrefl,axiom,
% 0.34/0.51 ! [X2: nat] :
% 0.34/0.51 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_irrefl
% 0.34/0.51 thf(fact_496_order__less__irrefl,axiom,
% 0.34/0.51 ! [X2: int] :
% 0.34/0.51 ~ ( ord_less_int @ X2 @ X2 ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_irrefl
% 0.34/0.51 thf(fact_497_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: real,F: real > real,B3: real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_498_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_499_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: real,F: num > real,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_500_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: real,F: nat > real,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_501_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: real,F: int > real,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.51 => ( ! [X5: int,Y4: int] :
% 0.34/0.51 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_502_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_503_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_504_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.51 => ( ! [X5: num,Y4: num] :
% 0.34/0.51 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_505_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.51 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_506_order__less__subst1,axiom,
% 0.34/0.51 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.51 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.51 => ( ! [X5: int,Y4: int] :
% 0.34/0.51 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst1
% 0.34/0.51 thf(fact_507_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > real,C: real] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_508_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_509_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > num,C: num] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_510_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_511_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: real,B3: real,F: real > int,C: int] :
% 0.34/0.51 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: real,Y4: real] :
% 0.34/0.51 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_512_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_513_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_514_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_515_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_516_order__less__subst2,axiom,
% 0.34/0.51 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 0.34/0.51 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.51 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 0.34/0.51 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.51 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.51 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_subst2
% 0.34/0.51 thf(fact_517_order__less__not__sym,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_not_sym
% 0.34/0.51 thf(fact_518_order__less__not__sym,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_not_sym
% 0.34/0.51 thf(fact_519_order__less__not__sym,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_not_sym
% 0.34/0.51 thf(fact_520_order__less__not__sym,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_not_sym
% 0.34/0.51 thf(fact_521_order__less__not__sym,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_not_sym
% 0.34/0.51 thf(fact_522_order__less__imp__triv,axiom,
% 0.34/0.51 ! [X2: real,Y3: real,P: $o] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_real @ Y3 @ X2 )
% 0.34/0.51 => P ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_triv
% 0.34/0.51 thf(fact_523_order__less__imp__triv,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat,P: $o] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_rat @ Y3 @ X2 )
% 0.34/0.51 => P ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_triv
% 0.34/0.51 thf(fact_524_order__less__imp__triv,axiom,
% 0.34/0.51 ! [X2: num,Y3: num,P: $o] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_num @ Y3 @ X2 )
% 0.34/0.51 => P ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_triv
% 0.34/0.51 thf(fact_525_order__less__imp__triv,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat,P: $o] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_nat @ Y3 @ X2 )
% 0.34/0.51 => P ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_triv
% 0.34/0.51 thf(fact_526_order__less__imp__triv,axiom,
% 0.34/0.51 ! [X2: int,Y3: int,P: $o] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( ( ord_less_int @ Y3 @ X2 )
% 0.34/0.51 => P ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_triv
% 0.34/0.51 thf(fact_527_linorder__less__linear,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 | ( X2 = Y3 )
% 0.34/0.51 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_linear
% 0.34/0.51 thf(fact_528_linorder__less__linear,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 | ( X2 = Y3 )
% 0.34/0.51 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_linear
% 0.34/0.51 thf(fact_529_linorder__less__linear,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 | ( X2 = Y3 )
% 0.34/0.51 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_linear
% 0.34/0.51 thf(fact_530_linorder__less__linear,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 | ( X2 = Y3 )
% 0.34/0.51 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_linear
% 0.34/0.51 thf(fact_531_linorder__less__linear,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 | ( X2 = Y3 )
% 0.34/0.51 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % linorder_less_linear
% 0.34/0.51 thf(fact_532_order__less__imp__not__eq,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq
% 0.34/0.51 thf(fact_533_order__less__imp__not__eq,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq
% 0.34/0.51 thf(fact_534_order__less__imp__not__eq,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq
% 0.34/0.51 thf(fact_535_order__less__imp__not__eq,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq
% 0.34/0.51 thf(fact_536_order__less__imp__not__eq,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( X2 != Y3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq
% 0.34/0.51 thf(fact_537_order__less__imp__not__eq2,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ( Y3 != X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq2
% 0.34/0.51 thf(fact_538_order__less__imp__not__eq2,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ( Y3 != X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq2
% 0.34/0.51 thf(fact_539_order__less__imp__not__eq2,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ( Y3 != X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq2
% 0.34/0.51 thf(fact_540_order__less__imp__not__eq2,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ( Y3 != X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq2
% 0.34/0.51 thf(fact_541_order__less__imp__not__eq2,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ( Y3 != X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_eq2
% 0.34/0.51 thf(fact_542_order__less__imp__not__less,axiom,
% 0.34/0.51 ! [X2: real,Y3: real] :
% 0.34/0.51 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_less
% 0.34/0.51 thf(fact_543_order__less__imp__not__less,axiom,
% 0.34/0.51 ! [X2: rat,Y3: rat] :
% 0.34/0.51 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_less
% 0.34/0.51 thf(fact_544_order__less__imp__not__less,axiom,
% 0.34/0.51 ! [X2: num,Y3: num] :
% 0.34/0.51 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_less
% 0.34/0.51 thf(fact_545_order__less__imp__not__less,axiom,
% 0.34/0.51 ! [X2: nat,Y3: nat] :
% 0.34/0.51 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_less
% 0.34/0.51 thf(fact_546_order__less__imp__not__less,axiom,
% 0.34/0.51 ! [X2: int,Y3: int] :
% 0.34/0.51 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.51 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.51
% 0.34/0.51 % order_less_imp_not_less
% 0.34/0.51 thf(fact_547_ex__in__conv,axiom,
% 0.34/0.51 ! [A3: set_set_nat] :
% 0.34/0.51 ( ( ? [X: set_nat] : ( member_set_nat @ X @ A3 ) )
% 0.34/0.51 = ( A3 != bot_bot_set_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_548_ex__in__conv,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ( ? [X: real] : ( member_real @ X @ A3 ) )
% 0.34/0.51 = ( A3 != bot_bot_set_real ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_549_ex__in__conv,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ( ? [X: $o] : ( member_o @ X @ A3 ) )
% 0.34/0.51 = ( A3 != bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_550_ex__in__conv,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ( ? [X: nat] : ( member_nat @ X @ A3 ) )
% 0.34/0.51 = ( A3 != bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_551_ex__in__conv,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ( ? [X: int] : ( member_int @ X @ A3 ) )
% 0.34/0.51 = ( A3 != bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % ex_in_conv
% 0.34/0.51 thf(fact_552_equals0I,axiom,
% 0.34/0.51 ! [A3: set_set_nat] :
% 0.34/0.51 ( ! [Y4: set_nat] :
% 0.34/0.51 ~ ( member_set_nat @ Y4 @ A3 )
% 0.34/0.51 => ( A3 = bot_bot_set_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_553_equals0I,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ( ! [Y4: real] :
% 0.34/0.51 ~ ( member_real @ Y4 @ A3 )
% 0.34/0.51 => ( A3 = bot_bot_set_real ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_554_equals0I,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ( ! [Y4: $o] :
% 0.34/0.51 ~ ( member_o @ Y4 @ A3 )
% 0.34/0.51 => ( A3 = bot_bot_set_o ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_555_equals0I,axiom,
% 0.34/0.51 ! [A3: set_nat] :
% 0.34/0.51 ( ! [Y4: nat] :
% 0.34/0.51 ~ ( member_nat @ Y4 @ A3 )
% 0.34/0.51 => ( A3 = bot_bot_set_nat ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_556_equals0I,axiom,
% 0.34/0.51 ! [A3: set_int] :
% 0.34/0.51 ( ! [Y4: int] :
% 0.34/0.51 ~ ( member_int @ Y4 @ A3 )
% 0.34/0.51 => ( A3 = bot_bot_set_int ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0I
% 0.34/0.51 thf(fact_557_equals0D,axiom,
% 0.34/0.51 ! [A3: set_set_nat,A2: set_nat] :
% 0.34/0.51 ( ( A3 = bot_bot_set_set_nat )
% 0.34/0.51 => ~ ( member_set_nat @ A2 @ A3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_558_equals0D,axiom,
% 0.34/0.51 ! [A3: set_real,A2: real] :
% 0.34/0.51 ( ( A3 = bot_bot_set_real )
% 0.34/0.51 => ~ ( member_real @ A2 @ A3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_559_equals0D,axiom,
% 0.34/0.51 ! [A3: set_o,A2: $o] :
% 0.34/0.51 ( ( A3 = bot_bot_set_o )
% 0.34/0.51 => ~ ( member_o @ A2 @ A3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_560_equals0D,axiom,
% 0.34/0.51 ! [A3: set_nat,A2: nat] :
% 0.34/0.51 ( ( A3 = bot_bot_set_nat )
% 0.34/0.51 => ~ ( member_nat @ A2 @ A3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_561_equals0D,axiom,
% 0.34/0.51 ! [A3: set_int,A2: int] :
% 0.34/0.51 ( ( A3 = bot_bot_set_int )
% 0.34/0.51 => ~ ( member_int @ A2 @ A3 ) ) ).
% 0.34/0.51
% 0.34/0.51 % equals0D
% 0.34/0.51 thf(fact_562_emptyE,axiom,
% 0.34/0.51 ! [A2: set_nat] :
% 0.34/0.51 ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_563_emptyE,axiom,
% 0.34/0.51 ! [A2: real] :
% 0.34/0.51 ~ ( member_real @ A2 @ bot_bot_set_real ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_564_emptyE,axiom,
% 0.34/0.51 ! [A2: $o] :
% 0.34/0.51 ~ ( member_o @ A2 @ bot_bot_set_o ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_565_emptyE,axiom,
% 0.34/0.51 ! [A2: nat] :
% 0.34/0.51 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_566_emptyE,axiom,
% 0.34/0.51 ! [A2: int] :
% 0.34/0.51 ~ ( member_int @ A2 @ bot_bot_set_int ) ).
% 0.34/0.51
% 0.34/0.51 % emptyE
% 0.34/0.51 thf(fact_567_not__psubset__empty,axiom,
% 0.34/0.51 ! [A3: set_real] :
% 0.34/0.51 ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).
% 0.34/0.51
% 0.34/0.51 % not_psubset_empty
% 0.34/0.51 thf(fact_568_not__psubset__empty,axiom,
% 0.34/0.51 ! [A3: set_o] :
% 0.34/0.51 ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).
% 0.34/0.51
% 0.34/0.51 % not_psubset_empty
% 0.34/0.51 thf(fact_569_not__psubset__empty,axiom,
% 0.34/0.52 ! [A3: set_nat] :
% 0.34/0.52 ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % not_psubset_empty
% 0.34/0.52 thf(fact_570_not__psubset__empty,axiom,
% 0.34/0.52 ! [A3: set_int] :
% 0.34/0.52 ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % not_psubset_empty
% 0.34/0.52 thf(fact_571_finite__psubset__induct,axiom,
% 0.34/0.52 ! [A3: set_nat,P: set_nat > $o] :
% 0.34/0.52 ( ( finite_finite_nat @ A3 )
% 0.34/0.52 => ( ! [A5: set_nat] :
% 0.34/0.52 ( ( finite_finite_nat @ A5 )
% 0.34/0.52 => ( ! [B5: set_nat] :
% 0.34/0.52 ( ( ord_less_set_nat @ B5 @ A5 )
% 0.34/0.52 => ( P @ B5 ) )
% 0.34/0.52 => ( P @ A5 ) ) )
% 0.34/0.52 => ( P @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_psubset_induct
% 0.34/0.52 thf(fact_572_finite__psubset__induct,axiom,
% 0.34/0.52 ! [A3: set_int,P: set_int > $o] :
% 0.34/0.52 ( ( finite_finite_int @ A3 )
% 0.34/0.52 => ( ! [A5: set_int] :
% 0.34/0.52 ( ( finite_finite_int @ A5 )
% 0.34/0.52 => ( ! [B5: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ B5 @ A5 )
% 0.34/0.52 => ( P @ B5 ) )
% 0.34/0.52 => ( P @ A5 ) ) )
% 0.34/0.52 => ( P @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_psubset_induct
% 0.34/0.52 thf(fact_573_finite__psubset__induct,axiom,
% 0.34/0.52 ! [A3: set_complex,P: set_complex > $o] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.52 => ( ! [A5: set_complex] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ A5 )
% 0.34/0.52 => ( ! [B5: set_complex] :
% 0.34/0.52 ( ( ord_less_set_complex @ B5 @ A5 )
% 0.34/0.52 => ( P @ B5 ) )
% 0.34/0.52 => ( P @ A5 ) ) )
% 0.34/0.52 => ( P @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_psubset_induct
% 0.34/0.52 thf(fact_574_finite__psubset__induct,axiom,
% 0.34/0.52 ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.52 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.52 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( finite6177210948735845034at_nat @ A5 )
% 0.34/0.52 => ( ! [B5: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ( ord_le7866589430770878221at_nat @ B5 @ A5 )
% 0.34/0.52 => ( P @ B5 ) )
% 0.34/0.52 => ( P @ A5 ) ) )
% 0.34/0.52 => ( P @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_psubset_induct
% 0.34/0.52 thf(fact_575_finite__psubset__induct,axiom,
% 0.34/0.52 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.52 => ( ! [A5: set_Extended_enat] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.52 => ( ! [B5: set_Extended_enat] :
% 0.34/0.52 ( ( ord_le2529575680413868914d_enat @ B5 @ A5 )
% 0.34/0.52 => ( P @ B5 ) )
% 0.34/0.52 => ( P @ A5 ) ) )
% 0.34/0.52 => ( P @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_psubset_induct
% 0.34/0.52 thf(fact_576_leD,axiom,
% 0.34/0.52 ! [Y3: real,X2: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_577_leD,axiom,
% 0.34/0.52 ! [Y3: set_int,X2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_set_int @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_578_leD,axiom,
% 0.34/0.52 ! [Y3: rat,X2: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_579_leD,axiom,
% 0.34/0.52 ! [Y3: num,X2: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_num @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_580_leD,axiom,
% 0.34/0.52 ! [Y3: nat,X2: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_581_leD,axiom,
% 0.34/0.52 ! [Y3: int,X2: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.34/0.52 => ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leD
% 0.34/0.52 thf(fact_582_leI,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leI
% 0.34/0.52 thf(fact_583_leI,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leI
% 0.34/0.52 thf(fact_584_leI,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leI
% 0.34/0.52 thf(fact_585_leI,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leI
% 0.34/0.52 thf(fact_586_leI,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % leI
% 0.34/0.52 thf(fact_587_nless__le,axiom,
% 0.34/0.52 ! [A2: real,B3: real] :
% 0.34/0.52 ( ( ~ ( ord_less_real @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_588_nless__le,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int] :
% 0.34/0.52 ( ( ~ ( ord_less_set_int @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_589_nless__le,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat] :
% 0.34/0.52 ( ( ~ ( ord_less_rat @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_590_nless__le,axiom,
% 0.34/0.52 ! [A2: num,B3: num] :
% 0.34/0.52 ( ( ~ ( ord_less_num @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_591_nless__le,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat] :
% 0.34/0.52 ( ( ~ ( ord_less_nat @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_592_nless__le,axiom,
% 0.34/0.52 ! [A2: int,B3: int] :
% 0.34/0.52 ( ( ~ ( ord_less_int @ A2 @ B3 ) )
% 0.34/0.52 = ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.52 | ( A2 = B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nless_le
% 0.34/0.52 thf(fact_593_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_594_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int] :
% 0.34/0.52 ( ~ ( ord_less_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_595_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_596_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ~ ( ord_less_num @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_597_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_598_antisym__conv1,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv1
% 0.34/0.52 thf(fact_599_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_600_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_set_int @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_601_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_602_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_603_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_604_antisym__conv2,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.34/0.52 = ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % antisym_conv2
% 0.34/0.52 thf(fact_605_dense__ge,axiom,
% 0.34/0.52 ! [Z: real,Y3: real] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ( ( ord_less_real @ Z @ X5 )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_ge
% 0.34/0.52 thf(fact_606_dense__ge,axiom,
% 0.34/0.52 ! [Z: rat,Y3: rat] :
% 0.34/0.52 ( ! [X5: rat] :
% 0.34/0.52 ( ( ord_less_rat @ Z @ X5 )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_ge
% 0.34/0.52 thf(fact_607_dense__le,axiom,
% 0.34/0.52 ! [Y3: real,Z: real] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ( ( ord_less_real @ X5 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_real @ X5 @ Z ) )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_le
% 0.34/0.52 thf(fact_608_dense__le,axiom,
% 0.34/0.52 ! [Y3: rat,Z: rat] :
% 0.34/0.52 ( ! [X5: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X5 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_rat @ X5 @ Z ) )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_le
% 0.34/0.52 thf(fact_609_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [X: real,Y: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_610_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [X: set_int,Y: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_611_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [X: rat,Y: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_612_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [X: num,Y: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_613_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [X: nat,Y: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_614_less__le__not__le,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [X: int,Y: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.52 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % less_le_not_le
% 0.34/0.52 thf(fact_615_not__le__imp__less,axiom,
% 0.34/0.52 ! [Y3: real,X2: real] :
% 0.34/0.52 ( ~ ( ord_less_eq_real @ Y3 @ X2 )
% 0.34/0.52 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % not_le_imp_less
% 0.34/0.52 thf(fact_616_not__le__imp__less,axiom,
% 0.34/0.52 ! [Y3: rat,X2: rat] :
% 0.34/0.52 ( ~ ( ord_less_eq_rat @ Y3 @ X2 )
% 0.34/0.52 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % not_le_imp_less
% 0.34/0.52 thf(fact_617_not__le__imp__less,axiom,
% 0.34/0.52 ! [Y3: num,X2: num] :
% 0.34/0.52 ( ~ ( ord_less_eq_num @ Y3 @ X2 )
% 0.34/0.52 => ( ord_less_num @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % not_le_imp_less
% 0.34/0.52 thf(fact_618_not__le__imp__less,axiom,
% 0.34/0.52 ! [Y3: nat,X2: nat] :
% 0.34/0.52 ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.52 => ( ord_less_nat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % not_le_imp_less
% 0.34/0.52 thf(fact_619_not__le__imp__less,axiom,
% 0.34/0.52 ! [Y3: int,X2: int] :
% 0.34/0.52 ( ~ ( ord_less_eq_int @ Y3 @ X2 )
% 0.34/0.52 => ( ord_less_int @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % not_le_imp_less
% 0.34/0.52 thf(fact_620_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_real
% 0.34/0.52 = ( ^ [A4: real,B4: real] :
% 0.34/0.52 ( ( ord_less_real @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_621_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [A4: set_int,B4: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_622_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_rat
% 0.34/0.52 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_623_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_num
% 0.34/0.52 = ( ^ [A4: num,B4: num] :
% 0.34/0.52 ( ( ord_less_num @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_624_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_nat
% 0.34/0.52 = ( ^ [A4: nat,B4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_625_order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_int
% 0.34/0.52 = ( ^ [A4: int,B4: int] :
% 0.34/0.52 ( ( ord_less_int @ A4 @ B4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.order_iff_strict
% 0.34/0.52 thf(fact_626_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [A4: real,B4: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_627_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [A4: set_int,B4: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_628_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_629_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [A4: num,B4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_630_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [A4: nat,B4: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_631_order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [A4: int,B4: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ A4 @ B4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_order
% 0.34/0.52 thf(fact_632_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: real,B3: real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.52 => ( ord_less_real @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_633_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_set_int @ B3 @ C )
% 0.34/0.52 => ( ord_less_set_int @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_634_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.52 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_635_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: num,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.52 => ( ord_less_num @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_636_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.52 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_637_order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [A2: int,B3: int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.52 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans1
% 0.34/0.52 thf(fact_638_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: real,B3: real,C: real] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ B3 @ C )
% 0.34/0.52 => ( ord_less_real @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_639_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ B3 @ C )
% 0.34/0.52 => ( ord_less_set_int @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_640_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_641_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ord_less_num @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_642_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.52 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_643_order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [A2: int,B3: int,C: int] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ B3 @ C )
% 0.34/0.52 => ( ord_less_int @ A2 @ C ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_trans2
% 0.34/0.52 thf(fact_644_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [A4: real,B4: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_645_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [A4: set_int,B4: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_646_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_647_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [A4: num,B4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_648_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [A4: nat,B4: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_649_order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [A4: int,B4: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ A4 @ B4 )
% 0.34/0.52 & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_iff_not
% 0.34/0.52 thf(fact_650_dense__ge__bounded,axiom,
% 0.34/0.52 ! [Z: real,X2: real,Y3: real] :
% 0.34/0.52 ( ( ord_less_real @ Z @ X2 )
% 0.34/0.52 => ( ! [W: real] :
% 0.34/0.52 ( ( ord_less_real @ Z @ W )
% 0.34/0.52 => ( ( ord_less_real @ W @ X2 )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ W ) ) )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_ge_bounded
% 0.34/0.52 thf(fact_651_dense__ge__bounded,axiom,
% 0.34/0.52 ! [Z: rat,X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ord_less_rat @ Z @ X2 )
% 0.34/0.52 => ( ! [W: rat] :
% 0.34/0.52 ( ( ord_less_rat @ Z @ W )
% 0.34/0.52 => ( ( ord_less_rat @ W @ X2 )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ W ) ) )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_ge_bounded
% 0.34/0.52 thf(fact_652_dense__le__bounded,axiom,
% 0.34/0.52 ! [X2: real,Y3: real,Z: real] :
% 0.34/0.52 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 => ( ! [W: real] :
% 0.34/0.52 ( ( ord_less_real @ X2 @ W )
% 0.34/0.52 => ( ( ord_less_real @ W @ Y3 )
% 0.34/0.52 => ( ord_less_eq_real @ W @ Z ) ) )
% 0.34/0.52 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_le_bounded
% 0.34/0.52 thf(fact_653_dense__le__bounded,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ! [W: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X2 @ W )
% 0.34/0.52 => ( ( ord_less_rat @ W @ Y3 )
% 0.34/0.52 => ( ord_less_eq_rat @ W @ Z ) ) )
% 0.34/0.52 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dense_le_bounded
% 0.34/0.52 thf(fact_654_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_real
% 0.34/0.52 = ( ^ [B4: real,A4: real] :
% 0.34/0.52 ( ( ord_less_real @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_655_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [B4: set_int,A4: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_656_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_rat
% 0.34/0.52 = ( ^ [B4: rat,A4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_657_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_num
% 0.34/0.52 = ( ^ [B4: num,A4: num] :
% 0.34/0.52 ( ( ord_less_num @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_658_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_nat
% 0.34/0.52 = ( ^ [B4: nat,A4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_659_dual__order_Oorder__iff__strict,axiom,
% 0.34/0.52 ( ord_less_eq_int
% 0.34/0.52 = ( ^ [B4: int,A4: int] :
% 0.34/0.52 ( ( ord_less_int @ B4 @ A4 )
% 0.34/0.52 | ( A4 = B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.order_iff_strict
% 0.34/0.52 thf(fact_660_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [B4: real,A4: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_661_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [B4: set_int,A4: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_662_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [B4: rat,A4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_663_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [B4: num,A4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_664_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [B4: nat,A4: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_665_dual__order_Ostrict__iff__order,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [B4: int,A4: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ B4 @ A4 )
% 0.34/0.52 & ( A4 != B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_order
% 0.34/0.52 thf(fact_666_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: real,A2: real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_real @ C @ B3 )
% 0.34/0.52 => ( ord_less_real @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_667_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: set_int,A2: set_int,C: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_set_int @ C @ B3 )
% 0.34/0.52 => ( ord_less_set_int @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_668_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_rat @ C @ B3 )
% 0.34/0.52 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_669_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: num,A2: num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_num @ C @ B3 )
% 0.34/0.52 => ( ord_less_num @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_670_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: nat,A2: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_nat @ C @ B3 )
% 0.34/0.52 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_671_dual__order_Ostrict__trans1,axiom,
% 0.34/0.52 ! [B3: int,A2: int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_int @ C @ B3 )
% 0.34/0.52 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans1
% 0.34/0.52 thf(fact_672_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: real,A2: real,C: real] :
% 0.34/0.52 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_real @ C @ B3 )
% 0.34/0.52 => ( ord_less_real @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_673_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: set_int,A2: set_int,C: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ C @ B3 )
% 0.34/0.52 => ( ord_less_set_int @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_674_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ C @ B3 )
% 0.34/0.52 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_675_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: num,A2: num,C: num] :
% 0.34/0.52 ( ( ord_less_num @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_num @ C @ B3 )
% 0.34/0.52 => ( ord_less_num @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_676_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: nat,A2: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_nat @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.34/0.52 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_677_dual__order_Ostrict__trans2,axiom,
% 0.34/0.52 ! [B3: int,A2: int,C: int] :
% 0.34/0.52 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.52 => ( ( ord_less_eq_int @ C @ B3 )
% 0.34/0.52 => ( ord_less_int @ C @ A2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_trans2
% 0.34/0.52 thf(fact_678_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [B4: real,A4: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_679_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [B4: set_int,A4: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_680_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [B4: rat,A4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_681_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [B4: num,A4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_682_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [B4: nat,A4: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_683_dual__order_Ostrict__iff__not,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [B4: int,A4: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ B4 @ A4 )
% 0.34/0.52 & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_iff_not
% 0.34/0.52 thf(fact_684_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: real,B3: real] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_685_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_686_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_687_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: num,B3: num] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_num @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_688_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_689_order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [A2: int,B3: int] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order.strict_implies_order
% 0.34/0.52 thf(fact_690_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: real,A2: real] :
% 0.34/0.52 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_real @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_691_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: set_int,A2: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_692_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: rat,A2: rat] :
% 0.34/0.52 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_693_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: num,A2: num] :
% 0.34/0.52 ( ( ord_less_num @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_694_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: nat,A2: nat] :
% 0.34/0.52 ( ( ord_less_nat @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_695_dual__order_Ostrict__implies__order,axiom,
% 0.34/0.52 ! [B3: int,A2: int] :
% 0.34/0.52 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.52 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % dual_order.strict_implies_order
% 0.34/0.52 thf(fact_696_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_real
% 0.34/0.52 = ( ^ [X: real,Y: real] :
% 0.34/0.52 ( ( ord_less_real @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_697_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [X: set_int,Y: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_698_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_rat
% 0.34/0.52 = ( ^ [X: rat,Y: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_699_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_num
% 0.34/0.52 = ( ^ [X: num,Y: num] :
% 0.34/0.52 ( ( ord_less_num @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_700_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_nat
% 0.34/0.52 = ( ^ [X: nat,Y: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_701_order__le__less,axiom,
% 0.34/0.52 ( ord_less_eq_int
% 0.34/0.52 = ( ^ [X: int,Y: int] :
% 0.34/0.52 ( ( ord_less_int @ X @ Y )
% 0.34/0.52 | ( X = Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less
% 0.34/0.52 thf(fact_702_order__less__le,axiom,
% 0.34/0.52 ( ord_less_real
% 0.34/0.52 = ( ^ [X: real,Y: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_703_order__less__le,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [X: set_int,Y: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_704_order__less__le,axiom,
% 0.34/0.52 ( ord_less_rat
% 0.34/0.52 = ( ^ [X: rat,Y: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_705_order__less__le,axiom,
% 0.34/0.52 ( ord_less_num
% 0.34/0.52 = ( ^ [X: num,Y: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_706_order__less__le,axiom,
% 0.34/0.52 ( ord_less_nat
% 0.34/0.52 = ( ^ [X: nat,Y: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_707_order__less__le,axiom,
% 0.34/0.52 ( ord_less_int
% 0.34/0.52 = ( ^ [X: int,Y: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X @ Y )
% 0.34/0.52 & ( X != Y ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le
% 0.34/0.52 thf(fact_708_linorder__not__le,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_le
% 0.34/0.52 thf(fact_709_linorder__not__le,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_rat @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_le
% 0.34/0.52 thf(fact_710_linorder__not__le,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_num @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_le
% 0.34/0.52 thf(fact_711_linorder__not__le,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_le
% 0.34/0.52 thf(fact_712_linorder__not__le,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_le
% 0.34/0.52 thf(fact_713_linorder__not__less,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_less
% 0.34/0.52 thf(fact_714_linorder__not__less,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_less
% 0.34/0.52 thf(fact_715_linorder__not__less,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_less
% 0.34/0.52 thf(fact_716_linorder__not__less,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_less
% 0.34/0.52 thf(fact_717_linorder__not__less,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 0.34/0.52 = ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_not_less
% 0.34/0.52 thf(fact_718_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_719_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_720_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_721_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_722_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_723_order__less__imp__le,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.52 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_imp_le
% 0.34/0.52 thf(fact_724_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: real,B3: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_725_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_726_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_727_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: num,B3: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_num @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_728_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_729_order__le__neq__trans,axiom,
% 0.34/0.52 ! [A2: int,B3: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.52 => ( ( A2 != B3 )
% 0.34/0.52 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_neq_trans
% 0.34/0.52 thf(fact_730_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: real,B3: real] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_731_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: set_int,B3: set_int] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_732_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_733_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: num,B3: num] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_num @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_734_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_735_order__neq__le__trans,axiom,
% 0.34/0.52 ! [A2: int,B3: int] :
% 0.34/0.52 ( ( A2 != B3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.52 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_neq_le_trans
% 0.34/0.52 thf(fact_736_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: real,Y3: real,Z: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_real @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_737_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_set_int @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_738_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_rat @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_739_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: num,Y3: num,Z: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_num @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_740_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat,Z: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_nat @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_741_order__le__less__trans,axiom,
% 0.34/0.52 ! [X2: int,Y3: int,Z: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_int @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_trans
% 0.34/0.52 thf(fact_742_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: real,Y3: real,Z: real] :
% 0.34/0.52 ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_743_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_744_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat,Z: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_745_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: num,Y3: num,Z: num] :
% 0.34/0.52 ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_746_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat,Z: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_747_order__less__le__trans,axiom,
% 0.34/0.52 ! [X2: int,Y3: int,Z: int] :
% 0.34/0.52 ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.34/0.52 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_trans
% 0.34/0.52 thf(fact_748_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: real > real,B3: real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.52 => ( ! [X5: real,Y4: real] :
% 0.34/0.52 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_749_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_750_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: num > real,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_751_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: nat > real,B3: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_752_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: int > real,B3: int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.52 => ( ! [X5: int,Y4: int] :
% 0.34/0.52 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_753_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_real @ B3 @ C )
% 0.34/0.52 => ( ! [X5: real,Y4: real] :
% 0.34/0.52 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_754_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_755_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_756_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_nat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_757_order__le__less__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_int @ B3 @ C )
% 0.34/0.52 => ( ! [X5: int,Y4: int] :
% 0.34/0.52 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst1
% 0.34/0.52 thf(fact_758_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_759_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_760_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_761_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_762_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_763_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_764_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_765_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > num,C: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_766_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_767_order__le__less__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_less_subst2
% 0.34/0.52 thf(fact_768_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_769_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_770_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_771_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_772_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_773_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: real,F: num > real,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_774_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_775_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: num,F: num > num,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_776_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_777_order__less__le__subst1,axiom,
% 0.34/0.52 ! [A2: int,F: num > int,B3: num,C: num] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ ( F @ B3 ) )
% 0.34/0.52 => ( ( ord_less_eq_num @ B3 @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst1
% 0.34/0.52 thf(fact_778_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: real,B3: real,F: real > real,C: real] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: real,Y4: real] :
% 0.34/0.52 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_779_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_780_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > real,C: real] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_781_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat,F: nat > real,C: real] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_782_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: int,B3: int,F: int > real,C: real] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: int,Y4: int] :
% 0.34/0.52 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_783_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: real,Y4: real] :
% 0.34/0.52 ( ( ord_less_real @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_784_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: rat,Y4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_785_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_num @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: num,Y4: num] :
% 0.34/0.52 ( ( ord_less_num @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_786_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_787_order__less__le__subst2,axiom,
% 0.34/0.52 ! [A2: int,B3: int,F: int > rat,C: rat] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 0.34/0.52 => ( ! [X5: int,Y4: int] :
% 0.34/0.52 ( ( ord_less_int @ X5 @ Y4 )
% 0.34/0.52 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 0.34/0.52 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_less_le_subst2
% 0.34/0.52 thf(fact_788_linorder__le__less__linear,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.52 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_le_less_linear
% 0.34/0.52 thf(fact_789_linorder__le__less__linear,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.52 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_le_less_linear
% 0.34/0.52 thf(fact_790_linorder__le__less__linear,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.52 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_le_less_linear
% 0.34/0.52 thf(fact_791_linorder__le__less__linear,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.52 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_le_less_linear
% 0.34/0.52 thf(fact_792_linorder__le__less__linear,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.52 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % linorder_le_less_linear
% 0.34/0.52 thf(fact_793_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: real,Y3: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_794_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: set_int,Y3: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_set_int @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_795_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: rat,Y3: rat] :
% 0.34/0.52 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_796_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: num,Y3: num] :
% 0.34/0.52 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_num @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_797_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: nat,Y3: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_798_order__le__imp__less__or__eq,axiom,
% 0.34/0.52 ! [X2: int,Y3: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.34/0.52 => ( ( ord_less_int @ X2 @ Y3 )
% 0.34/0.52 | ( X2 = Y3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % order_le_imp_less_or_eq
% 0.34/0.52 thf(fact_799_bot_Oextremum__uniqueI,axiom,
% 0.34/0.52 ! [A2: set_real] :
% 0.34/0.52 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.34/0.52 => ( A2 = bot_bot_set_real ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_uniqueI
% 0.34/0.52 thf(fact_800_bot_Oextremum__uniqueI,axiom,
% 0.34/0.52 ! [A2: set_o] :
% 0.34/0.52 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 0.34/0.52 => ( A2 = bot_bot_set_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_uniqueI
% 0.34/0.52 thf(fact_801_bot_Oextremum__uniqueI,axiom,
% 0.34/0.52 ! [A2: set_nat] :
% 0.34/0.52 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.52 => ( A2 = bot_bot_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_uniqueI
% 0.34/0.52 thf(fact_802_bot_Oextremum__uniqueI,axiom,
% 0.34/0.52 ! [A2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.34/0.52 => ( A2 = bot_bot_set_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_uniqueI
% 0.34/0.52 thf(fact_803_bot_Oextremum__uniqueI,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.34/0.52 => ( A2 = bot_bot_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_uniqueI
% 0.34/0.52 thf(fact_804_bot_Oextremum__unique,axiom,
% 0.34/0.52 ! [A2: set_real] :
% 0.34/0.52 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.34/0.52 = ( A2 = bot_bot_set_real ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_unique
% 0.34/0.52 thf(fact_805_bot_Oextremum__unique,axiom,
% 0.34/0.52 ! [A2: set_o] :
% 0.34/0.52 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 0.34/0.52 = ( A2 = bot_bot_set_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_unique
% 0.34/0.52 thf(fact_806_bot_Oextremum__unique,axiom,
% 0.34/0.52 ! [A2: set_nat] :
% 0.34/0.52 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.52 = ( A2 = bot_bot_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_unique
% 0.34/0.52 thf(fact_807_bot_Oextremum__unique,axiom,
% 0.34/0.52 ! [A2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.34/0.52 = ( A2 = bot_bot_set_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_unique
% 0.34/0.52 thf(fact_808_bot_Oextremum__unique,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 0.34/0.52 = ( A2 = bot_bot_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_unique
% 0.34/0.52 thf(fact_809_bot_Oextremum,axiom,
% 0.34/0.52 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum
% 0.34/0.52 thf(fact_810_bot_Oextremum,axiom,
% 0.34/0.52 ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum
% 0.34/0.52 thf(fact_811_bot_Oextremum,axiom,
% 0.34/0.52 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum
% 0.34/0.52 thf(fact_812_bot_Oextremum,axiom,
% 0.34/0.52 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum
% 0.34/0.52 thf(fact_813_bot_Oextremum,axiom,
% 0.34/0.52 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum
% 0.34/0.52 thf(fact_814_bot_Oextremum__strict,axiom,
% 0.34/0.52 ! [A2: set_real] :
% 0.34/0.52 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_strict
% 0.34/0.52 thf(fact_815_bot_Oextremum__strict,axiom,
% 0.34/0.52 ! [A2: set_o] :
% 0.34/0.52 ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_strict
% 0.34/0.52 thf(fact_816_bot_Oextremum__strict,axiom,
% 0.34/0.52 ! [A2: set_nat] :
% 0.34/0.52 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_strict
% 0.34/0.52 thf(fact_817_bot_Oextremum__strict,axiom,
% 0.34/0.52 ! [A2: set_int] :
% 0.34/0.52 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_strict
% 0.34/0.52 thf(fact_818_bot_Oextremum__strict,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% 0.34/0.52
% 0.34/0.52 % bot.extremum_strict
% 0.34/0.52 thf(fact_819_bot_Onot__eq__extremum,axiom,
% 0.34/0.52 ! [A2: set_real] :
% 0.34/0.52 ( ( A2 != bot_bot_set_real )
% 0.34/0.52 = ( ord_less_set_real @ bot_bot_set_real @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.not_eq_extremum
% 0.34/0.52 thf(fact_820_bot_Onot__eq__extremum,axiom,
% 0.34/0.52 ! [A2: set_o] :
% 0.34/0.52 ( ( A2 != bot_bot_set_o )
% 0.34/0.52 = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.not_eq_extremum
% 0.34/0.52 thf(fact_821_bot_Onot__eq__extremum,axiom,
% 0.34/0.52 ! [A2: set_nat] :
% 0.34/0.52 ( ( A2 != bot_bot_set_nat )
% 0.34/0.52 = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.not_eq_extremum
% 0.34/0.52 thf(fact_822_bot_Onot__eq__extremum,axiom,
% 0.34/0.52 ! [A2: set_int] :
% 0.34/0.52 ( ( A2 != bot_bot_set_int )
% 0.34/0.52 = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.not_eq_extremum
% 0.34/0.52 thf(fact_823_bot_Onot__eq__extremum,axiom,
% 0.34/0.52 ! [A2: nat] :
% 0.34/0.52 ( ( A2 != bot_bot_nat )
% 0.34/0.52 = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot.not_eq_extremum
% 0.34/0.52 thf(fact_824_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_real,A2: real] :
% 0.34/0.52 ( ( finite_finite_real @ A3 )
% 0.34/0.52 => ( ( member_real @ A2 @ A3 )
% 0.34/0.52 => ? [X5: real] :
% 0.34/0.52 ( ( member_real @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_real @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: real] :
% 0.34/0.52 ( ( member_real @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_825_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_o,A2: $o] :
% 0.34/0.52 ( ( finite_finite_o @ A3 )
% 0.34/0.52 => ( ( member_o @ A2 @ A3 )
% 0.34/0.52 => ? [X5: $o] :
% 0.34/0.52 ( ( member_o @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_o @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: $o] :
% 0.34/0.52 ( ( member_o @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_o @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_826_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_set_nat,A2: set_nat] :
% 0.34/0.52 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.52 => ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_set_nat @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_set_nat @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_827_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_Extended_enat,A2: extended_enat] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.52 => ( ( member_Extended_enat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ X5 @ A3 )
% 0.34/0.52 & ( ord_le2932123472753598470d_enat @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_828_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_set_int,A2: set_int] :
% 0.34/0.52 ( ( finite6197958912794628473et_int @ A3 )
% 0.34/0.52 => ( ( member_set_int @ A2 @ A3 )
% 0.34/0.52 => ? [X5: set_int] :
% 0.34/0.52 ( ( member_set_int @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_set_int @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: set_int] :
% 0.34/0.52 ( ( member_set_int @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_829_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_rat,A2: rat] :
% 0.34/0.52 ( ( finite_finite_rat @ A3 )
% 0.34/0.52 => ( ( member_rat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: rat] :
% 0.34/0.52 ( ( member_rat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_rat @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: rat] :
% 0.34/0.52 ( ( member_rat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_830_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_num,A2: num] :
% 0.34/0.52 ( ( finite_finite_num @ A3 )
% 0.34/0.52 => ( ( member_num @ A2 @ A3 )
% 0.34/0.52 => ? [X5: num] :
% 0.34/0.52 ( ( member_num @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_num @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: num] :
% 0.34/0.52 ( ( member_num @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_831_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_nat,A2: nat] :
% 0.34/0.52 ( ( finite_finite_nat @ A3 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: nat] :
% 0.34/0.52 ( ( member_nat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_nat @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: nat] :
% 0.34/0.52 ( ( member_nat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_832_finite__has__minimal2,axiom,
% 0.34/0.52 ! [A3: set_int,A2: int] :
% 0.34/0.52 ( ( finite_finite_int @ A3 )
% 0.34/0.52 => ( ( member_int @ A2 @ A3 )
% 0.34/0.52 => ? [X5: int] :
% 0.34/0.52 ( ( member_int @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_int @ X5 @ A2 )
% 0.34/0.52 & ! [Xa: int] :
% 0.34/0.52 ( ( member_int @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ Xa @ X5 )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_minimal2
% 0.34/0.52 thf(fact_833_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_real,A2: real] :
% 0.34/0.52 ( ( finite_finite_real @ A3 )
% 0.34/0.52 => ( ( member_real @ A2 @ A3 )
% 0.34/0.52 => ? [X5: real] :
% 0.34/0.52 ( ( member_real @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_real @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: real] :
% 0.34/0.52 ( ( member_real @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_real @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_834_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_o,A2: $o] :
% 0.34/0.52 ( ( finite_finite_o @ A3 )
% 0.34/0.52 => ( ( member_o @ A2 @ A3 )
% 0.34/0.52 => ? [X5: $o] :
% 0.34/0.52 ( ( member_o @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_o @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: $o] :
% 0.34/0.52 ( ( member_o @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_o @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_835_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_set_nat,A2: set_nat] :
% 0.34/0.52 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.52 => ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_set_nat @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_set_nat @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_836_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_Extended_enat,A2: extended_enat] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.52 => ( ( member_Extended_enat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ X5 @ A3 )
% 0.34/0.52 & ( ord_le2932123472753598470d_enat @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_837_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_set_int,A2: set_int] :
% 0.34/0.52 ( ( finite6197958912794628473et_int @ A3 )
% 0.34/0.52 => ( ( member_set_int @ A2 @ A3 )
% 0.34/0.52 => ? [X5: set_int] :
% 0.34/0.52 ( ( member_set_int @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_set_int @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: set_int] :
% 0.34/0.52 ( ( member_set_int @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_838_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_rat,A2: rat] :
% 0.34/0.52 ( ( finite_finite_rat @ A3 )
% 0.34/0.52 => ( ( member_rat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: rat] :
% 0.34/0.52 ( ( member_rat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_rat @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: rat] :
% 0.34/0.52 ( ( member_rat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_839_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_num,A2: num] :
% 0.34/0.52 ( ( finite_finite_num @ A3 )
% 0.34/0.52 => ( ( member_num @ A2 @ A3 )
% 0.34/0.52 => ? [X5: num] :
% 0.34/0.52 ( ( member_num @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_num @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: num] :
% 0.34/0.52 ( ( member_num @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_num @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_840_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_nat,A2: nat] :
% 0.34/0.52 ( ( finite_finite_nat @ A3 )
% 0.34/0.52 => ( ( member_nat @ A2 @ A3 )
% 0.34/0.52 => ? [X5: nat] :
% 0.34/0.52 ( ( member_nat @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_nat @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: nat] :
% 0.34/0.52 ( ( member_nat @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_841_finite__has__maximal2,axiom,
% 0.34/0.52 ! [A3: set_int,A2: int] :
% 0.34/0.52 ( ( finite_finite_int @ A3 )
% 0.34/0.52 => ( ( member_int @ A2 @ A3 )
% 0.34/0.52 => ? [X5: int] :
% 0.34/0.52 ( ( member_int @ X5 @ A3 )
% 0.34/0.52 & ( ord_less_eq_int @ A2 @ X5 )
% 0.34/0.52 & ! [Xa: int] :
% 0.34/0.52 ( ( member_int @ Xa @ A3 )
% 0.34/0.52 => ( ( ord_less_eq_int @ X5 @ Xa )
% 0.34/0.52 => ( X5 = Xa ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_has_maximal2
% 0.34/0.52 thf(fact_842_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_complex] :
% 0.34/0.52 ( ~ ( finite3207457112153483333omplex @ S )
% 0.34/0.52 => ( S != bot_bot_set_complex ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_843_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_Pr1261947904930325089at_nat] :
% 0.34/0.52 ( ~ ( finite6177210948735845034at_nat @ S )
% 0.34/0.52 => ( S != bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_844_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_Extended_enat] :
% 0.34/0.52 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.52 => ( S != bot_bo7653980558646680370d_enat ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_845_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_real] :
% 0.34/0.52 ( ~ ( finite_finite_real @ S )
% 0.34/0.52 => ( S != bot_bot_set_real ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_846_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_o] :
% 0.34/0.52 ( ~ ( finite_finite_o @ S )
% 0.34/0.52 => ( S != bot_bot_set_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_847_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_nat] :
% 0.34/0.52 ( ~ ( finite_finite_nat @ S )
% 0.34/0.52 => ( S != bot_bot_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_848_infinite__imp__nonempty,axiom,
% 0.34/0.52 ! [S: set_int] :
% 0.34/0.52 ( ~ ( finite_finite_int @ S )
% 0.34/0.52 => ( S != bot_bot_set_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_imp_nonempty
% 0.34/0.52 thf(fact_849_finite_OemptyI,axiom,
% 0.34/0.52 finite3207457112153483333omplex @ bot_bot_set_complex ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_850_finite_OemptyI,axiom,
% 0.34/0.52 finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_851_finite_OemptyI,axiom,
% 0.34/0.52 finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_852_finite_OemptyI,axiom,
% 0.34/0.52 finite_finite_real @ bot_bot_set_real ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_853_finite_OemptyI,axiom,
% 0.34/0.52 finite_finite_o @ bot_bot_set_o ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_854_finite_OemptyI,axiom,
% 0.34/0.52 finite_finite_nat @ bot_bot_set_nat ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_855_finite_OemptyI,axiom,
% 0.34/0.52 finite_finite_int @ bot_bot_set_int ).
% 0.34/0.52
% 0.34/0.52 % finite.emptyI
% 0.34/0.52 thf(fact_856_member__valid__both__member__options,axiom,
% 0.34/0.52 ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.52 ( ( vEBT_invar_vebt @ Tree @ N )
% 0.34/0.52 => ( ( vEBT_vebt_member @ Tree @ X2 )
% 0.34/0.52 => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
% 0.34/0.52 | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % member_valid_both_member_options
% 0.34/0.52 thf(fact_857_buildup__nothing__in__min__max,axiom,
% 0.34/0.52 ! [N: nat,X2: nat] :
% 0.34/0.52 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 0.34/0.52
% 0.34/0.52 % buildup_nothing_in_min_max
% 0.34/0.52 thf(fact_858_dele__member__cont__corr,axiom,
% 0.34/0.52 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.34/0.52 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.52 => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
% 0.34/0.52 = ( ( X2 != Y3 )
% 0.34/0.52 & ( vEBT_vebt_member @ T @ Y3 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % dele_member_cont_corr
% 0.34/0.52 thf(fact_859_finite__nat__set__iff__bounded__le,axiom,
% 0.34/0.52 ( finite_finite_nat
% 0.34/0.52 = ( ^ [N4: set_nat] :
% 0.34/0.52 ? [M2: nat] :
% 0.34/0.52 ! [X: nat] :
% 0.34/0.52 ( ( member_nat @ X @ N4 )
% 0.34/0.52 => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_nat_set_iff_bounded_le
% 0.34/0.52 thf(fact_860_infinite__nat__iff__unbounded__le,axiom,
% 0.34/0.52 ! [S: set_nat] :
% 0.34/0.52 ( ( ~ ( finite_finite_nat @ S ) )
% 0.34/0.52 = ( ! [M2: nat] :
% 0.34/0.52 ? [N2: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.34/0.52 & ( member_nat @ N2 @ S ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_nat_iff_unbounded_le
% 0.34/0.52 thf(fact_861_unbounded__k__infinite,axiom,
% 0.34/0.52 ! [K: nat,S: set_nat] :
% 0.34/0.52 ( ! [M4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ K @ M4 )
% 0.34/0.52 => ? [N5: nat] :
% 0.34/0.52 ( ( ord_less_nat @ M4 @ N5 )
% 0.34/0.52 & ( member_nat @ N5 @ S ) ) )
% 0.34/0.52 => ~ ( finite_finite_nat @ S ) ) ).
% 0.34/0.52
% 0.34/0.52 % unbounded_k_infinite
% 0.34/0.52 thf(fact_862_bounded__nat__set__is__finite,axiom,
% 0.34/0.52 ! [N6: set_nat,N: nat] :
% 0.34/0.52 ( ! [X5: nat] :
% 0.34/0.52 ( ( member_nat @ X5 @ N6 )
% 0.34/0.52 => ( ord_less_nat @ X5 @ N ) )
% 0.34/0.52 => ( finite_finite_nat @ N6 ) ) ).
% 0.34/0.52
% 0.34/0.52 % bounded_nat_set_is_finite
% 0.34/0.52 thf(fact_863_infinite__nat__iff__unbounded,axiom,
% 0.34/0.52 ! [S: set_nat] :
% 0.34/0.52 ( ( ~ ( finite_finite_nat @ S ) )
% 0.34/0.52 = ( ! [M2: nat] :
% 0.34/0.52 ? [N2: nat] :
% 0.34/0.52 ( ( ord_less_nat @ M2 @ N2 )
% 0.34/0.52 & ( member_nat @ N2 @ S ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % infinite_nat_iff_unbounded
% 0.34/0.52 thf(fact_864_finite__nat__set__iff__bounded,axiom,
% 0.34/0.52 ( finite_finite_nat
% 0.34/0.52 = ( ^ [N4: set_nat] :
% 0.34/0.52 ? [M2: nat] :
% 0.34/0.52 ! [X: nat] :
% 0.34/0.52 ( ( member_nat @ X @ N4 )
% 0.34/0.52 => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_nat_set_iff_bounded
% 0.34/0.52 thf(fact_865_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_complex,F: complex > real] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_complex )
% 0.34/0.52 => ~ ? [X4: complex] :
% 0.34/0.52 ( ( member_complex @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_866_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_Extended_enat,F: extended_enat > real] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.52 => ( ( S != bot_bo7653980558646680370d_enat )
% 0.34/0.52 => ~ ? [X4: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic1189837152898106425t_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_867_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_real,F: real > real] :
% 0.34/0.52 ( ( finite_finite_real @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_real )
% 0.34/0.52 => ~ ? [X4: real] :
% 0.34/0.52 ( ( member_real @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_868_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_o,F: $o > real] :
% 0.34/0.52 ( ( finite_finite_o @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_o )
% 0.34/0.52 => ~ ? [X4: $o] :
% 0.34/0.52 ( ( member_o @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_869_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_nat,F: nat > real] :
% 0.34/0.52 ( ( finite_finite_nat @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_nat )
% 0.34/0.52 => ~ ? [X4: nat] :
% 0.34/0.52 ( ( member_nat @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_870_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_int,F: int > real] :
% 0.34/0.52 ( ( finite_finite_int @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_int )
% 0.34/0.52 => ~ ? [X4: int] :
% 0.34/0.52 ( ( member_int @ X4 @ S )
% 0.34/0.52 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_871_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_complex,F: complex > rat] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_complex )
% 0.34/0.52 => ~ ? [X4: complex] :
% 0.34/0.52 ( ( member_complex @ X4 @ S )
% 0.34/0.52 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_872_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_Extended_enat,F: extended_enat > rat] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.52 => ( ( S != bot_bo7653980558646680370d_enat )
% 0.34/0.52 => ~ ? [X4: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ X4 @ S )
% 0.34/0.52 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_873_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_real,F: real > rat] :
% 0.34/0.52 ( ( finite_finite_real @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_real )
% 0.34/0.52 => ~ ? [X4: real] :
% 0.34/0.52 ( ( member_real @ X4 @ S )
% 0.34/0.52 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_874_arg__min__if__finite_I2_J,axiom,
% 0.34/0.52 ! [S: set_o,F: $o > rat] :
% 0.34/0.52 ( ( finite_finite_o @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_o )
% 0.34/0.52 => ~ ? [X4: $o] :
% 0.34/0.52 ( ( member_o @ X4 @ S )
% 0.34/0.52 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_if_finite(2)
% 0.34/0.52 thf(fact_875_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_complex,Y3: complex,F: complex > rat] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_complex )
% 0.34/0.52 => ( ( member_complex @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_876_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > rat] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.52 => ( ( S != bot_bo7653980558646680370d_enat )
% 0.34/0.52 => ( ( member_Extended_enat @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_877_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_real,Y3: real,F: real > rat] :
% 0.34/0.52 ( ( finite_finite_real @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_real )
% 0.34/0.52 => ( ( member_real @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_878_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_o,Y3: $o,F: $o > rat] :
% 0.34/0.52 ( ( finite_finite_o @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_o )
% 0.34/0.52 => ( ( member_o @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_879_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_nat,Y3: nat,F: nat > rat] :
% 0.34/0.52 ( ( finite_finite_nat @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_nat )
% 0.34/0.52 => ( ( member_nat @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_880_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_int,Y3: int,F: int > rat] :
% 0.34/0.52 ( ( finite_finite_int @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_int )
% 0.34/0.52 => ( ( member_int @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_881_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_complex,Y3: complex,F: complex > num] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_complex )
% 0.34/0.52 => ( ( member_complex @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_882_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > num] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.52 => ( ( S != bot_bo7653980558646680370d_enat )
% 0.34/0.52 => ( ( member_Extended_enat @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ ( lattic402713867396545063at_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_883_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_real,Y3: real,F: real > num] :
% 0.34/0.52 ( ( finite_finite_real @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_real )
% 0.34/0.52 => ( ( member_real @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_884_arg__min__least,axiom,
% 0.34/0.52 ! [S: set_o,Y3: $o,F: $o > num] :
% 0.34/0.52 ( ( finite_finite_o @ S )
% 0.34/0.52 => ( ( S != bot_bot_set_o )
% 0.34/0.52 => ( ( member_o @ Y3 @ S )
% 0.34/0.52 => ( ord_less_eq_num @ ( F @ ( lattic8556559851467007577_o_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % arg_min_least
% 0.34/0.52 thf(fact_885_nat__descend__induct,axiom,
% 0.34/0.52 ! [N: nat,P: nat > $o,M: nat] :
% 0.34/0.52 ( ! [K2: nat] :
% 0.34/0.52 ( ( ord_less_nat @ N @ K2 )
% 0.34/0.52 => ( P @ K2 ) )
% 0.34/0.52 => ( ! [K2: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ K2 @ N )
% 0.34/0.52 => ( ! [I3: nat] :
% 0.34/0.52 ( ( ord_less_nat @ K2 @ I3 )
% 0.34/0.52 => ( P @ I3 ) )
% 0.34/0.52 => ( P @ K2 ) ) )
% 0.34/0.52 => ( P @ M ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % nat_descend_induct
% 0.34/0.52 thf(fact_886_delete__pres__valid,axiom,
% 0.34/0.52 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.52 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.52 => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X2 ) @ N ) ) ).
% 0.34/0.52
% 0.34/0.52 % delete_pres_valid
% 0.34/0.52 thf(fact_887_subsetI,axiom,
% 0.34/0.52 ! [A3: set_real,B2: set_real] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ( ( member_real @ X5 @ A3 )
% 0.34/0.52 => ( member_real @ X5 @ B2 ) )
% 0.34/0.52 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetI
% 0.34/0.52 thf(fact_888_subsetI,axiom,
% 0.34/0.52 ! [A3: set_o,B2: set_o] :
% 0.34/0.52 ( ! [X5: $o] :
% 0.34/0.52 ( ( member_o @ X5 @ A3 )
% 0.34/0.52 => ( member_o @ X5 @ B2 ) )
% 0.34/0.52 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetI
% 0.34/0.52 thf(fact_889_subsetI,axiom,
% 0.34/0.52 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.52 ( ! [X5: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ X5 @ A3 )
% 0.34/0.52 => ( member_set_nat @ X5 @ B2 ) )
% 0.34/0.52 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetI
% 0.34/0.52 thf(fact_890_subsetI,axiom,
% 0.34/0.52 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.52 ( ! [X5: nat] :
% 0.34/0.52 ( ( member_nat @ X5 @ A3 )
% 0.34/0.52 => ( member_nat @ X5 @ B2 ) )
% 0.34/0.52 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetI
% 0.34/0.52 thf(fact_891_subsetI,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ! [X5: int] :
% 0.34/0.52 ( ( member_int @ X5 @ A3 )
% 0.34/0.52 => ( member_int @ X5 @ B2 ) )
% 0.34/0.52 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetI
% 0.34/0.52 thf(fact_892_subset__antisym,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.34/0.52 => ( A3 = B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_antisym
% 0.34/0.52 thf(fact_893_psubsetI,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( A3 != B2 )
% 0.34/0.52 => ( ord_less_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetI
% 0.34/0.52 thf(fact_894_in__mono,axiom,
% 0.34/0.52 ! [A3: set_real,B2: set_real,X2: real] :
% 0.34/0.52 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.34/0.52 => ( ( member_real @ X2 @ A3 )
% 0.34/0.52 => ( member_real @ X2 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_mono
% 0.34/0.52 thf(fact_895_in__mono,axiom,
% 0.34/0.52 ! [A3: set_o,B2: set_o,X2: $o] :
% 0.34/0.52 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.34/0.52 => ( ( member_o @ X2 @ A3 )
% 0.34/0.52 => ( member_o @ X2 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_mono
% 0.34/0.52 thf(fact_896_in__mono,axiom,
% 0.34/0.52 ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat] :
% 0.34/0.52 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.52 => ( member_set_nat @ X2 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_mono
% 0.34/0.52 thf(fact_897_in__mono,axiom,
% 0.34/0.52 ! [A3: set_nat,B2: set_nat,X2: nat] :
% 0.34/0.52 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_nat @ X2 @ A3 )
% 0.34/0.52 => ( member_nat @ X2 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_mono
% 0.34/0.52 thf(fact_898_in__mono,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,X2: int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( member_int @ X2 @ A3 )
% 0.34/0.52 => ( member_int @ X2 @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % in_mono
% 0.34/0.52 thf(fact_899_subsetD,axiom,
% 0.34/0.52 ! [A3: set_real,B2: set_real,C: real] :
% 0.34/0.52 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.34/0.52 => ( ( member_real @ C @ A3 )
% 0.34/0.52 => ( member_real @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetD
% 0.34/0.52 thf(fact_900_subsetD,axiom,
% 0.34/0.52 ! [A3: set_o,B2: set_o,C: $o] :
% 0.34/0.52 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.34/0.52 => ( ( member_o @ C @ A3 )
% 0.34/0.52 => ( member_o @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetD
% 0.34/0.52 thf(fact_901_subsetD,axiom,
% 0.34/0.52 ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.34/0.52 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_set_nat @ C @ A3 )
% 0.34/0.52 => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetD
% 0.34/0.52 thf(fact_902_subsetD,axiom,
% 0.34/0.52 ! [A3: set_nat,B2: set_nat,C: nat] :
% 0.34/0.52 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_nat @ C @ A3 )
% 0.34/0.52 => ( member_nat @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetD
% 0.34/0.52 thf(fact_903_subsetD,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,C: int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( member_int @ C @ A3 )
% 0.34/0.52 => ( member_int @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subsetD
% 0.34/0.52 thf(fact_904_psubsetD,axiom,
% 0.34/0.52 ! [A3: set_real,B2: set_real,C: real] :
% 0.34/0.52 ( ( ord_less_set_real @ A3 @ B2 )
% 0.34/0.52 => ( ( member_real @ C @ A3 )
% 0.34/0.52 => ( member_real @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetD
% 0.34/0.52 thf(fact_905_psubsetD,axiom,
% 0.34/0.52 ! [A3: set_o,B2: set_o,C: $o] :
% 0.34/0.52 ( ( ord_less_set_o @ A3 @ B2 )
% 0.34/0.52 => ( ( member_o @ C @ A3 )
% 0.34/0.52 => ( member_o @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetD
% 0.34/0.52 thf(fact_906_psubsetD,axiom,
% 0.34/0.52 ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.34/0.52 ( ( ord_less_set_set_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_set_nat @ C @ A3 )
% 0.34/0.52 => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetD
% 0.34/0.52 thf(fact_907_psubsetD,axiom,
% 0.34/0.52 ! [A3: set_nat,B2: set_nat,C: nat] :
% 0.34/0.52 ( ( ord_less_set_nat @ A3 @ B2 )
% 0.34/0.52 => ( ( member_nat @ C @ A3 )
% 0.34/0.52 => ( member_nat @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetD
% 0.34/0.52 thf(fact_908_psubsetD,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,C: int] :
% 0.34/0.52 ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( member_int @ C @ A3 )
% 0.34/0.52 => ( member_int @ C @ B2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetD
% 0.34/0.52 thf(fact_909_psubsetE,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.52 => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubsetE
% 0.34/0.52 thf(fact_910_equalityE,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( A3 = B2 )
% 0.34/0.52 => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % equalityE
% 0.34/0.52 thf(fact_911_subset__eq,axiom,
% 0.34/0.52 ( ord_less_eq_set_real
% 0.34/0.52 = ( ^ [A6: set_real,B6: set_real] :
% 0.34/0.52 ! [X: real] :
% 0.34/0.52 ( ( member_real @ X @ A6 )
% 0.34/0.52 => ( member_real @ X @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_eq
% 0.34/0.52 thf(fact_912_subset__eq,axiom,
% 0.34/0.52 ( ord_less_eq_set_o
% 0.34/0.52 = ( ^ [A6: set_o,B6: set_o] :
% 0.34/0.52 ! [X: $o] :
% 0.34/0.52 ( ( member_o @ X @ A6 )
% 0.34/0.52 => ( member_o @ X @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_eq
% 0.34/0.52 thf(fact_913_subset__eq,axiom,
% 0.34/0.52 ( ord_le6893508408891458716et_nat
% 0.34/0.52 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.34/0.52 ! [X: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ X @ A6 )
% 0.34/0.52 => ( member_set_nat @ X @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_eq
% 0.34/0.52 thf(fact_914_subset__eq,axiom,
% 0.34/0.52 ( ord_less_eq_set_nat
% 0.34/0.52 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.52 ! [X: nat] :
% 0.34/0.52 ( ( member_nat @ X @ A6 )
% 0.34/0.52 => ( member_nat @ X @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_eq
% 0.34/0.52 thf(fact_915_subset__eq,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ! [X: int] :
% 0.34/0.52 ( ( member_int @ X @ A6 )
% 0.34/0.52 => ( member_int @ X @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_eq
% 0.34/0.52 thf(fact_916_equalityD1,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( A3 = B2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % equalityD1
% 0.34/0.52 thf(fact_917_equalityD2,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( A3 = B2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ B2 @ A3 ) ) ).
% 0.34/0.52
% 0.34/0.52 % equalityD2
% 0.34/0.52 thf(fact_918_psubset__eq,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.34/0.52 & ( A6 != B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubset_eq
% 0.34/0.52 thf(fact_919_subset__iff,axiom,
% 0.34/0.52 ( ord_less_eq_set_real
% 0.34/0.52 = ( ^ [A6: set_real,B6: set_real] :
% 0.34/0.52 ! [T3: real] :
% 0.34/0.52 ( ( member_real @ T3 @ A6 )
% 0.34/0.52 => ( member_real @ T3 @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff
% 0.34/0.52 thf(fact_920_subset__iff,axiom,
% 0.34/0.52 ( ord_less_eq_set_o
% 0.34/0.52 = ( ^ [A6: set_o,B6: set_o] :
% 0.34/0.52 ! [T3: $o] :
% 0.34/0.52 ( ( member_o @ T3 @ A6 )
% 0.34/0.52 => ( member_o @ T3 @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff
% 0.34/0.52 thf(fact_921_subset__iff,axiom,
% 0.34/0.52 ( ord_le6893508408891458716et_nat
% 0.34/0.52 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.34/0.52 ! [T3: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ T3 @ A6 )
% 0.34/0.52 => ( member_set_nat @ T3 @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff
% 0.34/0.52 thf(fact_922_subset__iff,axiom,
% 0.34/0.52 ( ord_less_eq_set_nat
% 0.34/0.52 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.34/0.52 ! [T3: nat] :
% 0.34/0.52 ( ( member_nat @ T3 @ A6 )
% 0.34/0.52 => ( member_nat @ T3 @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff
% 0.34/0.52 thf(fact_923_subset__iff,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ! [T3: int] :
% 0.34/0.52 ( ( member_int @ T3 @ A6 )
% 0.34/0.52 => ( member_int @ T3 @ B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff
% 0.34/0.52 thf(fact_924_subset__refl,axiom,
% 0.34/0.52 ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% 0.34/0.52
% 0.34/0.52 % subset_refl
% 0.34/0.52 thf(fact_925_Collect__mono,axiom,
% 0.34/0.52 ! [P: real > $o,Q: real > $o] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( Q @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono
% 0.34/0.52 thf(fact_926_Collect__mono,axiom,
% 0.34/0.52 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.34/0.52 ( ! [X5: list_nat] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( Q @ X5 ) )
% 0.34/0.52 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono
% 0.34/0.52 thf(fact_927_Collect__mono,axiom,
% 0.34/0.52 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.34/0.52 ( ! [X5: set_nat] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( Q @ X5 ) )
% 0.34/0.52 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono
% 0.34/0.52 thf(fact_928_Collect__mono,axiom,
% 0.34/0.52 ! [P: nat > $o,Q: nat > $o] :
% 0.34/0.52 ( ! [X5: nat] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( Q @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono
% 0.34/0.52 thf(fact_929_Collect__mono,axiom,
% 0.34/0.52 ! [P: int > $o,Q: int > $o] :
% 0.34/0.52 ( ! [X5: int] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( Q @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono
% 0.34/0.52 thf(fact_930_subset__trans,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_trans
% 0.34/0.52 thf(fact_931_set__eq__subset,axiom,
% 0.34/0.52 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.34/0.52 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % set_eq_subset
% 0.34/0.52 thf(fact_932_Collect__mono__iff,axiom,
% 0.34/0.52 ! [P: real > $o,Q: real > $o] :
% 0.34/0.52 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.34/0.52 = ( ! [X: real] :
% 0.34/0.52 ( ( P @ X )
% 0.34/0.52 => ( Q @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono_iff
% 0.34/0.52 thf(fact_933_Collect__mono__iff,axiom,
% 0.34/0.52 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.34/0.52 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.34/0.52 = ( ! [X: list_nat] :
% 0.34/0.52 ( ( P @ X )
% 0.34/0.52 => ( Q @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono_iff
% 0.34/0.52 thf(fact_934_Collect__mono__iff,axiom,
% 0.34/0.52 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.34/0.52 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.34/0.52 = ( ! [X: set_nat] :
% 0.34/0.52 ( ( P @ X )
% 0.34/0.52 => ( Q @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono_iff
% 0.34/0.52 thf(fact_935_Collect__mono__iff,axiom,
% 0.34/0.52 ! [P: nat > $o,Q: nat > $o] :
% 0.34/0.52 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.34/0.52 = ( ! [X: nat] :
% 0.34/0.52 ( ( P @ X )
% 0.34/0.52 => ( Q @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono_iff
% 0.34/0.52 thf(fact_936_Collect__mono__iff,axiom,
% 0.34/0.52 ! [P: int > $o,Q: int > $o] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.34/0.52 = ( ! [X: int] :
% 0.34/0.52 ( ( P @ X )
% 0.34/0.52 => ( Q @ X ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_mono_iff
% 0.34/0.52 thf(fact_937_psubset__imp__subset,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubset_imp_subset
% 0.34/0.52 thf(fact_938_psubset__subset__trans,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.34/0.52 => ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % psubset_subset_trans
% 0.34/0.52 thf(fact_939_subset__not__subset__eq,axiom,
% 0.34/0.52 ( ord_less_set_int
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 0.34/0.52 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_not_subset_eq
% 0.34/0.52 thf(fact_940_subset__psubset__trans,axiom,
% 0.34/0.52 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.34/0.52 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.52 => ( ( ord_less_set_int @ B2 @ C2 )
% 0.34/0.52 => ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_psubset_trans
% 0.34/0.52 thf(fact_941_subset__iff__psubset__eq,axiom,
% 0.34/0.52 ( ord_less_eq_set_int
% 0.34/0.52 = ( ^ [A6: set_int,B6: set_int] :
% 0.34/0.52 ( ( ord_less_set_int @ A6 @ B6 )
% 0.34/0.52 | ( A6 = B6 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_iff_psubset_eq
% 0.34/0.52 thf(fact_942_bounded__Max__nat,axiom,
% 0.34/0.52 ! [P: nat > $o,X2: nat,M5: nat] :
% 0.34/0.52 ( ( P @ X2 )
% 0.34/0.52 => ( ! [X5: nat] :
% 0.34/0.52 ( ( P @ X5 )
% 0.34/0.52 => ( ord_less_eq_nat @ X5 @ M5 ) )
% 0.34/0.52 => ~ ! [M4: nat] :
% 0.34/0.52 ( ( P @ M4 )
% 0.34/0.52 => ~ ! [X4: nat] :
% 0.34/0.52 ( ( P @ X4 )
% 0.34/0.52 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bounded_Max_nat
% 0.34/0.52 thf(fact_943_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_set_nat,R: set_nat > set_nat > $o] :
% 0.34/0.52 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.52 => ( ! [X5: set_nat] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: set_nat] :
% 0.34/0.52 ( ( member_set_nat @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_944_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_complex,R: complex > complex > $o] :
% 0.34/0.52 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.52 => ( ! [X5: complex] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: complex,Y4: complex,Z4: complex] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: complex] :
% 0.34/0.52 ( ( member_complex @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: complex] :
% 0.34/0.52 ( ( member_complex @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_complex ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_945_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.34/0.52 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.52 => ( ! [X5: product_prod_nat_nat] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: product_prod_nat_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: product_prod_nat_nat] :
% 0.34/0.52 ( ( member8440522571783428010at_nat @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_946_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_Extended_enat,R: extended_enat > extended_enat > $o] :
% 0.34/0.52 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.52 => ( ! [X5: extended_enat] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: extended_enat,Y4: extended_enat,Z4: extended_enat] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: extended_enat] :
% 0.34/0.52 ( ( member_Extended_enat @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_947_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_real,R: real > real > $o] :
% 0.34/0.52 ( ( finite_finite_real @ A3 )
% 0.34/0.52 => ( ! [X5: real] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: real,Y4: real,Z4: real] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: real] :
% 0.34/0.52 ( ( member_real @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: real] :
% 0.34/0.52 ( ( member_real @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_real ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_948_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_o,R: $o > $o > $o] :
% 0.34/0.52 ( ( finite_finite_o @ A3 )
% 0.34/0.52 => ( ! [X5: $o] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: $o,Y4: $o,Z4: $o] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: $o] :
% 0.34/0.52 ( ( member_o @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: $o] :
% 0.34/0.52 ( ( member_o @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_949_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_nat,R: nat > nat > $o] :
% 0.34/0.52 ( ( finite_finite_nat @ A3 )
% 0.34/0.52 => ( ! [X5: nat] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: nat] :
% 0.34/0.52 ( ( member_nat @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: nat] :
% 0.34/0.52 ( ( member_nat @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_950_finite__transitivity__chain,axiom,
% 0.34/0.52 ! [A3: set_int,R: int > int > $o] :
% 0.34/0.52 ( ( finite_finite_int @ A3 )
% 0.34/0.52 => ( ! [X5: int] :
% 0.34/0.52 ~ ( R @ X5 @ X5 )
% 0.34/0.52 => ( ! [X5: int,Y4: int,Z4: int] :
% 0.34/0.52 ( ( R @ X5 @ Y4 )
% 0.34/0.52 => ( ( R @ Y4 @ Z4 )
% 0.34/0.52 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.52 => ( ! [X5: int] :
% 0.34/0.52 ( ( member_int @ X5 @ A3 )
% 0.34/0.52 => ? [Y5: int] :
% 0.34/0.52 ( ( member_int @ Y5 @ A3 )
% 0.34/0.52 & ( R @ X5 @ Y5 ) ) )
% 0.34/0.52 => ( A3 = bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % finite_transitivity_chain
% 0.34/0.52 thf(fact_951_bot__empty__eq,axiom,
% 0.34/0.52 ( bot_bot_set_nat_o
% 0.34/0.52 = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot_empty_eq
% 0.34/0.52 thf(fact_952_bot__empty__eq,axiom,
% 0.34/0.52 ( bot_bot_real_o
% 0.34/0.52 = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot_empty_eq
% 0.34/0.52 thf(fact_953_bot__empty__eq,axiom,
% 0.34/0.52 ( bot_bot_o_o
% 0.34/0.52 = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot_empty_eq
% 0.34/0.52 thf(fact_954_bot__empty__eq,axiom,
% 0.34/0.52 ( bot_bot_nat_o
% 0.34/0.52 = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot_empty_eq
% 0.34/0.52 thf(fact_955_bot__empty__eq,axiom,
% 0.34/0.52 ( bot_bot_int_o
% 0.34/0.52 = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % bot_empty_eq
% 0.34/0.52 thf(fact_956_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: list_nat > $o] :
% 0.34/0.52 ( ( ( collect_list_nat @ P )
% 0.34/0.52 = bot_bot_set_list_nat )
% 0.34/0.52 = ( P = bot_bot_list_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_957_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: set_nat > $o] :
% 0.34/0.52 ( ( ( collect_set_nat @ P )
% 0.34/0.52 = bot_bot_set_set_nat )
% 0.34/0.52 = ( P = bot_bot_set_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_958_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: real > $o] :
% 0.34/0.52 ( ( ( collect_real @ P )
% 0.34/0.52 = bot_bot_set_real )
% 0.34/0.52 = ( P = bot_bot_real_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_959_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: $o > $o] :
% 0.34/0.52 ( ( ( collect_o @ P )
% 0.34/0.52 = bot_bot_set_o )
% 0.34/0.52 = ( P = bot_bot_o_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_960_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: nat > $o] :
% 0.34/0.52 ( ( ( collect_nat @ P )
% 0.34/0.52 = bot_bot_set_nat )
% 0.34/0.52 = ( P = bot_bot_nat_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_961_Collect__empty__eq__bot,axiom,
% 0.34/0.52 ! [P: int > $o] :
% 0.34/0.52 ( ( ( collect_int @ P )
% 0.34/0.52 = bot_bot_set_int )
% 0.34/0.52 = ( P = bot_bot_int_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % Collect_empty_eq_bot
% 0.34/0.52 thf(fact_962_subset__emptyI,axiom,
% 0.34/0.52 ! [A3: set_set_nat] :
% 0.34/0.52 ( ! [X5: set_nat] :
% 0.34/0.52 ~ ( member_set_nat @ X5 @ A3 )
% 0.34/0.52 => ( ord_le6893508408891458716et_nat @ A3 @ bot_bot_set_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_emptyI
% 0.34/0.52 thf(fact_963_subset__emptyI,axiom,
% 0.34/0.52 ! [A3: set_real] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ~ ( member_real @ X5 @ A3 )
% 0.34/0.52 => ( ord_less_eq_set_real @ A3 @ bot_bot_set_real ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_emptyI
% 0.34/0.52 thf(fact_964_subset__emptyI,axiom,
% 0.34/0.52 ! [A3: set_o] :
% 0.34/0.52 ( ! [X5: $o] :
% 0.34/0.52 ~ ( member_o @ X5 @ A3 )
% 0.34/0.52 => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_emptyI
% 0.34/0.52 thf(fact_965_subset__emptyI,axiom,
% 0.34/0.52 ! [A3: set_nat] :
% 0.34/0.52 ( ! [X5: nat] :
% 0.34/0.52 ~ ( member_nat @ X5 @ A3 )
% 0.34/0.52 => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_emptyI
% 0.34/0.52 thf(fact_966_subset__emptyI,axiom,
% 0.34/0.52 ! [A3: set_int] :
% 0.34/0.52 ( ! [X5: int] :
% 0.34/0.52 ~ ( member_int @ X5 @ A3 )
% 0.34/0.52 => ( ord_less_eq_set_int @ A3 @ bot_bot_set_int ) ) ).
% 0.34/0.52
% 0.34/0.52 % subset_emptyI
% 0.34/0.52 thf(fact_967_field__lbound__gt__zero,axiom,
% 0.34/0.52 ! [D1: real,D2: real] :
% 0.34/0.52 ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.34/0.52 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.34/0.52 => ? [E: real] :
% 0.34/0.52 ( ( ord_less_real @ zero_zero_real @ E )
% 0.34/0.52 & ( ord_less_real @ E @ D1 )
% 0.34/0.52 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % field_lbound_gt_zero
% 0.34/0.52 thf(fact_968_field__lbound__gt__zero,axiom,
% 0.34/0.52 ! [D1: rat,D2: rat] :
% 0.34/0.52 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.34/0.52 => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 0.34/0.52 => ? [E: rat] :
% 0.34/0.52 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.34/0.52 & ( ord_less_rat @ E @ D1 )
% 0.34/0.52 & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % field_lbound_gt_zero
% 0.34/0.52 thf(fact_969_less__numeral__extra_I3_J,axiom,
% 0.34/0.52 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.34/0.52
% 0.34/0.52 % less_numeral_extra(3)
% 0.34/0.52 thf(fact_970_less__numeral__extra_I3_J,axiom,
% 0.34/0.52 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.34/0.52
% 0.34/0.52 % less_numeral_extra(3)
% 0.34/0.52 thf(fact_971_less__numeral__extra_I3_J,axiom,
% 0.34/0.52 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.34/0.52
% 0.34/0.52 % less_numeral_extra(3)
% 0.34/0.52 thf(fact_972_less__numeral__extra_I3_J,axiom,
% 0.34/0.52 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.34/0.52
% 0.34/0.52 % less_numeral_extra(3)
% 0.34/0.52 thf(fact_973_complete__interval,axiom,
% 0.34/0.52 ! [A2: real,B3: real,P: real > $o] :
% 0.34/0.52 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.52 => ( ( P @ A2 )
% 0.34/0.52 => ( ~ ( P @ B3 )
% 0.34/0.52 => ? [C3: real] :
% 0.34/0.52 ( ( ord_less_eq_real @ A2 @ C3 )
% 0.34/0.52 & ( ord_less_eq_real @ C3 @ B3 )
% 0.34/0.52 & ! [X4: real] :
% 0.34/0.52 ( ( ( ord_less_eq_real @ A2 @ X4 )
% 0.34/0.52 & ( ord_less_real @ X4 @ C3 ) )
% 0.34/0.52 => ( P @ X4 ) )
% 0.34/0.52 & ! [D3: real] :
% 0.34/0.52 ( ! [X5: real] :
% 0.34/0.52 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 0.34/0.52 & ( ord_less_real @ X5 @ D3 ) )
% 0.34/0.52 => ( P @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % complete_interval
% 0.34/0.52 thf(fact_974_complete__interval,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat,P: nat > $o] :
% 0.34/0.52 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.52 => ( ( P @ A2 )
% 0.34/0.52 => ( ~ ( P @ B3 )
% 0.34/0.52 => ? [C3: nat] :
% 0.34/0.52 ( ( ord_less_eq_nat @ A2 @ C3 )
% 0.34/0.52 & ( ord_less_eq_nat @ C3 @ B3 )
% 0.34/0.52 & ! [X4: nat] :
% 0.34/0.52 ( ( ( ord_less_eq_nat @ A2 @ X4 )
% 0.34/0.52 & ( ord_less_nat @ X4 @ C3 ) )
% 0.34/0.52 => ( P @ X4 ) )
% 0.34/0.52 & ! [D3: nat] :
% 0.34/0.52 ( ! [X5: nat] :
% 0.34/0.52 ( ( ( ord_less_eq_nat @ A2 @ X5 )
% 0.34/0.52 & ( ord_less_nat @ X5 @ D3 ) )
% 0.34/0.52 => ( P @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % complete_interval
% 0.34/0.52 thf(fact_975_complete__interval,axiom,
% 0.34/0.52 ! [A2: int,B3: int,P: int > $o] :
% 0.34/0.52 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.52 => ( ( P @ A2 )
% 0.34/0.52 => ( ~ ( P @ B3 )
% 0.34/0.52 => ? [C3: int] :
% 0.34/0.52 ( ( ord_less_eq_int @ A2 @ C3 )
% 0.34/0.52 & ( ord_less_eq_int @ C3 @ B3 )
% 0.34/0.52 & ! [X4: int] :
% 0.34/0.52 ( ( ( ord_less_eq_int @ A2 @ X4 )
% 0.34/0.52 & ( ord_less_int @ X4 @ C3 ) )
% 0.34/0.52 => ( P @ X4 ) )
% 0.34/0.52 & ! [D3: int] :
% 0.34/0.52 ( ! [X5: int] :
% 0.34/0.52 ( ( ( ord_less_eq_int @ A2 @ X5 )
% 0.34/0.52 & ( ord_less_int @ X5 @ D3 ) )
% 0.34/0.52 => ( P @ X5 ) )
% 0.34/0.52 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% 0.34/0.52
% 0.34/0.52 % complete_interval
% 0.34/0.52 thf(fact_976_verit__comp__simplify1_I3_J,axiom,
% 0.34/0.52 ! [B7: real,A7: real] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
% 0.34/0.52 = ( ord_less_real @ A7 @ B7 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(3)
% 0.34/0.52 thf(fact_977_verit__comp__simplify1_I3_J,axiom,
% 0.34/0.52 ! [B7: rat,A7: rat] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
% 0.34/0.52 = ( ord_less_rat @ A7 @ B7 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(3)
% 0.34/0.52 thf(fact_978_verit__comp__simplify1_I3_J,axiom,
% 0.34/0.52 ! [B7: num,A7: num] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
% 0.34/0.52 = ( ord_less_num @ A7 @ B7 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(3)
% 0.34/0.52 thf(fact_979_verit__comp__simplify1_I3_J,axiom,
% 0.34/0.52 ! [B7: nat,A7: nat] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
% 0.34/0.52 = ( ord_less_nat @ A7 @ B7 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(3)
% 0.34/0.52 thf(fact_980_verit__comp__simplify1_I3_J,axiom,
% 0.34/0.52 ! [B7: int,A7: int] :
% 0.34/0.52 ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
% 0.34/0.52 = ( ord_less_int @ A7 @ B7 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(3)
% 0.34/0.52 thf(fact_981_pinf_I6_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.52 => ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(6)
% 0.34/0.52 thf(fact_982_pinf_I6_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.52 => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(6)
% 0.34/0.52 thf(fact_983_pinf_I6_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.52 => ~ ( ord_less_eq_num @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(6)
% 0.34/0.52 thf(fact_984_pinf_I6_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.52 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(6)
% 0.34/0.52 thf(fact_985_pinf_I6_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.52 => ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(6)
% 0.34/0.52 thf(fact_986_pinf_I8_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.52 => ( ord_less_eq_real @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(8)
% 0.34/0.52 thf(fact_987_pinf_I8_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.52 => ( ord_less_eq_rat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(8)
% 0.34/0.52 thf(fact_988_pinf_I8_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.52 => ( ord_less_eq_num @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(8)
% 0.34/0.52 thf(fact_989_pinf_I8_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.52 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(8)
% 0.34/0.52 thf(fact_990_pinf_I8_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.52 => ( ord_less_eq_int @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % pinf(8)
% 0.34/0.52 thf(fact_991_minf_I6_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_eq_real @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(6)
% 0.34/0.52 thf(fact_992_minf_I6_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_eq_rat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(6)
% 0.34/0.52 thf(fact_993_minf_I6_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_eq_num @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(6)
% 0.34/0.52 thf(fact_994_minf_I6_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(6)
% 0.34/0.52 thf(fact_995_minf_I6_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_eq_int @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(6)
% 0.34/0.52 thf(fact_996_minf_I8_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(8)
% 0.34/0.52 thf(fact_997_minf_I8_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(8)
% 0.34/0.52 thf(fact_998_minf_I8_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_eq_num @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(8)
% 0.34/0.52 thf(fact_999_minf_I8_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(8)
% 0.34/0.52 thf(fact_1000_minf_I8_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(8)
% 0.34/0.52 thf(fact_1001_verit__la__disequality,axiom,
% 0.34/0.52 ! [A2: rat,B3: rat] :
% 0.34/0.52 ( ( A2 = B3 )
% 0.34/0.52 | ~ ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.52 | ~ ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_la_disequality
% 0.34/0.52 thf(fact_1002_verit__la__disequality,axiom,
% 0.34/0.52 ! [A2: num,B3: num] :
% 0.34/0.52 ( ( A2 = B3 )
% 0.34/0.52 | ~ ( ord_less_eq_num @ A2 @ B3 )
% 0.34/0.52 | ~ ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_la_disequality
% 0.34/0.52 thf(fact_1003_verit__la__disequality,axiom,
% 0.34/0.52 ! [A2: nat,B3: nat] :
% 0.34/0.52 ( ( A2 = B3 )
% 0.34/0.52 | ~ ( ord_less_eq_nat @ A2 @ B3 )
% 0.34/0.52 | ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_la_disequality
% 0.34/0.52 thf(fact_1004_verit__la__disequality,axiom,
% 0.34/0.52 ! [A2: int,B3: int] :
% 0.34/0.52 ( ( A2 = B3 )
% 0.34/0.52 | ~ ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.52 | ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 0.34/0.52
% 0.34/0.52 % verit_la_disequality
% 0.34/0.52 thf(fact_1005_verit__comp__simplify1_I2_J,axiom,
% 0.34/0.52 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(2)
% 0.34/0.52 thf(fact_1006_verit__comp__simplify1_I2_J,axiom,
% 0.34/0.52 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(2)
% 0.34/0.52 thf(fact_1007_verit__comp__simplify1_I2_J,axiom,
% 0.34/0.52 ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(2)
% 0.34/0.52 thf(fact_1008_verit__comp__simplify1_I2_J,axiom,
% 0.34/0.52 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(2)
% 0.34/0.52 thf(fact_1009_verit__comp__simplify1_I2_J,axiom,
% 0.34/0.52 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 0.34/0.52
% 0.34/0.52 % verit_comp_simplify1(2)
% 0.34/0.52 thf(fact_1010_minf_I7_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_real @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(7)
% 0.34/0.52 thf(fact_1011_minf_I7_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_rat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(7)
% 0.34/0.52 thf(fact_1012_minf_I7_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_num @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(7)
% 0.34/0.52 thf(fact_1013_minf_I7_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_nat @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(7)
% 0.34/0.52 thf(fact_1014_minf_I7_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.52 => ~ ( ord_less_int @ T @ X4 ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(7)
% 0.34/0.52 thf(fact_1015_minf_I5_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_real @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(5)
% 0.34/0.52 thf(fact_1016_minf_I5_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_rat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(5)
% 0.34/0.52 thf(fact_1017_minf_I5_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_num @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(5)
% 0.34/0.52 thf(fact_1018_minf_I5_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_nat @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(5)
% 0.34/0.52 thf(fact_1019_minf_I5_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.52 => ( ord_less_int @ X4 @ T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(5)
% 0.34/0.52 thf(fact_1020_minf_I4_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(4)
% 0.34/0.52 thf(fact_1021_minf_I4_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(4)
% 0.34/0.52 thf(fact_1022_minf_I4_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(4)
% 0.34/0.52 thf(fact_1023_minf_I4_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(4)
% 0.34/0.52 thf(fact_1024_minf_I4_J,axiom,
% 0.34/0.52 ! [T: int] :
% 0.34/0.52 ? [Z4: int] :
% 0.34/0.52 ! [X4: int] :
% 0.34/0.52 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(4)
% 0.34/0.52 thf(fact_1025_minf_I3_J,axiom,
% 0.34/0.52 ! [T: real] :
% 0.34/0.52 ? [Z4: real] :
% 0.34/0.52 ! [X4: real] :
% 0.34/0.52 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(3)
% 0.34/0.52 thf(fact_1026_minf_I3_J,axiom,
% 0.34/0.52 ! [T: rat] :
% 0.34/0.52 ? [Z4: rat] :
% 0.34/0.52 ! [X4: rat] :
% 0.34/0.52 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(3)
% 0.34/0.52 thf(fact_1027_minf_I3_J,axiom,
% 0.34/0.52 ! [T: num] :
% 0.34/0.52 ? [Z4: num] :
% 0.34/0.52 ! [X4: num] :
% 0.34/0.52 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.52 => ( X4 != T ) ) ).
% 0.34/0.52
% 0.34/0.52 % minf(3)
% 0.34/0.52 thf(fact_1028_minf_I3_J,axiom,
% 0.34/0.52 ! [T: nat] :
% 0.34/0.52 ? [Z4: nat] :
% 0.34/0.52 ! [X4: nat] :
% 0.34/0.52 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(3)
% 0.34/0.53 thf(fact_1029_minf_I3_J,axiom,
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(3)
% 0.34/0.53 thf(fact_1030_minf_I2_J,axiom,
% 0.34/0.53 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 0.34/0.53 ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(2)
% 0.34/0.53 thf(fact_1031_minf_I2_J,axiom,
% 0.34/0.53 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.34/0.53 ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(2)
% 0.34/0.53 thf(fact_1032_minf_I2_J,axiom,
% 0.34/0.53 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 0.34/0.53 ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(2)
% 0.34/0.53 thf(fact_1033_minf_I2_J,axiom,
% 0.34/0.53 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(2)
% 0.34/0.53 thf(fact_1034_minf_I2_J,axiom,
% 0.34/0.53 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(2)
% 0.34/0.53 thf(fact_1035_minf_I1_J,axiom,
% 0.34/0.53 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 0.34/0.53 ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(1)
% 0.34/0.53 thf(fact_1036_minf_I1_J,axiom,
% 0.34/0.53 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.34/0.53 ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(1)
% 0.34/0.53 thf(fact_1037_minf_I1_J,axiom,
% 0.34/0.53 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 0.34/0.53 ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(1)
% 0.34/0.53 thf(fact_1038_minf_I1_J,axiom,
% 0.34/0.53 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(1)
% 0.34/0.53 thf(fact_1039_minf_I1_J,axiom,
% 0.34/0.53 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ X5 @ Z5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ X5 @ Z5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ X4 @ Z4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % minf(1)
% 0.34/0.53 thf(fact_1040_pinf_I7_J,axiom,
% 0.34/0.53 ! [T: real] :
% 0.34/0.53 ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ( ord_less_real @ T @ X4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(7)
% 0.34/0.53 thf(fact_1041_pinf_I7_J,axiom,
% 0.34/0.53 ! [T: rat] :
% 0.34/0.53 ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ( ord_less_rat @ T @ X4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(7)
% 0.34/0.53 thf(fact_1042_pinf_I7_J,axiom,
% 0.34/0.53 ! [T: num] :
% 0.34/0.53 ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ( ord_less_num @ T @ X4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(7)
% 0.34/0.53 thf(fact_1043_pinf_I7_J,axiom,
% 0.34/0.53 ! [T: nat] :
% 0.34/0.53 ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ( ord_less_nat @ T @ X4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(7)
% 0.34/0.53 thf(fact_1044_pinf_I7_J,axiom,
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ( ord_less_int @ T @ X4 ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(7)
% 0.34/0.53 thf(fact_1045_pinf_I5_J,axiom,
% 0.34/0.53 ! [T: real] :
% 0.34/0.53 ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ~ ( ord_less_real @ X4 @ T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(5)
% 0.34/0.53 thf(fact_1046_pinf_I5_J,axiom,
% 0.34/0.53 ! [T: rat] :
% 0.34/0.53 ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ~ ( ord_less_rat @ X4 @ T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(5)
% 0.34/0.53 thf(fact_1047_pinf_I5_J,axiom,
% 0.34/0.53 ! [T: num] :
% 0.34/0.53 ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ~ ( ord_less_num @ X4 @ T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(5)
% 0.34/0.53 thf(fact_1048_pinf_I5_J,axiom,
% 0.34/0.53 ! [T: nat] :
% 0.34/0.53 ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ~ ( ord_less_nat @ X4 @ T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(5)
% 0.34/0.53 thf(fact_1049_pinf_I5_J,axiom,
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ~ ( ord_less_int @ X4 @ T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(5)
% 0.34/0.53 thf(fact_1050_pinf_I4_J,axiom,
% 0.34/0.53 ! [T: real] :
% 0.34/0.53 ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(4)
% 0.34/0.53 thf(fact_1051_pinf_I4_J,axiom,
% 0.34/0.53 ! [T: rat] :
% 0.34/0.53 ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(4)
% 0.34/0.53 thf(fact_1052_pinf_I4_J,axiom,
% 0.34/0.53 ! [T: num] :
% 0.34/0.53 ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(4)
% 0.34/0.53 thf(fact_1053_pinf_I4_J,axiom,
% 0.34/0.53 ! [T: nat] :
% 0.34/0.53 ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(4)
% 0.34/0.53 thf(fact_1054_pinf_I4_J,axiom,
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(4)
% 0.34/0.53 thf(fact_1055_pinf_I3_J,axiom,
% 0.34/0.53 ! [T: real] :
% 0.34/0.53 ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(3)
% 0.34/0.53 thf(fact_1056_pinf_I3_J,axiom,
% 0.34/0.53 ! [T: rat] :
% 0.34/0.53 ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(3)
% 0.34/0.53 thf(fact_1057_pinf_I3_J,axiom,
% 0.34/0.53 ! [T: num] :
% 0.34/0.53 ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(3)
% 0.34/0.53 thf(fact_1058_pinf_I3_J,axiom,
% 0.34/0.53 ! [T: nat] :
% 0.34/0.53 ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(3)
% 0.34/0.53 thf(fact_1059_pinf_I3_J,axiom,
% 0.34/0.53 ! [T: int] :
% 0.34/0.53 ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ( X4 != T ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(3)
% 0.34/0.53 thf(fact_1060_pinf_I2_J,axiom,
% 0.34/0.53 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 0.34/0.53 ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(2)
% 0.34/0.53 thf(fact_1061_pinf_I2_J,axiom,
% 0.34/0.53 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.34/0.53 ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(2)
% 0.34/0.53 thf(fact_1062_pinf_I2_J,axiom,
% 0.34/0.53 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 0.34/0.53 ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(2)
% 0.34/0.53 thf(fact_1063_pinf_I2_J,axiom,
% 0.34/0.53 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(2)
% 0.34/0.53 thf(fact_1064_pinf_I2_J,axiom,
% 0.34/0.53 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 | ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(2)
% 0.34/0.53 thf(fact_1065_pinf_I1_J,axiom,
% 0.34/0.53 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 0.34/0.53 ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: real] :
% 0.34/0.53 ! [X5: real] :
% 0.34/0.53 ( ( ord_less_real @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: real] :
% 0.34/0.53 ! [X4: real] :
% 0.34/0.53 ( ( ord_less_real @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(1)
% 0.34/0.53 thf(fact_1066_pinf_I1_J,axiom,
% 0.34/0.53 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 0.34/0.53 ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: rat] :
% 0.34/0.53 ! [X5: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: rat] :
% 0.34/0.53 ! [X4: rat] :
% 0.34/0.53 ( ( ord_less_rat @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(1)
% 0.34/0.53 thf(fact_1067_pinf_I1_J,axiom,
% 0.34/0.53 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 0.34/0.53 ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: num] :
% 0.34/0.53 ! [X5: num] :
% 0.34/0.53 ( ( ord_less_num @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: num] :
% 0.34/0.53 ! [X4: num] :
% 0.34/0.53 ( ( ord_less_num @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(1)
% 0.34/0.53 thf(fact_1068_pinf_I1_J,axiom,
% 0.34/0.53 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 0.34/0.53 ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: nat] :
% 0.34/0.53 ! [X5: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: nat] :
% 0.34/0.53 ! [X4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(1)
% 0.34/0.53 thf(fact_1069_pinf_I1_J,axiom,
% 0.34/0.53 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 0.34/0.53 ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ Z5 @ X5 )
% 0.34/0.53 => ( ( P @ X5 )
% 0.34/0.53 = ( P4 @ X5 ) ) )
% 0.34/0.53 => ( ? [Z5: int] :
% 0.34/0.53 ! [X5: int] :
% 0.34/0.53 ( ( ord_less_int @ Z5 @ X5 )
% 0.34/0.53 => ( ( Q @ X5 )
% 0.34/0.53 = ( Q2 @ X5 ) ) )
% 0.34/0.53 => ? [Z4: int] :
% 0.34/0.53 ! [X4: int] :
% 0.34/0.53 ( ( ord_less_int @ Z4 @ X4 )
% 0.34/0.53 => ( ( ( P @ X4 )
% 0.34/0.53 & ( Q @ X4 ) )
% 0.34/0.53 = ( ( P4 @ X4 )
% 0.34/0.53 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pinf(1)
% 0.34/0.53 thf(fact_1070_verit__comp__simplify1_I1_J,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ~ ( ord_less_real @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % verit_comp_simplify1(1)
% 0.34/0.53 thf(fact_1071_verit__comp__simplify1_I1_J,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % verit_comp_simplify1(1)
% 0.34/0.53 thf(fact_1072_verit__comp__simplify1_I1_J,axiom,
% 0.34/0.53 ! [A2: num] :
% 0.34/0.53 ~ ( ord_less_num @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % verit_comp_simplify1(1)
% 0.34/0.53 thf(fact_1073_verit__comp__simplify1_I1_J,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % verit_comp_simplify1(1)
% 0.34/0.53 thf(fact_1074_verit__comp__simplify1_I1_J,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ~ ( ord_less_int @ A2 @ A2 ) ).
% 0.34/0.53
% 0.34/0.53 % verit_comp_simplify1(1)
% 0.34/0.53 thf(fact_1075_ex__gt__or__lt,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ? [B: real] :
% 0.34/0.53 ( ( ord_less_real @ A2 @ B )
% 0.34/0.53 | ( ord_less_real @ B @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % ex_gt_or_lt
% 0.34/0.53 thf(fact_1076_le__numeral__extra_I3_J,axiom,
% 0.34/0.53 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(3)
% 0.34/0.53 thf(fact_1077_le__numeral__extra_I3_J,axiom,
% 0.34/0.53 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(3)
% 0.34/0.53 thf(fact_1078_le__numeral__extra_I3_J,axiom,
% 0.34/0.53 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(3)
% 0.34/0.53 thf(fact_1079_le__numeral__extra_I3_J,axiom,
% 0.34/0.53 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(3)
% 0.34/0.53 thf(fact_1080_both__member__options__def,axiom,
% 0.34/0.53 ( vEBT_V8194947554948674370ptions
% 0.34/0.53 = ( ^ [T3: vEBT_VEBT,X: nat] :
% 0.34/0.53 ( ( vEBT_V5719532721284313246member @ T3 @ X )
% 0.34/0.53 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % both_member_options_def
% 0.34/0.53 thf(fact_1081_valid__member__both__member__options,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.53 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.34/0.53 => ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % valid_member_both_member_options
% 0.34/0.53 thf(fact_1082_both__member__options__equiv__member,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.53 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.34/0.53 = ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % both_member_options_equiv_member
% 0.34/0.53 thf(fact_1083_Leaf__0__not,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o] :
% 0.34/0.53 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % Leaf_0_not
% 0.34/0.53 thf(fact_1084_dele__bmo__cont__corr,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.53 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
% 0.34/0.53 = ( ( X2 != Y3 )
% 0.34/0.53 & ( vEBT_V8194947554948674370ptions @ T @ Y3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dele_bmo_cont_corr
% 0.34/0.53 thf(fact_1085_maxbmo,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT,X2: nat] :
% 0.34/0.53 ( ( ( vEBT_vebt_maxt @ T )
% 0.34/0.53 = ( some_nat @ X2 ) )
% 0.34/0.53 => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % maxbmo
% 0.34/0.53 thf(fact_1086_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.34/0.53 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.34/0.53 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.34/0.53 = none_P5556105721700978146at_nat ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(2)
% 0.34/0.53 thf(fact_1087_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.34/0.53 ! [Uw: num > num > num,V: num] :
% 0.34/0.53 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.34/0.53 = none_num ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(2)
% 0.34/0.53 thf(fact_1088_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.34/0.53 ! [Uw: nat > nat > nat,V: nat] :
% 0.34/0.53 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.34/0.53 = none_nat ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(2)
% 0.34/0.53 thf(fact_1089_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.34/0.53 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.34/0.53 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 0.34/0.53 = Y3 )
% 0.34/0.53 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.34/0.53 => ( Y3 != none_P5556105721700978146at_nat ) )
% 0.34/0.53 => ( ( ? [V2: product_prod_nat_nat] :
% 0.34/0.53 ( Xa2
% 0.34/0.53 = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.34/0.53 => ( ( Xb = none_P5556105721700978146at_nat )
% 0.34/0.53 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 0.34/0.53 => ~ ! [A: product_prod_nat_nat] :
% 0.34/0.53 ( ( Xa2
% 0.34/0.53 = ( some_P7363390416028606310at_nat @ A ) )
% 0.34/0.53 => ! [B: product_prod_nat_nat] :
% 0.34/0.53 ( ( Xb
% 0.34/0.53 = ( some_P7363390416028606310at_nat @ B ) )
% 0.34/0.53 => ( Y3
% 0.34/0.53 != ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.elims
% 0.34/0.53 thf(fact_1090_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.34/0.53 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 0.34/0.53 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 0.34/0.53 = Y3 )
% 0.34/0.53 => ( ( ( Xa2 = none_num )
% 0.34/0.53 => ( Y3 != none_num ) )
% 0.34/0.53 => ( ( ? [V2: num] :
% 0.34/0.53 ( Xa2
% 0.34/0.53 = ( some_num @ V2 ) )
% 0.34/0.53 => ( ( Xb = none_num )
% 0.34/0.53 => ( Y3 != none_num ) ) )
% 0.34/0.53 => ~ ! [A: num] :
% 0.34/0.53 ( ( Xa2
% 0.34/0.53 = ( some_num @ A ) )
% 0.34/0.53 => ! [B: num] :
% 0.34/0.53 ( ( Xb
% 0.34/0.53 = ( some_num @ B ) )
% 0.34/0.53 => ( Y3
% 0.34/0.53 != ( some_num @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.elims
% 0.34/0.53 thf(fact_1091_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.34/0.53 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.34/0.53 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 0.34/0.53 = Y3 )
% 0.34/0.53 => ( ( ( Xa2 = none_nat )
% 0.34/0.53 => ( Y3 != none_nat ) )
% 0.34/0.53 => ( ( ? [V2: nat] :
% 0.34/0.53 ( Xa2
% 0.34/0.53 = ( some_nat @ V2 ) )
% 0.34/0.53 => ( ( Xb = none_nat )
% 0.34/0.53 => ( Y3 != none_nat ) ) )
% 0.34/0.53 => ~ ! [A: nat] :
% 0.34/0.53 ( ( Xa2
% 0.34/0.53 = ( some_nat @ A ) )
% 0.34/0.53 => ! [B: nat] :
% 0.34/0.53 ( ( Xb
% 0.34/0.53 = ( some_nat @ B ) )
% 0.34/0.53 => ( Y3
% 0.34/0.53 != ( some_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.elims
% 0.34/0.53 thf(fact_1092_Set_Ois__empty__def,axiom,
% 0.34/0.53 ( is_empty_real
% 0.34/0.53 = ( ^ [A6: set_real] : ( A6 = bot_bot_set_real ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Set.is_empty_def
% 0.34/0.53 thf(fact_1093_Set_Ois__empty__def,axiom,
% 0.34/0.53 ( is_empty_o
% 0.34/0.53 = ( ^ [A6: set_o] : ( A6 = bot_bot_set_o ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Set.is_empty_def
% 0.34/0.53 thf(fact_1094_Set_Ois__empty__def,axiom,
% 0.34/0.53 ( is_empty_nat
% 0.34/0.53 = ( ^ [A6: set_nat] : ( A6 = bot_bot_set_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Set.is_empty_def
% 0.34/0.53 thf(fact_1095_Set_Ois__empty__def,axiom,
% 0.34/0.53 ( is_empty_int
% 0.34/0.53 = ( ^ [A6: set_int] : ( A6 = bot_bot_set_int ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Set.is_empty_def
% 0.34/0.53 thf(fact_1096_of__nat__0__less__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_less_iff
% 0.34/0.53 thf(fact_1097_of__nat__0__less__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_less_iff
% 0.34/0.53 thf(fact_1098_of__nat__0__less__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_less_iff
% 0.34/0.53 thf(fact_1099_of__nat__0__less__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_less_iff
% 0.34/0.53 thf(fact_1100_enumerate__mono__iff,axiom,
% 0.34/0.53 ! [S: set_Extended_enat,M: nat,N: nat] :
% 0.34/0.53 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.53 => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_mono_iff
% 0.34/0.53 thf(fact_1101_enumerate__mono__iff,axiom,
% 0.34/0.53 ! [S: set_nat,M: nat,N: nat] :
% 0.34/0.53 ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_mono_iff
% 0.34/0.53 thf(fact_1102_not__min__Null__member,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT] :
% 0.34/0.53 ( ~ ( vEBT_VEBT_minNull @ T )
% 0.34/0.53 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.34/0.53
% 0.34/0.53 % not_min_Null_member
% 0.34/0.53 thf(fact_1103_of__nat__eq__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ( semiri1314217659103216013at_int @ M )
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( M = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_iff
% 0.34/0.53 thf(fact_1104_of__nat__eq__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ( semiri5074537144036343181t_real @ M )
% 0.34/0.53 = ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( M = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_iff
% 0.34/0.53 thf(fact_1105_of__nat__0,axiom,
% 0.34/0.53 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0
% 0.34/0.53 thf(fact_1106_of__nat__0,axiom,
% 0.34/0.53 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0
% 0.34/0.53 thf(fact_1107_of__nat__0,axiom,
% 0.34/0.53 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0
% 0.34/0.53 thf(fact_1108_of__nat__0,axiom,
% 0.34/0.53 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0
% 0.34/0.53 thf(fact_1109_of__nat__0__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( zero_zero_rat
% 0.34/0.53 = ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 = ( zero_zero_nat = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_eq_iff
% 0.34/0.53 thf(fact_1110_of__nat__0__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( zero_zero_nat
% 0.34/0.53 = ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 = ( zero_zero_nat = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_eq_iff
% 0.34/0.53 thf(fact_1111_of__nat__0__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( zero_zero_int
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( zero_zero_nat = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_eq_iff
% 0.34/0.53 thf(fact_1112_of__nat__0__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( zero_zero_real
% 0.34/0.53 = ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( zero_zero_nat = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_eq_iff
% 0.34/0.53 thf(fact_1113_of__nat__eq__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ( semiri681578069525770553at_rat @ M )
% 0.34/0.53 = zero_zero_rat )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_0_iff
% 0.34/0.53 thf(fact_1114_of__nat__eq__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ( semiri1316708129612266289at_nat @ M )
% 0.34/0.53 = zero_zero_nat )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_0_iff
% 0.34/0.53 thf(fact_1115_of__nat__eq__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ( semiri1314217659103216013at_int @ M )
% 0.34/0.53 = zero_zero_int )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_0_iff
% 0.34/0.53 thf(fact_1116_of__nat__eq__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ( semiri5074537144036343181t_real @ M )
% 0.34/0.53 = zero_zero_real )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_0_iff
% 0.34/0.53 thf(fact_1117_of__nat__less__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_iff
% 0.34/0.53 thf(fact_1118_of__nat__less__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_iff
% 0.34/0.53 thf(fact_1119_of__nat__less__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_iff
% 0.34/0.53 thf(fact_1120_of__nat__less__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_iff
% 0.34/0.53 thf(fact_1121_of__nat__le__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_iff
% 0.34/0.53 thf(fact_1122_of__nat__le__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_iff
% 0.34/0.53 thf(fact_1123_of__nat__le__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_iff
% 0.34/0.53 thf(fact_1124_of__nat__le__iff,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_iff
% 0.34/0.53 thf(fact_1125_of__nat__le__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_0_iff
% 0.34/0.53 thf(fact_1126_of__nat__le__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_0_iff
% 0.34/0.53 thf(fact_1127_of__nat__le__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_0_iff
% 0.34/0.53 thf(fact_1128_of__nat__le__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.34/0.53 = ( M = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_le_0_iff
% 0.34/0.53 thf(fact_1129_int__ops_I1_J,axiom,
% 0.34/0.53 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % int_ops(1)
% 0.34/0.53 thf(fact_1130_nat__int__comparison_I2_J,axiom,
% 0.34/0.53 ( ord_less_nat
% 0.34/0.53 = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat_int_comparison(2)
% 0.34/0.53 thf(fact_1131_nat__int__comparison_I3_J,axiom,
% 0.34/0.53 ( ord_less_eq_nat
% 0.34/0.53 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat_int_comparison(3)
% 0.34/0.53 thf(fact_1132_enumerate__in__set,axiom,
% 0.34/0.53 ! [S: set_Extended_enat,N: nat] :
% 0.34/0.53 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.53 => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_in_set
% 0.34/0.53 thf(fact_1133_enumerate__in__set,axiom,
% 0.34/0.53 ! [S: set_nat,N: nat] :
% 0.34/0.53 ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_in_set
% 0.34/0.53 thf(fact_1134_enumerate__Ex,axiom,
% 0.34/0.53 ! [S: set_nat,S2: nat] :
% 0.34/0.53 ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( ( member_nat @ S2 @ S )
% 0.34/0.53 => ? [N3: nat] :
% 0.34/0.53 ( ( infini8530281810654367211te_nat @ S @ N3 )
% 0.34/0.53 = S2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_Ex
% 0.34/0.53 thf(fact_1135_of__nat__0__le__iff,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_le_iff
% 0.34/0.53 thf(fact_1136_of__nat__0__le__iff,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_le_iff
% 0.34/0.53 thf(fact_1137_of__nat__0__le__iff,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_le_iff
% 0.34/0.53 thf(fact_1138_of__nat__0__le__iff,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_0_le_iff
% 0.34/0.53 thf(fact_1139_of__nat__less__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_0_iff
% 0.34/0.53 thf(fact_1140_of__nat__less__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_0_iff
% 0.34/0.53 thf(fact_1141_of__nat__less__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_0_iff
% 0.34/0.53 thf(fact_1142_of__nat__less__0__iff,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_0_iff
% 0.34/0.53 thf(fact_1143_of__nat__less__imp__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_imp_less
% 0.34/0.53 thf(fact_1144_of__nat__less__imp__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_imp_less
% 0.34/0.53 thf(fact_1145_of__nat__less__imp__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_imp_less
% 0.34/0.53 thf(fact_1146_of__nat__less__imp__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_less_imp_less
% 0.34/0.53 thf(fact_1147_less__imp__of__nat__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_imp_of_nat_less
% 0.34/0.53 thf(fact_1148_less__imp__of__nat__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_imp_of_nat_less
% 0.34/0.53 thf(fact_1149_less__imp__of__nat__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_imp_of_nat_less
% 0.34/0.53 thf(fact_1150_less__imp__of__nat__less,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_imp_of_nat_less
% 0.34/0.53 thf(fact_1151_of__nat__mono,axiom,
% 0.34/0.53 ! [I: nat,J: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_mono
% 0.34/0.53 thf(fact_1152_of__nat__mono,axiom,
% 0.34/0.53 ! [I: nat,J: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_mono
% 0.34/0.53 thf(fact_1153_of__nat__mono,axiom,
% 0.34/0.53 ! [I: nat,J: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_mono
% 0.34/0.53 thf(fact_1154_of__nat__mono,axiom,
% 0.34/0.53 ! [I: nat,J: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_mono
% 0.34/0.53 thf(fact_1155_le__enumerate,axiom,
% 0.34/0.53 ! [S: set_nat,N: nat] :
% 0.34/0.53 ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_enumerate
% 0.34/0.53 thf(fact_1156_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.34/0.53 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.34/0.53 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B3 ) )
% 0.34/0.53 = ( some_P7363390416028606310at_nat @ ( F @ A2 @ B3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(3)
% 0.34/0.53 thf(fact_1157_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.34/0.53 ! [F: num > num > num,A2: num,B3: num] :
% 0.34/0.53 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A2 ) @ ( some_num @ B3 ) )
% 0.34/0.53 = ( some_num @ ( F @ A2 @ B3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(3)
% 0.34/0.53 thf(fact_1158_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.34/0.53 ! [F: nat > nat > nat,A2: nat,B3: nat] :
% 0.34/0.53 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A2 ) @ ( some_nat @ B3 ) )
% 0.34/0.53 = ( some_nat @ ( F @ A2 @ B3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(3)
% 0.34/0.53 thf(fact_1159_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.34/0.53 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.34/0.53 = none_P5556105721700978146at_nat ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(1)
% 0.34/0.53 thf(fact_1160_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: num > num > num,Uv: option_num] :
% 0.34/0.53 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.34/0.53 = none_num ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(1)
% 0.34/0.53 thf(fact_1161_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.34/0.53 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.34/0.53 = none_nat ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.option_shift.simps(1)
% 0.34/0.53 thf(fact_1162_vebt__pred_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: $o,Uv: $o] :
% 0.34/0.53 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 0.34/0.53 = none_nat ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_pred.simps(1)
% 0.34/0.53 thf(fact_1163_enumerate__mono,axiom,
% 0.34/0.53 ! [M: nat,N: nat,S: set_Extended_enat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.53 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_mono
% 0.34/0.53 thf(fact_1164_enumerate__mono,axiom,
% 0.34/0.53 ! [M: nat,N: nat,S: set_nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_mono
% 0.34/0.53 thf(fact_1165_vebt__buildup_Osimps_I1_J,axiom,
% 0.34/0.53 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.34/0.53 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_buildup.simps(1)
% 0.34/0.53 thf(fact_1166_vebt__delete_Osimps_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o] :
% 0.34/0.53 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat )
% 0.34/0.53 = ( vEBT_Leaf @ $false @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_delete.simps(1)
% 0.34/0.53 thf(fact_1167_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.34/0.53 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.membermima.simps(1)
% 0.34/0.53 thf(fact_1168_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.34/0.53 ! [Uu: $o] :
% 0.34/0.53 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.minNull.simps(3)
% 0.34/0.53 thf(fact_1169_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.34/0.53 ! [Uv: $o] :
% 0.34/0.53 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.minNull.simps(2)
% 0.34/0.53 thf(fact_1170_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.34/0.53 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.minNull.simps(1)
% 0.34/0.53 thf(fact_1171_deg1Leaf,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.34/0.53 = ( ? [A4: $o,B4: $o] :
% 0.34/0.53 ( T
% 0.34/0.53 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % deg1Leaf
% 0.34/0.53 thf(fact_1172_deg__1__Leaf,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.34/0.53 => ? [A: $o,B: $o] :
% 0.34/0.53 ( T
% 0.34/0.53 = ( vEBT_Leaf @ A @ B ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % deg_1_Leaf
% 0.34/0.53 thf(fact_1173_deg__1__Leafy,axiom,
% 0.34/0.53 ! [T: vEBT_VEBT,N: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.53 => ( ( N = one_one_nat )
% 0.34/0.53 => ? [A: $o,B: $o] :
% 0.34/0.53 ( T
% 0.34/0.53 = ( vEBT_Leaf @ A @ B ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % deg_1_Leafy
% 0.34/0.53 thf(fact_1174_pos__int__cases,axiom,
% 0.34/0.53 ! [K: int] :
% 0.34/0.53 ( ( ord_less_int @ zero_zero_int @ K )
% 0.34/0.53 => ~ ! [N3: nat] :
% 0.34/0.53 ( ( K
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N3 ) )
% 0.34/0.53 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % pos_int_cases
% 0.34/0.53 thf(fact_1175_zero__less__imp__eq__int,axiom,
% 0.34/0.53 ! [K: int] :
% 0.34/0.53 ( ( ord_less_int @ zero_zero_int @ K )
% 0.34/0.53 => ? [N3: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.34/0.53 & ( K
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % zero_less_imp_eq_int
% 0.34/0.53 thf(fact_1176_of__nat__1,axiom,
% 0.34/0.53 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.34/0.53 = one_one_complex ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1
% 0.34/0.53 thf(fact_1177_of__nat__1,axiom,
% 0.34/0.53 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.34/0.53 = one_one_rat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1
% 0.34/0.53 thf(fact_1178_of__nat__1,axiom,
% 0.34/0.53 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.34/0.53 = one_one_nat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1
% 0.34/0.53 thf(fact_1179_of__nat__1,axiom,
% 0.34/0.53 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.34/0.53 = one_one_int ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1
% 0.34/0.53 thf(fact_1180_of__nat__1,axiom,
% 0.34/0.53 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.34/0.53 = one_one_real ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1
% 0.34/0.53 thf(fact_1181_of__nat__1__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( one_one_complex
% 0.34/0.53 = ( semiri8010041392384452111omplex @ N ) )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1_eq_iff
% 0.34/0.53 thf(fact_1182_of__nat__1__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( one_one_rat
% 0.34/0.53 = ( semiri681578069525770553at_rat @ N ) )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1_eq_iff
% 0.34/0.53 thf(fact_1183_of__nat__1__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( one_one_nat
% 0.34/0.53 = ( semiri1316708129612266289at_nat @ N ) )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1_eq_iff
% 0.34/0.53 thf(fact_1184_of__nat__1__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( one_one_int
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1_eq_iff
% 0.34/0.53 thf(fact_1185_of__nat__1__eq__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( one_one_real
% 0.34/0.53 = ( semiri5074537144036343181t_real @ N ) )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_1_eq_iff
% 0.34/0.53 thf(fact_1186_of__nat__eq__1__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ( semiri8010041392384452111omplex @ N )
% 0.34/0.53 = one_one_complex )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_1_iff
% 0.34/0.53 thf(fact_1187_of__nat__eq__1__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ( semiri681578069525770553at_rat @ N )
% 0.34/0.53 = one_one_rat )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_1_iff
% 0.34/0.53 thf(fact_1188_of__nat__eq__1__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ( semiri1316708129612266289at_nat @ N )
% 0.34/0.53 = one_one_nat )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_1_iff
% 0.34/0.53 thf(fact_1189_of__nat__eq__1__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ( semiri1314217659103216013at_int @ N )
% 0.34/0.53 = one_one_int )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_1_iff
% 0.34/0.53 thf(fact_1190_of__nat__eq__1__iff,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ( semiri5074537144036343181t_real @ N )
% 0.34/0.53 = one_one_real )
% 0.34/0.53 = ( N = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_eq_1_iff
% 0.34/0.53 thf(fact_1191_less__one,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ N @ one_one_nat )
% 0.34/0.53 = ( N = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_one
% 0.34/0.53 thf(fact_1192_nonneg__int__cases,axiom,
% 0.34/0.53 ! [K: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.34/0.53 => ~ ! [N3: nat] :
% 0.34/0.53 ( K
% 0.34/0.53 != ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nonneg_int_cases
% 0.34/0.53 thf(fact_1193_zero__le__imp__eq__int,axiom,
% 0.34/0.53 ! [K: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.34/0.53 => ? [N3: nat] :
% 0.34/0.53 ( K
% 0.34/0.53 = ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % zero_le_imp_eq_int
% 0.34/0.53 thf(fact_1194_less__eq__int__code_I1_J,axiom,
% 0.34/0.53 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.34/0.53
% 0.34/0.53 % less_eq_int_code(1)
% 0.34/0.53 thf(fact_1195_one__reorient,axiom,
% 0.34/0.53 ! [X2: complex] :
% 0.34/0.53 ( ( one_one_complex = X2 )
% 0.34/0.53 = ( X2 = one_one_complex ) ) ).
% 0.34/0.53
% 0.34/0.53 % one_reorient
% 0.34/0.53 thf(fact_1196_one__reorient,axiom,
% 0.34/0.53 ! [X2: real] :
% 0.34/0.53 ( ( one_one_real = X2 )
% 0.34/0.53 = ( X2 = one_one_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % one_reorient
% 0.34/0.53 thf(fact_1197_one__reorient,axiom,
% 0.34/0.53 ! [X2: rat] :
% 0.34/0.53 ( ( one_one_rat = X2 )
% 0.34/0.53 = ( X2 = one_one_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % one_reorient
% 0.34/0.53 thf(fact_1198_one__reorient,axiom,
% 0.34/0.53 ! [X2: nat] :
% 0.34/0.53 ( ( one_one_nat = X2 )
% 0.34/0.53 = ( X2 = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % one_reorient
% 0.34/0.53 thf(fact_1199_one__reorient,axiom,
% 0.34/0.53 ! [X2: int] :
% 0.34/0.53 ( ( one_one_int = X2 )
% 0.34/0.53 = ( X2 = one_one_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % one_reorient
% 0.34/0.53 thf(fact_1200_less__int__code_I1_J,axiom,
% 0.34/0.53 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % less_int_code(1)
% 0.34/0.53 thf(fact_1201_verit__la__generic,axiom,
% 0.34/0.53 ! [A2: int,X2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ X2 )
% 0.34/0.53 | ( A2 = X2 )
% 0.34/0.53 | ( ord_less_eq_int @ X2 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_la_generic
% 0.34/0.53 thf(fact_1202_imp__le__cong,axiom,
% 0.34/0.53 ! [X2: int,X7: int,P: $o,P4: $o] :
% 0.34/0.53 ( ( X2 = X7 )
% 0.34/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.34/0.53 => ( P = P4 ) )
% 0.34/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.34/0.53 => P )
% 0.34/0.53 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.34/0.53 => P4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % imp_le_cong
% 0.34/0.53 thf(fact_1203_conj__le__cong,axiom,
% 0.34/0.53 ! [X2: int,X7: int,P: $o,P4: $o] :
% 0.34/0.53 ( ( X2 = X7 )
% 0.34/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.34/0.53 => ( P = P4 ) )
% 0.34/0.53 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.34/0.53 & P )
% 0.34/0.53 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 0.34/0.53 & P4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % conj_le_cong
% 0.34/0.53 thf(fact_1204_le__numeral__extra_I4_J,axiom,
% 0.34/0.53 ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(4)
% 0.34/0.53 thf(fact_1205_le__numeral__extra_I4_J,axiom,
% 0.34/0.53 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(4)
% 0.34/0.53 thf(fact_1206_le__numeral__extra_I4_J,axiom,
% 0.34/0.53 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(4)
% 0.34/0.53 thf(fact_1207_le__numeral__extra_I4_J,axiom,
% 0.34/0.53 ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.34/0.53
% 0.34/0.53 % le_numeral_extra(4)
% 0.34/0.53 thf(fact_1208_less__numeral__extra_I4_J,axiom,
% 0.34/0.53 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(4)
% 0.34/0.53 thf(fact_1209_less__numeral__extra_I4_J,axiom,
% 0.34/0.53 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(4)
% 0.34/0.53 thf(fact_1210_less__numeral__extra_I4_J,axiom,
% 0.34/0.53 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(4)
% 0.34/0.53 thf(fact_1211_less__numeral__extra_I4_J,axiom,
% 0.34/0.53 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(4)
% 0.34/0.53 thf(fact_1212_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.34/0.53 ! [Uu: $o,Uv: $o,D: nat] :
% 0.34/0.53 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.34/0.53 = ( D = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.valid'.simps(1)
% 0.34/0.53 thf(fact_1213_less__numeral__extra_I1_J,axiom,
% 0.34/0.53 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(1)
% 0.34/0.53 thf(fact_1214_less__numeral__extra_I1_J,axiom,
% 0.34/0.53 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(1)
% 0.34/0.53 thf(fact_1215_less__numeral__extra_I1_J,axiom,
% 0.34/0.53 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(1)
% 0.34/0.53 thf(fact_1216_less__numeral__extra_I1_J,axiom,
% 0.34/0.53 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.34/0.53
% 0.34/0.53 % less_numeral_extra(1)
% 0.34/0.53 thf(fact_1217_vebt__member_Osimps_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o,X2: nat] :
% 0.34/0.53 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.34/0.53 = ( ( ( X2 = zero_zero_nat )
% 0.34/0.53 => A2 )
% 0.34/0.53 & ( ( X2 != zero_zero_nat )
% 0.34/0.53 => ( ( ( X2 = one_one_nat )
% 0.34/0.53 => B3 )
% 0.34/0.53 & ( X2 = one_one_nat ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_member.simps(1)
% 0.34/0.53 thf(fact_1218_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o,X2: nat] :
% 0.34/0.53 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.34/0.53 = ( ( ( X2 = zero_zero_nat )
% 0.34/0.53 => A2 )
% 0.34/0.53 & ( ( X2 != zero_zero_nat )
% 0.34/0.53 => ( ( ( X2 = one_one_nat )
% 0.34/0.53 => B3 )
% 0.34/0.53 & ( X2 = one_one_nat ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT_internal.naive_member.simps(1)
% 0.34/0.53 thf(fact_1219_zle__int,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % zle_int
% 0.34/0.53 thf(fact_1220_vebt__mint_Osimps_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.53 & ( ~ A2
% 0.34/0.53 => ( ( B3
% 0.34/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.53 & ( ~ B3
% 0.34/0.53 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = none_nat ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_mint.simps(1)
% 0.34/0.53 thf(fact_1221_vebt__maxt_Osimps_I1_J,axiom,
% 0.34/0.53 ! [B3: $o,A2: $o] :
% 0.34/0.53 ( ( B3
% 0.34/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.53 & ( ~ B3
% 0.34/0.53 => ( ( A2
% 0.34/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.53 & ( ~ A2
% 0.34/0.53 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 0.34/0.53 = none_nat ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_maxt.simps(1)
% 0.34/0.53 thf(fact_1222_vebt__succ_Osimps_I1_J,axiom,
% 0.34/0.53 ! [B3: $o,Uu: $o] :
% 0.34/0.53 ( ( B3
% 0.34/0.53 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
% 0.34/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.53 & ( ~ B3
% 0.34/0.53 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
% 0.34/0.53 = none_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_succ.simps(1)
% 0.34/0.53 thf(fact_1223_not__one__less__zero,axiom,
% 0.34/0.53 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_less_zero
% 0.34/0.53 thf(fact_1224_not__one__less__zero,axiom,
% 0.34/0.53 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_less_zero
% 0.34/0.53 thf(fact_1225_not__one__less__zero,axiom,
% 0.34/0.53 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_less_zero
% 0.34/0.53 thf(fact_1226_not__one__less__zero,axiom,
% 0.34/0.53 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_less_zero
% 0.34/0.53 thf(fact_1227_zero__less__one,axiom,
% 0.34/0.53 ord_less_real @ zero_zero_real @ one_one_real ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one
% 0.34/0.53 thf(fact_1228_zero__less__one,axiom,
% 0.34/0.53 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one
% 0.34/0.53 thf(fact_1229_zero__less__one,axiom,
% 0.34/0.53 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one
% 0.34/0.53 thf(fact_1230_zero__less__one,axiom,
% 0.34/0.53 ord_less_int @ zero_zero_int @ one_one_int ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one
% 0.34/0.53 thf(fact_1231_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one_class.zero_le_one
% 0.34/0.53 thf(fact_1232_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one_class.zero_le_one
% 0.34/0.53 thf(fact_1233_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one_class.zero_le_one
% 0.34/0.53 thf(fact_1234_zero__less__one__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.34/0.53
% 0.34/0.53 % zero_less_one_class.zero_le_one
% 0.34/0.53 thf(fact_1235_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.34/0.53
% 0.34/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.53 thf(fact_1236_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.53 thf(fact_1237_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.53 thf(fact_1238_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.34/0.53 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.34/0.53
% 0.34/0.53 % linordered_nonzero_semiring_class.zero_le_one
% 0.34/0.53 thf(fact_1239_not__one__le__zero,axiom,
% 0.34/0.53 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_le_zero
% 0.34/0.53 thf(fact_1240_not__one__le__zero,axiom,
% 0.34/0.53 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_le_zero
% 0.34/0.53 thf(fact_1241_not__one__le__zero,axiom,
% 0.34/0.53 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_le_zero
% 0.34/0.53 thf(fact_1242_not__one__le__zero,axiom,
% 0.34/0.53 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % not_one_le_zero
% 0.34/0.53 thf(fact_1243_reals__Archimedean2,axiom,
% 0.34/0.53 ! [X2: rat] :
% 0.34/0.53 ? [N3: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % reals_Archimedean2
% 0.34/0.53 thf(fact_1244_reals__Archimedean2,axiom,
% 0.34/0.53 ! [X2: real] :
% 0.34/0.53 ? [N3: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % reals_Archimedean2
% 0.34/0.53 thf(fact_1245_real__arch__simple,axiom,
% 0.34/0.53 ! [X2: real] :
% 0.34/0.53 ? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % real_arch_simple
% 0.34/0.53 thf(fact_1246_real__arch__simple,axiom,
% 0.34/0.53 ! [X2: rat] :
% 0.34/0.53 ? [N3: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % real_arch_simple
% 0.34/0.53 thf(fact_1247_vebt__pred_Osimps_I3_J,axiom,
% 0.34/0.53 ! [B3: $o,A2: $o,Va: nat] :
% 0.34/0.53 ( ( B3
% 0.34/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 0.34/0.53 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.53 & ( ~ B3
% 0.34/0.53 => ( ( A2
% 0.34/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 0.34/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.53 & ( ~ A2
% 0.34/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 0.34/0.53 = none_nat ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_pred.simps(3)
% 0.34/0.53 thf(fact_1248_zero__neq__one,axiom,
% 0.34/0.53 zero_zero_complex != one_one_complex ).
% 0.34/0.53
% 0.34/0.53 % zero_neq_one
% 0.34/0.53 thf(fact_1249_zero__neq__one,axiom,
% 0.34/0.53 zero_zero_real != one_one_real ).
% 0.34/0.53
% 0.34/0.53 % zero_neq_one
% 0.34/0.53 thf(fact_1250_zero__neq__one,axiom,
% 0.34/0.53 zero_zero_rat != one_one_rat ).
% 0.34/0.53
% 0.34/0.53 % zero_neq_one
% 0.34/0.53 thf(fact_1251_zero__neq__one,axiom,
% 0.34/0.53 zero_zero_nat != one_one_nat ).
% 0.34/0.53
% 0.34/0.53 % zero_neq_one
% 0.34/0.53 thf(fact_1252_zero__neq__one,axiom,
% 0.34/0.53 zero_zero_int != one_one_int ).
% 0.34/0.53
% 0.34/0.53 % zero_neq_one
% 0.34/0.53 thf(fact_1253_dbl__inc__simps_I2_J,axiom,
% 0.34/0.53 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 0.34/0.53 = one_one_complex ) ).
% 0.34/0.53
% 0.34/0.53 % dbl_inc_simps(2)
% 0.34/0.53 thf(fact_1254_dbl__inc__simps_I2_J,axiom,
% 0.34/0.53 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 0.34/0.53 = one_one_real ) ).
% 0.34/0.53
% 0.34/0.53 % dbl_inc_simps(2)
% 0.34/0.53 thf(fact_1255_dbl__inc__simps_I2_J,axiom,
% 0.34/0.53 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 0.34/0.53 = one_one_rat ) ).
% 0.34/0.53
% 0.34/0.53 % dbl_inc_simps(2)
% 0.34/0.53 thf(fact_1256_dbl__inc__simps_I2_J,axiom,
% 0.34/0.53 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 0.34/0.53 = one_one_int ) ).
% 0.34/0.53
% 0.34/0.53 % dbl_inc_simps(2)
% 0.34/0.53 thf(fact_1257_VEBT_Osize__gen_I2_J,axiom,
% 0.34/0.53 ! [X21: $o,X222: $o] :
% 0.34/0.53 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % VEBT.size_gen(2)
% 0.34/0.53 thf(fact_1258_old_Onat_Oinject,axiom,
% 0.34/0.53 ! [Nat: nat,Nat2: nat] :
% 0.34/0.53 ( ( ( suc @ Nat )
% 0.34/0.53 = ( suc @ Nat2 ) )
% 0.34/0.53 = ( Nat = Nat2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % old.nat.inject
% 0.34/0.53 thf(fact_1259_nat_Oinject,axiom,
% 0.34/0.53 ! [X22: nat,Y2: nat] :
% 0.34/0.53 ( ( ( suc @ X22 )
% 0.34/0.53 = ( suc @ Y2 ) )
% 0.34/0.53 = ( X22 = Y2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat.inject
% 0.34/0.53 thf(fact_1260_Suc__less__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_less_eq
% 0.34/0.53 thf(fact_1261_Suc__mono,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_mono
% 0.34/0.53 thf(fact_1262_lessI,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % lessI
% 0.34/0.53 thf(fact_1263_Suc__le__mono,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 0.34/0.53 = ( ord_less_eq_nat @ N @ M ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_le_mono
% 0.34/0.53 thf(fact_1264_less__Suc0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.34/0.53 = ( N = zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_Suc0
% 0.34/0.53 thf(fact_1265_zero__less__Suc,axiom,
% 0.34/0.53 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % zero_less_Suc
% 0.34/0.53 thf(fact_1266_n__not__Suc__n,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( N
% 0.34/0.53 != ( suc @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % n_not_Suc_n
% 0.34/0.53 thf(fact_1267_Suc__inject,axiom,
% 0.34/0.53 ! [X2: nat,Y3: nat] :
% 0.34/0.53 ( ( ( suc @ X2 )
% 0.34/0.53 = ( suc @ Y3 ) )
% 0.34/0.53 => ( X2 = Y3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_inject
% 0.34/0.53 thf(fact_1268_exists__least__lemma,axiom,
% 0.34/0.53 ! [P: nat > $o] :
% 0.34/0.53 ( ~ ( P @ zero_zero_nat )
% 0.34/0.53 => ( ? [X_12: nat] : ( P @ X_12 )
% 0.34/0.53 => ? [N3: nat] :
% 0.34/0.53 ( ~ ( P @ N3 )
% 0.34/0.53 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % exists_least_lemma
% 0.34/0.53 thf(fact_1269_nat_Odistinct_I1_J,axiom,
% 0.34/0.53 ! [X22: nat] :
% 0.34/0.53 ( zero_zero_nat
% 0.34/0.53 != ( suc @ X22 ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat.distinct(1)
% 0.34/0.53 thf(fact_1270_old_Onat_Odistinct_I2_J,axiom,
% 0.34/0.53 ! [Nat2: nat] :
% 0.34/0.53 ( ( suc @ Nat2 )
% 0.34/0.53 != zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % old.nat.distinct(2)
% 0.34/0.53 thf(fact_1271_old_Onat_Odistinct_I1_J,axiom,
% 0.34/0.53 ! [Nat2: nat] :
% 0.34/0.53 ( zero_zero_nat
% 0.34/0.53 != ( suc @ Nat2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % old.nat.distinct(1)
% 0.34/0.53 thf(fact_1272_nat_OdiscI,axiom,
% 0.34/0.53 ! [Nat: nat,X22: nat] :
% 0.34/0.53 ( ( Nat
% 0.34/0.53 = ( suc @ X22 ) )
% 0.34/0.53 => ( Nat != zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat.discI
% 0.34/0.53 thf(fact_1273_old_Onat_Oexhaust,axiom,
% 0.34/0.53 ! [Y3: nat] :
% 0.34/0.53 ( ( Y3 != zero_zero_nat )
% 0.34/0.53 => ~ ! [Nat3: nat] :
% 0.34/0.53 ( Y3
% 0.34/0.53 != ( suc @ Nat3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % old.nat.exhaust
% 0.34/0.53 thf(fact_1274_nat__induct,axiom,
% 0.34/0.53 ! [P: nat > $o,N: nat] :
% 0.34/0.53 ( ( P @ zero_zero_nat )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( P @ N3 )
% 0.34/0.53 => ( P @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( P @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat_induct
% 0.34/0.53 thf(fact_1275_diff__induct,axiom,
% 0.34/0.53 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.34/0.53 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 0.34/0.53 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 0.34/0.53 => ( ! [X5: nat,Y4: nat] :
% 0.34/0.53 ( ( P @ X5 @ Y4 )
% 0.34/0.53 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 0.34/0.53 => ( P @ M @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_induct
% 0.34/0.53 thf(fact_1276_zero__induct,axiom,
% 0.34/0.53 ! [P: nat > $o,K: nat] :
% 0.34/0.53 ( ( P @ K )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( P @ ( suc @ N3 ) )
% 0.34/0.53 => ( P @ N3 ) )
% 0.34/0.53 => ( P @ zero_zero_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % zero_induct
% 0.34/0.53 thf(fact_1277_Suc__neq__Zero,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( suc @ M )
% 0.34/0.53 != zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_neq_Zero
% 0.34/0.53 thf(fact_1278_Zero__neq__Suc,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( zero_zero_nat
% 0.34/0.53 != ( suc @ M ) ) ).
% 0.34/0.53
% 0.34/0.53 % Zero_neq_Suc
% 0.34/0.53 thf(fact_1279_Zero__not__Suc,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( zero_zero_nat
% 0.34/0.53 != ( suc @ M ) ) ).
% 0.34/0.53
% 0.34/0.53 % Zero_not_Suc
% 0.34/0.53 thf(fact_1280_not0__implies__Suc,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( N != zero_zero_nat )
% 0.34/0.53 => ? [M4: nat] :
% 0.34/0.53 ( N
% 0.34/0.53 = ( suc @ M4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % not0_implies_Suc
% 0.34/0.53 thf(fact_1281_vebt__buildup_Ocases,axiom,
% 0.34/0.53 ! [X2: nat] :
% 0.34/0.53 ( ( X2 != zero_zero_nat )
% 0.34/0.53 => ( ( X2
% 0.34/0.53 != ( suc @ zero_zero_nat ) )
% 0.34/0.53 => ~ ! [Va2: nat] :
% 0.34/0.53 ( X2
% 0.34/0.53 != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_buildup.cases
% 0.34/0.53 thf(fact_1282_not__less__less__Suc__eq,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ~ ( ord_less_nat @ N @ M )
% 0.34/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.34/0.53 = ( N = M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % not_less_less_Suc_eq
% 0.34/0.53 thf(fact_1283_strict__inc__induct,axiom,
% 0.34/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_nat @ I @ J )
% 0.34/0.53 => ( ! [I2: nat] :
% 0.34/0.53 ( ( J
% 0.34/0.53 = ( suc @ I2 ) )
% 0.34/0.53 => ( P @ I2 ) )
% 0.34/0.53 => ( ! [I2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I2 @ J )
% 0.34/0.53 => ( ( P @ ( suc @ I2 ) )
% 0.34/0.53 => ( P @ I2 ) ) )
% 0.34/0.53 => ( P @ I ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % strict_inc_induct
% 0.34/0.53 thf(fact_1284_less__Suc__induct,axiom,
% 0.34/0.53 ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.34/0.53 ( ( ord_less_nat @ I @ J )
% 0.34/0.53 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 0.34/0.53 => ( ! [I2: nat,J2: nat,K2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I2 @ J2 )
% 0.34/0.53 => ( ( ord_less_nat @ J2 @ K2 )
% 0.34/0.53 => ( ( P @ I2 @ J2 )
% 0.34/0.53 => ( ( P @ J2 @ K2 )
% 0.34/0.53 => ( P @ I2 @ K2 ) ) ) ) )
% 0.34/0.53 => ( P @ I @ J ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_Suc_induct
% 0.34/0.53 thf(fact_1285_less__trans__Suc,axiom,
% 0.34/0.53 ! [I: nat,J: nat,K: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I @ J )
% 0.34/0.53 => ( ( ord_less_nat @ J @ K )
% 0.34/0.53 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_trans_Suc
% 0.34/0.53 thf(fact_1286_Suc__less__SucD,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_less_SucD
% 0.34/0.53 thf(fact_1287_less__antisym,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ~ ( ord_less_nat @ N @ M )
% 0.34/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.34/0.53 => ( M = N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_antisym
% 0.34/0.53 thf(fact_1288_Suc__less__eq2,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.34/0.53 = ( ? [M6: nat] :
% 0.34/0.53 ( ( M
% 0.34/0.53 = ( suc @ M6 ) )
% 0.34/0.53 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_less_eq2
% 0.34/0.53 thf(fact_1289_All__less__Suc,axiom,
% 0.34/0.53 ! [N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ! [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.34/0.53 => ( P @ I4 ) ) )
% 0.34/0.53 = ( ( P @ N )
% 0.34/0.53 & ! [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.34/0.53 => ( P @ I4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % All_less_Suc
% 0.34/0.53 thf(fact_1290_not__less__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ~ ( ord_less_nat @ M @ N ) )
% 0.34/0.53 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % not_less_eq
% 0.34/0.53 thf(fact_1291_less__Suc__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.34/0.53 = ( ( ord_less_nat @ M @ N )
% 0.34/0.53 | ( M = N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_Suc_eq
% 0.34/0.53 thf(fact_1292_Ex__less__Suc,axiom,
% 0.34/0.53 ! [N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ? [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.34/0.53 & ( P @ I4 ) ) )
% 0.34/0.53 = ( ( P @ N )
% 0.34/0.53 | ? [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.34/0.53 & ( P @ I4 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Ex_less_Suc
% 0.34/0.53 thf(fact_1293_less__SucI,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_SucI
% 0.34/0.53 thf(fact_1294_less__SucE,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.34/0.53 => ( ~ ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( M = N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_SucE
% 0.34/0.53 thf(fact_1295_Suc__lessI,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ( ( suc @ M )
% 0.34/0.53 != N )
% 0.34/0.53 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_lessI
% 0.34/0.53 thf(fact_1296_Suc__lessE,axiom,
% 0.34/0.53 ! [I: nat,K: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.34/0.53 => ~ ! [J2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I @ J2 )
% 0.34/0.53 => ( K
% 0.34/0.53 != ( suc @ J2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_lessE
% 0.34/0.53 thf(fact_1297_Suc__lessD,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_lessD
% 0.34/0.53 thf(fact_1298_Nat_OlessE,axiom,
% 0.34/0.53 ! [I: nat,K: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I @ K )
% 0.34/0.53 => ( ( K
% 0.34/0.53 != ( suc @ I ) )
% 0.34/0.53 => ~ ! [J2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I @ J2 )
% 0.34/0.53 => ( K
% 0.34/0.53 != ( suc @ J2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Nat.lessE
% 0.34/0.53 thf(fact_1299_Suc__leD,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.34/0.53 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_leD
% 0.34/0.53 thf(fact_1300_le__SucE,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.34/0.53 => ( ~ ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( M
% 0.34/0.53 = ( suc @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_SucE
% 0.34/0.53 thf(fact_1301_le__SucI,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_SucI
% 0.34/0.53 thf(fact_1302_Suc__le__D,axiom,
% 0.34/0.53 ! [N: nat,M7: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 0.34/0.53 => ? [M4: nat] :
% 0.34/0.53 ( M7
% 0.34/0.53 = ( suc @ M4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_le_D
% 0.34/0.53 thf(fact_1303_le__Suc__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.34/0.53 = ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 | ( M
% 0.34/0.53 = ( suc @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_Suc_eq
% 0.34/0.53 thf(fact_1304_Suc__n__not__le__n,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_n_not_le_n
% 0.34/0.53 thf(fact_1305_not__less__eq__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 0.34/0.53
% 0.34/0.53 % not_less_eq_eq
% 0.34/0.53 thf(fact_1306_full__nat__induct,axiom,
% 0.34/0.53 ! [P: nat > $o,N: nat] :
% 0.34/0.53 ( ! [N3: nat] :
% 0.34/0.53 ( ! [M3: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 0.34/0.53 => ( P @ M3 ) )
% 0.34/0.53 => ( P @ N3 ) )
% 0.34/0.53 => ( P @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % full_nat_induct
% 0.34/0.53 thf(fact_1307_nat__induct__at__least,axiom,
% 0.34/0.53 ! [M: nat,N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( ( P @ M )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N3 )
% 0.34/0.53 => ( ( P @ N3 )
% 0.34/0.53 => ( P @ ( suc @ N3 ) ) ) )
% 0.34/0.53 => ( P @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat_induct_at_least
% 0.34/0.53 thf(fact_1308_transitive__stepwise__le,axiom,
% 0.34/0.53 ! [M: nat,N: nat,R: nat > nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( ! [X5: nat] : ( R @ X5 @ X5 )
% 0.34/0.53 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 0.34/0.53 ( ( R @ X5 @ Y4 )
% 0.34/0.53 => ( ( R @ Y4 @ Z4 )
% 0.34/0.53 => ( R @ X5 @ Z4 ) ) )
% 0.34/0.53 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.34/0.53 => ( R @ M @ N ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % transitive_stepwise_le
% 0.34/0.53 thf(fact_1309_lift__Suc__mono__less__iff,axiom,
% 0.34/0.53 ! [F: nat > real,N: nat,M: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less_iff
% 0.34/0.53 thf(fact_1310_lift__Suc__mono__less__iff,axiom,
% 0.34/0.53 ! [F: nat > rat,N: nat,M: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less_iff
% 0.34/0.53 thf(fact_1311_lift__Suc__mono__less__iff,axiom,
% 0.34/0.53 ! [F: nat > num,N: nat,M: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less_iff
% 0.34/0.53 thf(fact_1312_lift__Suc__mono__less__iff,axiom,
% 0.34/0.53 ! [F: nat > nat,N: nat,M: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less_iff
% 0.34/0.53 thf(fact_1313_lift__Suc__mono__less__iff,axiom,
% 0.34/0.53 ! [F: nat > int,N: nat,M: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 0.34/0.53 = ( ord_less_nat @ N @ M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less_iff
% 0.34/0.53 thf(fact_1314_lift__Suc__mono__less,axiom,
% 0.34/0.53 ! [F: nat > real,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less
% 0.34/0.53 thf(fact_1315_lift__Suc__mono__less,axiom,
% 0.34/0.53 ! [F: nat > rat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less
% 0.34/0.53 thf(fact_1316_lift__Suc__mono__less,axiom,
% 0.34/0.53 ! [F: nat > num,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less
% 0.34/0.53 thf(fact_1317_lift__Suc__mono__less,axiom,
% 0.34/0.53 ! [F: nat > nat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less
% 0.34/0.53 thf(fact_1318_lift__Suc__mono__less,axiom,
% 0.34/0.53 ! [F: nat > int,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_less
% 0.34/0.53 thf(fact_1319_lift__Suc__mono__le,axiom,
% 0.34/0.53 ! [F: nat > set_int,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_le
% 0.34/0.53 thf(fact_1320_lift__Suc__mono__le,axiom,
% 0.34/0.53 ! [F: nat > rat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_le
% 0.34/0.53 thf(fact_1321_lift__Suc__mono__le,axiom,
% 0.34/0.53 ! [F: nat > num,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_le
% 0.34/0.53 thf(fact_1322_lift__Suc__mono__le,axiom,
% 0.34/0.53 ! [F: nat > nat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_le
% 0.34/0.53 thf(fact_1323_lift__Suc__mono__le,axiom,
% 0.34/0.53 ! [F: nat > int,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_mono_le
% 0.34/0.53 thf(fact_1324_lift__Suc__antimono__le,axiom,
% 0.34/0.53 ! [F: nat > set_int,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_antimono_le
% 0.34/0.53 thf(fact_1325_lift__Suc__antimono__le,axiom,
% 0.34/0.53 ! [F: nat > rat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_antimono_le
% 0.34/0.53 thf(fact_1326_lift__Suc__antimono__le,axiom,
% 0.34/0.53 ! [F: nat > num,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_antimono_le
% 0.34/0.53 thf(fact_1327_lift__Suc__antimono__le,axiom,
% 0.34/0.53 ! [F: nat > nat,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_antimono_le
% 0.34/0.53 thf(fact_1328_lift__Suc__antimono__le,axiom,
% 0.34/0.53 ! [F: nat > int,N: nat,N7: nat] :
% 0.34/0.53 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.34/0.53 => ( ( ord_less_eq_nat @ N @ N7 )
% 0.34/0.53 => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % lift_Suc_antimono_le
% 0.34/0.53 thf(fact_1329_of__nat__neq__0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 0.34/0.53 != zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_neq_0
% 0.34/0.53 thf(fact_1330_of__nat__neq__0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 0.34/0.53 != zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_neq_0
% 0.34/0.53 thf(fact_1331_of__nat__neq__0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 0.34/0.53 != zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_neq_0
% 0.34/0.53 thf(fact_1332_of__nat__neq__0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 0.34/0.53 != zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % of_nat_neq_0
% 0.34/0.53 thf(fact_1333_Ex__less__Suc2,axiom,
% 0.34/0.53 ! [N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ? [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.34/0.53 & ( P @ I4 ) ) )
% 0.34/0.53 = ( ( P @ zero_zero_nat )
% 0.34/0.53 | ? [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.34/0.53 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % Ex_less_Suc2
% 0.34/0.53 thf(fact_1334_gr0__conv__Suc,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.53 = ( ? [M2: nat] :
% 0.34/0.53 ( N
% 0.34/0.53 = ( suc @ M2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % gr0_conv_Suc
% 0.34/0.53 thf(fact_1335_All__less__Suc2,axiom,
% 0.34/0.53 ! [N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ! [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 0.34/0.53 => ( P @ I4 ) ) )
% 0.34/0.53 = ( ( P @ zero_zero_nat )
% 0.34/0.53 & ! [I4: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I4 @ N )
% 0.34/0.53 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % All_less_Suc2
% 0.34/0.53 thf(fact_1336_gr0__implies__Suc,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.53 => ? [M4: nat] :
% 0.34/0.53 ( N
% 0.34/0.53 = ( suc @ M4 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % gr0_implies_Suc
% 0.34/0.53 thf(fact_1337_less__Suc__eq__0__disj,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.34/0.53 = ( ( M = zero_zero_nat )
% 0.34/0.53 | ? [J3: nat] :
% 0.34/0.53 ( ( M
% 0.34/0.53 = ( suc @ J3 ) )
% 0.34/0.53 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_Suc_eq_0_disj
% 0.34/0.53 thf(fact_1338_Suc__leI,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ N )
% 0.34/0.53 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_leI
% 0.34/0.53 thf(fact_1339_Suc__le__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_le_eq
% 0.34/0.53 thf(fact_1340_dec__induct,axiom,
% 0.34/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ( P @ I )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ N3 )
% 0.34/0.53 => ( ( ord_less_nat @ N3 @ J )
% 0.34/0.53 => ( ( P @ N3 )
% 0.34/0.53 => ( P @ ( suc @ N3 ) ) ) ) )
% 0.34/0.53 => ( P @ J ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % dec_induct
% 0.34/0.53 thf(fact_1341_inc__induct,axiom,
% 0.34/0.53 ! [I: nat,J: nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ J )
% 0.34/0.53 => ( ( P @ J )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ N3 )
% 0.34/0.53 => ( ( ord_less_nat @ N3 @ J )
% 0.34/0.53 => ( ( P @ ( suc @ N3 ) )
% 0.34/0.53 => ( P @ N3 ) ) ) )
% 0.34/0.53 => ( P @ I ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % inc_induct
% 0.34/0.53 thf(fact_1342_Suc__le__lessD,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.34/0.53 => ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_le_lessD
% 0.34/0.53 thf(fact_1343_le__less__Suc__eq,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.34/0.53 = ( N = M ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_less_Suc_eq
% 0.34/0.53 thf(fact_1344_less__Suc__eq__le,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 0.34/0.53 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_Suc_eq_le
% 0.34/0.53 thf(fact_1345_less__eq__Suc__le,axiom,
% 0.34/0.53 ( ord_less_nat
% 0.34/0.53 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_eq_Suc_le
% 0.34/0.53 thf(fact_1346_le__imp__less__Suc,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.53 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % le_imp_less_Suc
% 0.34/0.53 thf(fact_1347_One__nat__def,axiom,
% 0.34/0.53 ( one_one_nat
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % One_nat_def
% 0.34/0.53 thf(fact_1348_vebt__delete_Osimps_I3_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o,N: nat] :
% 0.34/0.53 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ N ) ) )
% 0.34/0.53 = ( vEBT_Leaf @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_delete.simps(3)
% 0.34/0.53 thf(fact_1349_ex__least__nat__less,axiom,
% 0.34/0.53 ! [P: nat > $o,N: nat] :
% 0.34/0.53 ( ( P @ N )
% 0.34/0.53 => ( ~ ( P @ zero_zero_nat )
% 0.34/0.53 => ? [K2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ K2 @ N )
% 0.34/0.53 & ! [I3: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I3 @ K2 )
% 0.34/0.53 => ~ ( P @ I3 ) )
% 0.34/0.53 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % ex_least_nat_less
% 0.34/0.53 thf(fact_1350_linorder__neqE__linordered__idom,axiom,
% 0.34/0.53 ! [X2: real,Y3: real] :
% 0.34/0.53 ( ( X2 != Y3 )
% 0.34/0.53 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 0.34/0.53 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_neqE_linordered_idom
% 0.34/0.53 thf(fact_1351_linorder__neqE__linordered__idom,axiom,
% 0.34/0.53 ! [X2: rat,Y3: rat] :
% 0.34/0.53 ( ( X2 != Y3 )
% 0.34/0.53 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.53 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_neqE_linordered_idom
% 0.34/0.53 thf(fact_1352_linorder__neqE__linordered__idom,axiom,
% 0.34/0.53 ! [X2: int,Y3: int] :
% 0.34/0.53 ( ( X2 != Y3 )
% 0.34/0.53 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 0.34/0.53 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % linorder_neqE_linordered_idom
% 0.34/0.53 thf(fact_1353_nat__induct__non__zero,axiom,
% 0.34/0.53 ! [N: nat,P: nat > $o] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.53 => ( ( P @ one_one_nat )
% 0.34/0.53 => ( ! [N3: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.34/0.53 => ( ( P @ N3 )
% 0.34/0.53 => ( P @ ( suc @ N3 ) ) ) )
% 0.34/0.53 => ( P @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % nat_induct_non_zero
% 0.34/0.53 thf(fact_1354_enumerate__step,axiom,
% 0.34/0.53 ! [S: set_Extended_enat,N: nat] :
% 0.34/0.53 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.53 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_step
% 0.34/0.53 thf(fact_1355_enumerate__step,axiom,
% 0.34/0.53 ! [S: set_nat,N: nat] :
% 0.34/0.53 ( ~ ( finite_finite_nat @ S )
% 0.34/0.53 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % enumerate_step
% 0.34/0.53 thf(fact_1356_invar__vebt_Ointros_I1_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % invar_vebt.intros(1)
% 0.34/0.53 thf(fact_1357_vebt__delete_Osimps_I2_J,axiom,
% 0.34/0.53 ! [A2: $o,B3: $o] :
% 0.34/0.53 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.53 = ( vEBT_Leaf @ A2 @ $false ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_delete.simps(2)
% 0.34/0.53 thf(fact_1358_int__one__le__iff__zero__less,axiom,
% 0.34/0.53 ! [Z: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ one_one_int @ Z )
% 0.34/0.53 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.34/0.53
% 0.34/0.53 % int_one_le_iff_zero_less
% 0.34/0.53 thf(fact_1359_vebt__buildup_Osimps_I2_J,axiom,
% 0.34/0.53 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.34/0.53 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_buildup.simps(2)
% 0.34/0.53 thf(fact_1360_vebt__succ_Osimps_I2_J,axiom,
% 0.34/0.53 ! [Uv: $o,Uw: $o,N: nat] :
% 0.34/0.53 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 0.34/0.53 = none_nat ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_succ.simps(2)
% 0.34/0.53 thf(fact_1361_vebt__pred_Osimps_I2_J,axiom,
% 0.34/0.53 ! [A2: $o,Uw: $o] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.53 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.53 & ( ~ A2
% 0.34/0.53 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.53 = none_nat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % vebt_pred.simps(2)
% 0.34/0.53 thf(fact_1362_option_Osize__gen_I1_J,axiom,
% 0.34/0.53 ! [X2: nat > nat] :
% 0.34/0.53 ( ( size_option_nat @ X2 @ none_nat )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size_gen(1)
% 0.34/0.53 thf(fact_1363_option_Osize__gen_I1_J,axiom,
% 0.34/0.53 ! [X2: product_prod_nat_nat > nat] :
% 0.34/0.53 ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size_gen(1)
% 0.34/0.53 thf(fact_1364_option_Osize__gen_I1_J,axiom,
% 0.34/0.53 ! [X2: num > nat] :
% 0.34/0.53 ( ( size_option_num @ X2 @ none_num )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size_gen(1)
% 0.34/0.53 thf(fact_1365_option_Osize_I3_J,axiom,
% 0.34/0.53 ( ( size_size_option_nat @ none_nat )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(3)
% 0.34/0.53 thf(fact_1366_option_Osize_I3_J,axiom,
% 0.34/0.53 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(3)
% 0.34/0.53 thf(fact_1367_option_Osize_I3_J,axiom,
% 0.34/0.53 ( ( size_size_option_num @ none_num )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(3)
% 0.34/0.53 thf(fact_1368_option_Osize_I4_J,axiom,
% 0.34/0.53 ! [X22: nat] :
% 0.34/0.53 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(4)
% 0.34/0.53 thf(fact_1369_option_Osize_I4_J,axiom,
% 0.34/0.53 ! [X22: product_prod_nat_nat] :
% 0.34/0.53 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(4)
% 0.34/0.53 thf(fact_1370_option_Osize_I4_J,axiom,
% 0.34/0.53 ! [X22: num] :
% 0.34/0.53 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.34/0.53 = ( suc @ zero_zero_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % option.size(4)
% 0.34/0.53 thf(fact_1371_list__decode_Ocases,axiom,
% 0.34/0.53 ! [X2: nat] :
% 0.34/0.53 ( ( X2 != zero_zero_nat )
% 0.34/0.53 => ~ ! [N3: nat] :
% 0.34/0.53 ( X2
% 0.34/0.53 != ( suc @ N3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % list_decode.cases
% 0.34/0.53 thf(fact_1372_deg__SUcn__Node,axiom,
% 0.34/0.53 ! [Tree: vEBT_VEBT,N: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 0.34/0.53 => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.34/0.53 ( Tree
% 0.34/0.53 = ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % deg_SUcn_Node
% 0.34/0.53 thf(fact_1373_neg__int__cases,axiom,
% 0.34/0.53 ! [K: int] :
% 0.34/0.53 ( ( ord_less_int @ K @ zero_zero_int )
% 0.34/0.53 => ~ ! [N3: nat] :
% 0.34/0.53 ( ( K
% 0.34/0.53 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.34/0.53 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_int_cases
% 0.34/0.53 thf(fact_1374_frac__eq,axiom,
% 0.34/0.53 ! [X2: real] :
% 0.34/0.53 ( ( ( archim2898591450579166408c_real @ X2 )
% 0.34/0.53 = X2 )
% 0.34/0.53 = ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.53 & ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % frac_eq
% 0.34/0.53 thf(fact_1375_frac__eq,axiom,
% 0.34/0.53 ! [X2: rat] :
% 0.34/0.53 ( ( ( archimedean_frac_rat @ X2 )
% 0.34/0.53 = X2 )
% 0.34/0.53 = ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.53 & ( ord_less_rat @ X2 @ one_one_rat ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % frac_eq
% 0.34/0.53 thf(fact_1376_finite__enum__subset,axiom,
% 0.34/0.53 ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
% 0.34/0.53 ( ! [I2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
% 0.34/0.53 => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
% 0.34/0.53 = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
% 0.34/0.53 => ( ( finite4001608067531595151d_enat @ X6 )
% 0.34/0.53 => ( ( finite4001608067531595151d_enat @ Y7 )
% 0.34/0.53 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ X6 ) @ ( finite121521170596916366d_enat @ Y7 ) )
% 0.34/0.53 => ( ord_le7203529160286727270d_enat @ X6 @ Y7 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % finite_enum_subset
% 0.34/0.53 thf(fact_1377_finite__enum__subset,axiom,
% 0.34/0.53 ! [X6: set_nat,Y7: set_nat] :
% 0.34/0.53 ( ! [I2: nat] :
% 0.34/0.53 ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
% 0.34/0.53 => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
% 0.34/0.53 = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
% 0.34/0.53 => ( ( finite_finite_nat @ X6 )
% 0.34/0.53 => ( ( finite_finite_nat @ Y7 )
% 0.34/0.53 => ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y7 ) )
% 0.34/0.53 => ( ord_less_eq_set_nat @ X6 @ Y7 ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % finite_enum_subset
% 0.34/0.53 thf(fact_1378_Suc__diff__1,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.53 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.34/0.53 = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_diff_1
% 0.34/0.53 thf(fact_1379_inverse__of__nat__le,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.53 => ( ( N != zero_zero_nat )
% 0.34/0.53 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % inverse_of_nat_le
% 0.34/0.53 thf(fact_1380_inverse__of__nat__le,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.53 => ( ( N != zero_zero_nat )
% 0.34/0.53 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 0.34/0.53
% 0.34/0.53 % inverse_of_nat_le
% 0.34/0.53 thf(fact_1381_deg__deg__n,axiom,
% 0.34/0.53 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.34/0.53 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 0.34/0.53 => ( Deg = N ) ) ).
% 0.34/0.53
% 0.34/0.53 % deg_deg_n
% 0.34/0.53 thf(fact_1382_add_Oinverse__inverse,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_inverse
% 0.34/0.53 thf(fact_1383_add_Oinverse__inverse,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_inverse
% 0.34/0.53 thf(fact_1384_add_Oinverse__inverse,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_inverse
% 0.34/0.53 thf(fact_1385_add_Oinverse__inverse,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_inverse
% 0.34/0.53 thf(fact_1386_add_Oinverse__inverse,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_inverse
% 0.34/0.53 thf(fact_1387_neg__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: int,B3: int] :
% 0.34/0.53 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.53 = ( uminus_uminus_int @ B3 ) )
% 0.34/0.53 = ( A2 = B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_iff_equal
% 0.34/0.53 thf(fact_1388_neg__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: real,B3: real] :
% 0.34/0.53 ( ( ( uminus_uminus_real @ A2 )
% 0.34/0.53 = ( uminus_uminus_real @ B3 ) )
% 0.34/0.53 = ( A2 = B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_iff_equal
% 0.34/0.53 thf(fact_1389_neg__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: rat,B3: rat] :
% 0.34/0.53 ( ( ( uminus_uminus_rat @ A2 )
% 0.34/0.53 = ( uminus_uminus_rat @ B3 ) )
% 0.34/0.53 = ( A2 = B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_iff_equal
% 0.34/0.53 thf(fact_1390_neg__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.53 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.53 = ( uminus1351360451143612070nteger @ B3 ) )
% 0.34/0.53 = ( A2 = B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_iff_equal
% 0.34/0.53 thf(fact_1391_neg__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: complex,B3: complex] :
% 0.34/0.53 ( ( ( uminus1482373934393186551omplex @ A2 )
% 0.34/0.53 = ( uminus1482373934393186551omplex @ B3 ) )
% 0.34/0.53 = ( A2 = B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_iff_equal
% 0.34/0.53 thf(fact_1392_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( minus_minus_real @ A2 @ A2 )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.53 thf(fact_1393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ A2 @ A2 )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.53 thf(fact_1394_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ A2 @ A2 )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.53 thf(fact_1395_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( minus_minus_int @ A2 @ A2 )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % cancel_comm_monoid_add_class.diff_cancel
% 0.34/0.53 thf(fact_1396_diff__zero,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( minus_minus_real @ A2 @ zero_zero_real )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_zero
% 0.34/0.53 thf(fact_1397_diff__zero,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_zero
% 0.34/0.53 thf(fact_1398_diff__zero,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_zero
% 0.34/0.53 thf(fact_1399_diff__zero,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_zero
% 0.34/0.53 thf(fact_1400_zero__diff,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % zero_diff
% 0.34/0.53 thf(fact_1401_diff__0__right,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( minus_minus_real @ A2 @ zero_zero_real )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0_right
% 0.34/0.53 thf(fact_1402_diff__0__right,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0_right
% 0.34/0.53 thf(fact_1403_diff__0__right,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 0.34/0.53 = A2 ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0_right
% 0.34/0.53 thf(fact_1404_diff__self,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( minus_minus_real @ A2 @ A2 )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % diff_self
% 0.34/0.53 thf(fact_1405_diff__self,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ A2 @ A2 )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % diff_self
% 0.34/0.53 thf(fact_1406_diff__self,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( minus_minus_int @ A2 @ A2 )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % diff_self
% 0.34/0.53 thf(fact_1407_div__by__0,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( divide_divide_rat @ A2 @ zero_zero_rat )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % div_by_0
% 0.34/0.53 thf(fact_1408_div__by__0,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( divide_divide_int @ A2 @ zero_zero_int )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % div_by_0
% 0.34/0.53 thf(fact_1409_div__by__0,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( divide_divide_nat @ A2 @ zero_zero_nat )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % div_by_0
% 0.34/0.53 thf(fact_1410_div__by__0,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( divide_divide_real @ A2 @ zero_zero_real )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % div_by_0
% 0.34/0.53 thf(fact_1411_div__by__0,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
% 0.34/0.53 = zero_zero_complex ) ).
% 0.34/0.53
% 0.34/0.53 % div_by_0
% 0.34/0.53 thf(fact_1412_div__0,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( divide_divide_rat @ zero_zero_rat @ A2 )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % div_0
% 0.34/0.53 thf(fact_1413_div__0,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( divide_divide_int @ zero_zero_int @ A2 )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % div_0
% 0.34/0.53 thf(fact_1414_div__0,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( divide_divide_nat @ zero_zero_nat @ A2 )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % div_0
% 0.34/0.53 thf(fact_1415_div__0,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( divide_divide_real @ zero_zero_real @ A2 )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % div_0
% 0.34/0.53 thf(fact_1416_div__0,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A2 )
% 0.34/0.53 = zero_zero_complex ) ).
% 0.34/0.53
% 0.34/0.53 % div_0
% 0.34/0.53 thf(fact_1417_neg__le__iff__le,axiom,
% 0.34/0.53 ! [B3: real,A2: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_iff_le
% 0.34/0.53 thf(fact_1418_neg__le__iff__le,axiom,
% 0.34/0.53 ! [B3: code_integer,A2: code_integer] :
% 0.34/0.53 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_iff_le
% 0.34/0.53 thf(fact_1419_neg__le__iff__le,axiom,
% 0.34/0.53 ! [B3: rat,A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_iff_le
% 0.34/0.53 thf(fact_1420_neg__le__iff__le,axiom,
% 0.34/0.53 ! [B3: int,A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_iff_le
% 0.34/0.53 thf(fact_1421_neg__equal__zero,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.53 = A2 )
% 0.34/0.53 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_zero
% 0.34/0.53 thf(fact_1422_neg__equal__zero,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ( uminus_uminus_real @ A2 )
% 0.34/0.53 = A2 )
% 0.34/0.53 = ( A2 = zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_zero
% 0.34/0.53 thf(fact_1423_neg__equal__zero,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ( uminus_uminus_rat @ A2 )
% 0.34/0.53 = A2 )
% 0.34/0.53 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_zero
% 0.34/0.53 thf(fact_1424_neg__equal__zero,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.53 = A2 )
% 0.34/0.53 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_zero
% 0.34/0.53 thf(fact_1425_equal__neg__zero,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % equal_neg_zero
% 0.34/0.53 thf(fact_1426_equal__neg__zero,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( A2 = zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % equal_neg_zero
% 0.34/0.53 thf(fact_1427_equal__neg__zero,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % equal_neg_zero
% 0.34/0.53 thf(fact_1428_equal__neg__zero,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( A2
% 0.34/0.53 = ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % equal_neg_zero
% 0.34/0.53 thf(fact_1429_neg__equal__0__iff__equal,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.53 = zero_zero_int )
% 0.34/0.53 = ( A2 = zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_0_iff_equal
% 0.34/0.53 thf(fact_1430_neg__equal__0__iff__equal,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ( uminus_uminus_real @ A2 )
% 0.34/0.53 = zero_zero_real )
% 0.34/0.53 = ( A2 = zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_0_iff_equal
% 0.34/0.53 thf(fact_1431_neg__equal__0__iff__equal,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ( uminus_uminus_rat @ A2 )
% 0.34/0.53 = zero_zero_rat )
% 0.34/0.53 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_0_iff_equal
% 0.34/0.53 thf(fact_1432_neg__equal__0__iff__equal,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.53 = zero_z3403309356797280102nteger )
% 0.34/0.53 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_0_iff_equal
% 0.34/0.53 thf(fact_1433_neg__equal__0__iff__equal,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( ( uminus1482373934393186551omplex @ A2 )
% 0.34/0.53 = zero_zero_complex )
% 0.34/0.53 = ( A2 = zero_zero_complex ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_equal_0_iff_equal
% 0.34/0.53 thf(fact_1434_neg__0__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( zero_zero_int
% 0.34/0.53 = ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( zero_zero_int = A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_equal_iff_equal
% 0.34/0.53 thf(fact_1435_neg__0__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( zero_zero_real
% 0.34/0.53 = ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( zero_zero_real = A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_equal_iff_equal
% 0.34/0.53 thf(fact_1436_neg__0__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( zero_zero_rat
% 0.34/0.53 = ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( zero_zero_rat = A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_equal_iff_equal
% 0.34/0.53 thf(fact_1437_neg__0__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( zero_z3403309356797280102nteger
% 0.34/0.53 = ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( zero_z3403309356797280102nteger = A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_equal_iff_equal
% 0.34/0.53 thf(fact_1438_neg__0__equal__iff__equal,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( zero_zero_complex
% 0.34/0.53 = ( uminus1482373934393186551omplex @ A2 ) )
% 0.34/0.53 = ( zero_zero_complex = A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_equal_iff_equal
% 0.34/0.53 thf(fact_1439_add_Oinverse__neutral,axiom,
% 0.34/0.53 ( ( uminus_uminus_int @ zero_zero_int )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_neutral
% 0.34/0.53 thf(fact_1440_add_Oinverse__neutral,axiom,
% 0.34/0.53 ( ( uminus_uminus_real @ zero_zero_real )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_neutral
% 0.34/0.53 thf(fact_1441_add_Oinverse__neutral,axiom,
% 0.34/0.53 ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_neutral
% 0.34/0.53 thf(fact_1442_add_Oinverse__neutral,axiom,
% 0.34/0.53 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.34/0.53 = zero_z3403309356797280102nteger ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_neutral
% 0.34/0.53 thf(fact_1443_add_Oinverse__neutral,axiom,
% 0.34/0.53 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.34/0.53 = zero_zero_complex ) ).
% 0.34/0.53
% 0.34/0.53 % add.inverse_neutral
% 0.34/0.53 thf(fact_1444_neg__less__iff__less,axiom,
% 0.34/0.53 ! [B3: int,A2: int] :
% 0.34/0.53 ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_int @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_iff_less
% 0.34/0.53 thf(fact_1445_neg__less__iff__less,axiom,
% 0.34/0.53 ! [B3: real,A2: real] :
% 0.34/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_real @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_iff_less
% 0.34/0.53 thf(fact_1446_neg__less__iff__less,axiom,
% 0.34/0.53 ! [B3: rat,A2: rat] :
% 0.34/0.53 ( ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_iff_less
% 0.34/0.53 thf(fact_1447_neg__less__iff__less,axiom,
% 0.34/0.53 ! [B3: code_integer,A2: code_integer] :
% 0.34/0.53 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le6747313008572928689nteger @ A2 @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_iff_less
% 0.34/0.53 thf(fact_1448_minus__diff__eq,axiom,
% 0.34/0.53 ! [A2: int,B3: int] :
% 0.34/0.53 ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B3 ) )
% 0.34/0.53 = ( minus_minus_int @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % minus_diff_eq
% 0.34/0.53 thf(fact_1449_minus__diff__eq,axiom,
% 0.34/0.53 ! [A2: real,B3: real] :
% 0.34/0.53 ( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B3 ) )
% 0.34/0.53 = ( minus_minus_real @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % minus_diff_eq
% 0.34/0.53 thf(fact_1450_minus__diff__eq,axiom,
% 0.34/0.53 ! [A2: rat,B3: rat] :
% 0.34/0.53 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.34/0.53 = ( minus_minus_rat @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % minus_diff_eq
% 0.34/0.53 thf(fact_1451_minus__diff__eq,axiom,
% 0.34/0.53 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.53 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
% 0.34/0.53 = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % minus_diff_eq
% 0.34/0.53 thf(fact_1452_minus__diff__eq,axiom,
% 0.34/0.53 ! [A2: complex,B3: complex] :
% 0.34/0.53 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B3 ) )
% 0.34/0.53 = ( minus_minus_complex @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % minus_diff_eq
% 0.34/0.53 thf(fact_1453_diff__Suc__Suc,axiom,
% 0.34/0.53 ! [M: nat,N: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.34/0.53 = ( minus_minus_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_Suc_Suc
% 0.34/0.53 thf(fact_1454_Suc__diff__diff,axiom,
% 0.34/0.53 ! [M: nat,N: nat,K: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 0.34/0.53 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 0.34/0.53
% 0.34/0.53 % Suc_diff_diff
% 0.34/0.53 thf(fact_1455_diff__self__eq__0,axiom,
% 0.34/0.53 ! [M: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ M @ M )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % diff_self_eq_0
% 0.34/0.53 thf(fact_1456_diff__0__eq__0,axiom,
% 0.34/0.53 ! [N: nat] :
% 0.34/0.53 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0_eq_0
% 0.34/0.53 thf(fact_1457_diff__diff__cancel,axiom,
% 0.34/0.53 ! [I: nat,N: nat] :
% 0.34/0.53 ( ( ord_less_eq_nat @ I @ N )
% 0.34/0.53 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 0.34/0.53 = I ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_diff_cancel
% 0.34/0.53 thf(fact_1458_diff__ge__0__iff__ge,axiom,
% 0.34/0.53 ! [A2: real,B3: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_eq_real @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_ge_0_iff_ge
% 0.34/0.53 thf(fact_1459_diff__ge__0__iff__ge,axiom,
% 0.34/0.53 ! [A2: rat,B3: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_ge_0_iff_ge
% 0.34/0.53 thf(fact_1460_diff__ge__0__iff__ge,axiom,
% 0.34/0.53 ! [A2: int,B3: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_ge_0_iff_ge
% 0.34/0.53 thf(fact_1461_diff__gt__0__iff__gt,axiom,
% 0.34/0.53 ! [A2: real,B3: real] :
% 0.34/0.53 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_real @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_gt_0_iff_gt
% 0.34/0.53 thf(fact_1462_diff__gt__0__iff__gt,axiom,
% 0.34/0.53 ! [A2: rat,B3: rat] :
% 0.34/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_rat @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_gt_0_iff_gt
% 0.34/0.53 thf(fact_1463_diff__gt__0__iff__gt,axiom,
% 0.34/0.53 ! [A2: int,B3: int] :
% 0.34/0.53 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
% 0.34/0.53 = ( ord_less_int @ B3 @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_gt_0_iff_gt
% 0.34/0.53 thf(fact_1464_neg__0__le__iff__le,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_le_iff_le
% 0.34/0.53 thf(fact_1465_neg__0__le__iff__le,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_le_iff_le
% 0.34/0.53 thf(fact_1466_neg__0__le__iff__le,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_le_iff_le
% 0.34/0.53 thf(fact_1467_neg__0__le__iff__le,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_le_iff_le
% 0.34/0.53 thf(fact_1468_neg__le__0__iff__le,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
% 0.34/0.53 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_0_iff_le
% 0.34/0.53 thf(fact_1469_neg__le__0__iff__le,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.53 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_0_iff_le
% 0.34/0.53 thf(fact_1470_neg__le__0__iff__le,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_0_iff_le
% 0.34/0.53 thf(fact_1471_neg__le__0__iff__le,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 0.34/0.53 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_le_0_iff_le
% 0.34/0.53 thf(fact_1472_less__eq__neg__nonpos,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_eq_neg_nonpos
% 0.34/0.53 thf(fact_1473_less__eq__neg__nonpos,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_eq_neg_nonpos
% 0.34/0.53 thf(fact_1474_less__eq__neg__nonpos,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_eq_neg_nonpos
% 0.34/0.53 thf(fact_1475_less__eq__neg__nonpos,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_eq_neg_nonpos
% 0.34/0.53 thf(fact_1476_neg__less__eq__nonneg,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_eq_nonneg
% 0.34/0.53 thf(fact_1477_neg__less__eq__nonneg,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_eq_nonneg
% 0.34/0.53 thf(fact_1478_neg__less__eq__nonneg,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_eq_nonneg
% 0.34/0.53 thf(fact_1479_neg__less__eq__nonneg,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_eq_nonneg
% 0.34/0.53 thf(fact_1480_neg__less__0__iff__less,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 0.34/0.53 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_0_iff_less
% 0.34/0.53 thf(fact_1481_neg__less__0__iff__less,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
% 0.34/0.53 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_0_iff_less
% 0.34/0.53 thf(fact_1482_neg__less__0__iff__less,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.53 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_0_iff_less
% 0.34/0.53 thf(fact_1483_neg__less__0__iff__less,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 0.34/0.53 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_0_iff_less
% 0.34/0.53 thf(fact_1484_neg__0__less__iff__less,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_less_iff_less
% 0.34/0.53 thf(fact_1485_neg__0__less__iff__less,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_less_iff_less
% 0.34/0.53 thf(fact_1486_neg__0__less__iff__less,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_less_iff_less
% 0.34/0.53 thf(fact_1487_neg__0__less__iff__less,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_0_less_iff_less
% 0.34/0.53 thf(fact_1488_neg__less__pos,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_pos
% 0.34/0.53 thf(fact_1489_neg__less__pos,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_pos
% 0.34/0.53 thf(fact_1490_neg__less__pos,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_pos
% 0.34/0.53 thf(fact_1491_neg__less__pos,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.34/0.53 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % neg_less_pos
% 0.34/0.53 thf(fact_1492_less__neg__neg,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.34/0.53 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_neg_neg
% 0.34/0.53 thf(fact_1493_less__neg__neg,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 0.34/0.53 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_neg_neg
% 0.34/0.53 thf(fact_1494_less__neg__neg,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.34/0.53 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_neg_neg
% 0.34/0.53 thf(fact_1495_less__neg__neg,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.34/0.53 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.34/0.53
% 0.34/0.53 % less_neg_neg
% 0.34/0.53 thf(fact_1496_diff__numeral__special_I9_J,axiom,
% 0.34/0.53 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.34/0.53 = zero_zero_complex ) ).
% 0.34/0.53
% 0.34/0.53 % diff_numeral_special(9)
% 0.34/0.53 thf(fact_1497_diff__numeral__special_I9_J,axiom,
% 0.34/0.53 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.34/0.53 = zero_zero_real ) ).
% 0.34/0.53
% 0.34/0.53 % diff_numeral_special(9)
% 0.34/0.53 thf(fact_1498_diff__numeral__special_I9_J,axiom,
% 0.34/0.53 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.34/0.53 = zero_zero_rat ) ).
% 0.34/0.53
% 0.34/0.53 % diff_numeral_special(9)
% 0.34/0.53 thf(fact_1499_diff__numeral__special_I9_J,axiom,
% 0.34/0.53 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.34/0.53 = zero_zero_int ) ).
% 0.34/0.53
% 0.34/0.53 % diff_numeral_special(9)
% 0.34/0.53 thf(fact_1500_div__self,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( A2 != zero_zero_rat )
% 0.34/0.53 => ( ( divide_divide_rat @ A2 @ A2 )
% 0.34/0.53 = one_one_rat ) ) ).
% 0.34/0.53
% 0.34/0.53 % div_self
% 0.34/0.53 thf(fact_1501_div__self,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( A2 != zero_zero_int )
% 0.34/0.53 => ( ( divide_divide_int @ A2 @ A2 )
% 0.34/0.53 = one_one_int ) ) ).
% 0.34/0.53
% 0.34/0.53 % div_self
% 0.34/0.53 thf(fact_1502_div__self,axiom,
% 0.34/0.53 ! [A2: nat] :
% 0.34/0.53 ( ( A2 != zero_zero_nat )
% 0.34/0.53 => ( ( divide_divide_nat @ A2 @ A2 )
% 0.34/0.53 = one_one_nat ) ) ).
% 0.34/0.53
% 0.34/0.53 % div_self
% 0.34/0.53 thf(fact_1503_div__self,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( A2 != zero_zero_real )
% 0.34/0.53 => ( ( divide_divide_real @ A2 @ A2 )
% 0.34/0.53 = one_one_real ) ) ).
% 0.34/0.53
% 0.34/0.53 % div_self
% 0.34/0.53 thf(fact_1504_div__self,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( A2 != zero_zero_complex )
% 0.34/0.53 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 0.34/0.53 = one_one_complex ) ) ).
% 0.34/0.53
% 0.34/0.53 % div_self
% 0.34/0.53 thf(fact_1505_verit__minus__simplify_I3_J,axiom,
% 0.34/0.53 ! [B3: int] :
% 0.34/0.53 ( ( minus_minus_int @ zero_zero_int @ B3 )
% 0.34/0.53 = ( uminus_uminus_int @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_minus_simplify(3)
% 0.34/0.53 thf(fact_1506_verit__minus__simplify_I3_J,axiom,
% 0.34/0.53 ! [B3: real] :
% 0.34/0.53 ( ( minus_minus_real @ zero_zero_real @ B3 )
% 0.34/0.53 = ( uminus_uminus_real @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_minus_simplify(3)
% 0.34/0.53 thf(fact_1507_verit__minus__simplify_I3_J,axiom,
% 0.34/0.53 ! [B3: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ zero_zero_rat @ B3 )
% 0.34/0.53 = ( uminus_uminus_rat @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_minus_simplify(3)
% 0.34/0.53 thf(fact_1508_verit__minus__simplify_I3_J,axiom,
% 0.34/0.53 ! [B3: code_integer] :
% 0.34/0.53 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.34/0.53 = ( uminus1351360451143612070nteger @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_minus_simplify(3)
% 0.34/0.53 thf(fact_1509_verit__minus__simplify_I3_J,axiom,
% 0.34/0.53 ! [B3: complex] :
% 0.34/0.53 ( ( minus_minus_complex @ zero_zero_complex @ B3 )
% 0.34/0.53 = ( uminus1482373934393186551omplex @ B3 ) ) ).
% 0.34/0.53
% 0.34/0.53 % verit_minus_simplify(3)
% 0.34/0.53 thf(fact_1510_diff__0,axiom,
% 0.34/0.53 ! [A2: int] :
% 0.34/0.53 ( ( minus_minus_int @ zero_zero_int @ A2 )
% 0.34/0.53 = ( uminus_uminus_int @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0
% 0.34/0.53 thf(fact_1511_diff__0,axiom,
% 0.34/0.53 ! [A2: real] :
% 0.34/0.53 ( ( minus_minus_real @ zero_zero_real @ A2 )
% 0.34/0.53 = ( uminus_uminus_real @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0
% 0.34/0.53 thf(fact_1512_diff__0,axiom,
% 0.34/0.53 ! [A2: rat] :
% 0.34/0.53 ( ( minus_minus_rat @ zero_zero_rat @ A2 )
% 0.34/0.53 = ( uminus_uminus_rat @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0
% 0.34/0.53 thf(fact_1513_diff__0,axiom,
% 0.34/0.53 ! [A2: code_integer] :
% 0.34/0.53 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.34/0.53 = ( uminus1351360451143612070nteger @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0
% 0.34/0.53 thf(fact_1514_diff__0,axiom,
% 0.34/0.53 ! [A2: complex] :
% 0.34/0.53 ( ( minus_minus_complex @ zero_zero_complex @ A2 )
% 0.34/0.53 = ( uminus1482373934393186551omplex @ A2 ) ) ).
% 0.34/0.53
% 0.34/0.53 % diff_0
% 0.34/0.53 thf(fact_1515_zero__less__diff,axiom,
% 0.34/0.53 ! [N: nat,M: nat] :
% 0.34/0.53 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 0.34/0.53 = ( ord_less_nat @ M @ N ) ) ).
% 0.34/0.53
% 0.34/0.53 % zero_less_diff
% 0.34/0.53 thf(fact_1516_card_Oempty,axiom,
% 0.34/0.53 ( ( finite_card_complex @ bot_bot_set_complex )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % card.empty
% 0.34/0.53 thf(fact_1517_card_Oempty,axiom,
% 0.34/0.53 ( ( finite_card_list_nat @ bot_bot_set_list_nat )
% 0.34/0.53 = zero_zero_nat ) ).
% 0.34/0.53
% 0.34/0.53 % card.empty
% 0.34/0.53 thf(fact_1518_card_Oempty,axiom,
% 0.34/0.54 ( ( finite_card_set_nat @ bot_bot_set_set_nat )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % card.empty
% 0.34/0.54 thf(fact_1519_card_Oempty,axiom,
% 0.34/0.54 ( ( finite_card_real @ bot_bot_set_real )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % card.empty
% 0.34/0.54 thf(fact_1520_card_Oempty,axiom,
% 0.34/0.54 ( ( finite_card_o @ bot_bot_set_o )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % card.empty
% 0.34/0.54 thf(fact_1521_card_Oempty,axiom,
% 0.34/0.54 ( ( finite_card_nat @ bot_bot_set_nat )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % card.empty
% 0.34/0.54 thf(fact_1522_card_Oempty,axiom,
% 0.34/0.54 ( ( finite_card_int @ bot_bot_set_int )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % card.empty
% 0.34/0.54 thf(fact_1523_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ~ ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ( ( finite_card_list_nat @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1524_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ~ ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ( ( finite_card_set_nat @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1525_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ~ ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( ( finite_card_nat @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1526_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ~ ( finite_finite_int @ A3 )
% 0.34/0.54 => ( ( finite_card_int @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1527_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ~ ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( ( finite_card_complex @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1528_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ~ ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1529_card_Oinfinite,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % card.infinite
% 0.34/0.54 thf(fact_1530_diff__is__0__eq,axiom,
% 0.34/0.54 ! [M: nat,N: nat] :
% 0.34/0.54 ( ( ( minus_minus_nat @ M @ N )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_is_0_eq
% 0.34/0.54 thf(fact_1531_diff__is__0__eq_H,axiom,
% 0.34/0.54 ! [M: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.54 => ( ( minus_minus_nat @ M @ N )
% 0.34/0.54 = zero_zero_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_is_0_eq'
% 0.34/0.54 thf(fact_1532_diff__Suc__1,axiom,
% 0.34/0.54 ! [N: nat] :
% 0.34/0.54 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 0.34/0.54 = N ) ).
% 0.34/0.54
% 0.34/0.54 % diff_Suc_1
% 0.34/0.54 thf(fact_1533_negative__eq__positive,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.34/0.54 = ( semiri1314217659103216013at_int @ M ) )
% 0.34/0.54 = ( ( N = zero_zero_nat )
% 0.34/0.54 & ( M = zero_zero_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % negative_eq_positive
% 0.34/0.54 thf(fact_1534_negative__zle,axiom,
% 0.34/0.54 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.34/0.54
% 0.34/0.54 % negative_zle
% 0.34/0.54 thf(fact_1535_diff__numeral__special_I12_J,axiom,
% 0.34/0.54 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.34/0.54 = zero_zero_int ) ).
% 0.34/0.54
% 0.34/0.54 % diff_numeral_special(12)
% 0.34/0.54 thf(fact_1536_diff__numeral__special_I12_J,axiom,
% 0.34/0.54 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.34/0.54 = zero_zero_real ) ).
% 0.34/0.54
% 0.34/0.54 % diff_numeral_special(12)
% 0.34/0.54 thf(fact_1537_diff__numeral__special_I12_J,axiom,
% 0.34/0.54 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.34/0.54 = zero_zero_rat ) ).
% 0.34/0.54
% 0.34/0.54 % diff_numeral_special(12)
% 0.34/0.54 thf(fact_1538_diff__numeral__special_I12_J,axiom,
% 0.34/0.54 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.34/0.54 = zero_z3403309356797280102nteger ) ).
% 0.34/0.54
% 0.34/0.54 % diff_numeral_special(12)
% 0.34/0.54 thf(fact_1539_diff__numeral__special_I12_J,axiom,
% 0.34/0.54 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.34/0.54 = zero_zero_complex ) ).
% 0.34/0.54
% 0.34/0.54 % diff_numeral_special(12)
% 0.34/0.54 thf(fact_1540_Suc__pred,axiom,
% 0.34/0.54 ! [N: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.54 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.34/0.54 = N ) ) ).
% 0.34/0.54
% 0.34/0.54 % Suc_pred
% 0.34/0.54 thf(fact_1541_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_list_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1542_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1543_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( ( ( finite_card_complex @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_complex ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1544_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1545_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bo7653980558646680370d_enat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1546_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( finite_finite_real @ A3 )
% 0.34/0.54 => ( ( ( finite_card_real @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1547_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( finite_finite_o @ A3 )
% 0.34/0.54 => ( ( ( finite_card_o @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_o ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1548_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( ( ( finite_card_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1549_card__0__eq,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ A3 )
% 0.34/0.54 => ( ( ( finite_card_int @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( A3 = bot_bot_set_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_0_eq
% 0.34/0.54 thf(fact_1550_negative__zless,axiom,
% 0.34/0.54 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.34/0.54
% 0.34/0.54 % negative_zless
% 0.34/0.54 thf(fact_1551_finite__enumerate__mono__iff,axiom,
% 0.34/0.54 ! [S: set_Extended_enat,M: nat,N: nat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ M @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
% 0.34/0.54 = ( ord_less_nat @ M @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_mono_iff
% 0.34/0.54 thf(fact_1552_finite__enumerate__mono__iff,axiom,
% 0.34/0.54 ! [S: set_nat,M: nat,N: nat] :
% 0.34/0.54 ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
% 0.34/0.54 = ( ord_less_nat @ M @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_mono_iff
% 0.34/0.54 thf(fact_1553_diff__eq__diff__eq,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real,D: real] :
% 0.34/0.54 ( ( ( minus_minus_real @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_real @ C @ D ) )
% 0.34/0.54 => ( ( A2 = B3 )
% 0.34/0.54 = ( C = D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_eq
% 0.34/0.54 thf(fact_1554_diff__eq__diff__eq,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.34/0.54 ( ( ( minus_minus_rat @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_rat @ C @ D ) )
% 0.34/0.54 => ( ( A2 = B3 )
% 0.34/0.54 = ( C = D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_eq
% 0.34/0.54 thf(fact_1555_diff__eq__diff__eq,axiom,
% 0.34/0.54 ! [A2: int,B3: int,C: int,D: int] :
% 0.34/0.54 ( ( ( minus_minus_int @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_int @ C @ D ) )
% 0.34/0.54 => ( ( A2 = B3 )
% 0.34/0.54 = ( C = D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_eq
% 0.34/0.54 thf(fact_1556_equation__minus__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( A2
% 0.34/0.54 = ( uminus_uminus_int @ B3 ) )
% 0.34/0.54 = ( B3
% 0.34/0.54 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % equation_minus_iff
% 0.34/0.54 thf(fact_1557_equation__minus__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( A2
% 0.34/0.54 = ( uminus_uminus_real @ B3 ) )
% 0.34/0.54 = ( B3
% 0.34/0.54 = ( uminus_uminus_real @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % equation_minus_iff
% 0.34/0.54 thf(fact_1558_equation__minus__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( A2
% 0.34/0.54 = ( uminus_uminus_rat @ B3 ) )
% 0.34/0.54 = ( B3
% 0.34/0.54 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % equation_minus_iff
% 0.34/0.54 thf(fact_1559_equation__minus__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( A2
% 0.34/0.54 = ( uminus1351360451143612070nteger @ B3 ) )
% 0.34/0.54 = ( B3
% 0.34/0.54 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % equation_minus_iff
% 0.34/0.54 thf(fact_1560_equation__minus__iff,axiom,
% 0.34/0.54 ! [A2: complex,B3: complex] :
% 0.34/0.54 ( ( A2
% 0.34/0.54 = ( uminus1482373934393186551omplex @ B3 ) )
% 0.34/0.54 = ( B3
% 0.34/0.54 = ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % equation_minus_iff
% 0.34/0.54 thf(fact_1561_minus__equation__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ( uminus_uminus_int @ A2 )
% 0.34/0.54 = B3 )
% 0.34/0.54 = ( ( uminus_uminus_int @ B3 )
% 0.34/0.54 = A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_equation_iff
% 0.34/0.54 thf(fact_1562_minus__equation__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ( uminus_uminus_real @ A2 )
% 0.34/0.54 = B3 )
% 0.34/0.54 = ( ( uminus_uminus_real @ B3 )
% 0.34/0.54 = A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_equation_iff
% 0.34/0.54 thf(fact_1563_minus__equation__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ( uminus_uminus_rat @ A2 )
% 0.34/0.54 = B3 )
% 0.34/0.54 = ( ( uminus_uminus_rat @ B3 )
% 0.34/0.54 = A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_equation_iff
% 0.34/0.54 thf(fact_1564_minus__equation__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.34/0.54 = B3 )
% 0.34/0.54 = ( ( uminus1351360451143612070nteger @ B3 )
% 0.34/0.54 = A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_equation_iff
% 0.34/0.54 thf(fact_1565_minus__equation__iff,axiom,
% 0.34/0.54 ! [A2: complex,B3: complex] :
% 0.34/0.54 ( ( ( uminus1482373934393186551omplex @ A2 )
% 0.34/0.54 = B3 )
% 0.34/0.54 = ( ( uminus1482373934393186551omplex @ B3 )
% 0.34/0.54 = A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_equation_iff
% 0.34/0.54 thf(fact_1566_minus__diff__commute,axiom,
% 0.34/0.54 ! [B3: int,A2: int] :
% 0.34/0.54 ( ( minus_minus_int @ ( uminus_uminus_int @ B3 ) @ A2 )
% 0.34/0.54 = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_diff_commute
% 0.34/0.54 thf(fact_1567_minus__diff__commute,axiom,
% 0.34/0.54 ! [B3: real,A2: real] :
% 0.34/0.54 ( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A2 )
% 0.34/0.54 = ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_diff_commute
% 0.34/0.54 thf(fact_1568_minus__diff__commute,axiom,
% 0.34/0.54 ! [B3: rat,A2: rat] :
% 0.34/0.54 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B3 ) @ A2 )
% 0.34/0.54 = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_diff_commute
% 0.34/0.54 thf(fact_1569_minus__diff__commute,axiom,
% 0.34/0.54 ! [B3: code_integer,A2: code_integer] :
% 0.34/0.54 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 )
% 0.34/0.54 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_diff_commute
% 0.34/0.54 thf(fact_1570_minus__diff__commute,axiom,
% 0.34/0.54 ! [B3: complex,A2: complex] :
% 0.34/0.54 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 )
% 0.34/0.54 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_diff_commute
% 0.34/0.54 thf(fact_1571_diff__right__commute,axiom,
% 0.34/0.54 ! [A2: real,C: real,B3: real] :
% 0.34/0.54 ( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 )
% 0.34/0.54 = ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_commute
% 0.34/0.54 thf(fact_1572_diff__right__commute,axiom,
% 0.34/0.54 ! [A2: rat,C: rat,B3: rat] :
% 0.34/0.54 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 )
% 0.34/0.54 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_commute
% 0.34/0.54 thf(fact_1573_diff__right__commute,axiom,
% 0.34/0.54 ! [A2: nat,C: nat,B3: nat] :
% 0.34/0.54 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B3 )
% 0.34/0.54 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_commute
% 0.34/0.54 thf(fact_1574_diff__right__commute,axiom,
% 0.34/0.54 ! [A2: int,C: int,B3: int] :
% 0.34/0.54 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 )
% 0.34/0.54 = ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_commute
% 0.34/0.54 thf(fact_1575_diff__commute,axiom,
% 0.34/0.54 ! [I: nat,J: nat,K: nat] :
% 0.34/0.54 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.34/0.54 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_commute
% 0.34/0.54 thf(fact_1576_size__neq__size__imp__neq,axiom,
% 0.34/0.54 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.34/0.54 ( ( ( size_size_VEBT_VEBT @ X2 )
% 0.34/0.54 != ( size_size_VEBT_VEBT @ Y3 ) )
% 0.34/0.54 => ( X2 != Y3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % size_neq_size_imp_neq
% 0.34/0.54 thf(fact_1577_size__neq__size__imp__neq,axiom,
% 0.34/0.54 ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.34/0.54 ( ( ( size_s6755466524823107622T_VEBT @ X2 )
% 0.34/0.54 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.34/0.54 => ( X2 != Y3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % size_neq_size_imp_neq
% 0.34/0.54 thf(fact_1578_size__neq__size__imp__neq,axiom,
% 0.34/0.54 ! [X2: num,Y3: num] :
% 0.34/0.54 ( ( ( size_size_num @ X2 )
% 0.34/0.54 != ( size_size_num @ Y3 ) )
% 0.34/0.54 => ( X2 != Y3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % size_neq_size_imp_neq
% 0.34/0.54 thf(fact_1579_size__neq__size__imp__neq,axiom,
% 0.34/0.54 ! [X2: list_o,Y3: list_o] :
% 0.34/0.54 ( ( ( size_size_list_o @ X2 )
% 0.34/0.54 != ( size_size_list_o @ Y3 ) )
% 0.34/0.54 => ( X2 != Y3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % size_neq_size_imp_neq
% 0.34/0.54 thf(fact_1580_size__neq__size__imp__neq,axiom,
% 0.34/0.54 ! [X2: list_nat,Y3: list_nat] :
% 0.34/0.54 ( ( ( size_size_list_nat @ X2 )
% 0.34/0.54 != ( size_size_list_nat @ Y3 ) )
% 0.34/0.54 => ( X2 != Y3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % size_neq_size_imp_neq
% 0.34/0.54 thf(fact_1581_of__nat__diff,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.54 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 0.34/0.54 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % of_nat_diff
% 0.34/0.54 thf(fact_1582_of__nat__diff,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.54 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 0.34/0.54 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % of_nat_diff
% 0.34/0.54 thf(fact_1583_of__nat__diff,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.54 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 0.34/0.54 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % of_nat_diff
% 0.34/0.54 thf(fact_1584_of__nat__diff,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.54 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 0.34/0.54 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % of_nat_diff
% 0.34/0.54 thf(fact_1585_diff__eq__diff__less__eq,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real,D: real] :
% 0.34/0.54 ( ( ( minus_minus_real @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_real @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less_eq
% 0.34/0.54 thf(fact_1586_diff__eq__diff__less__eq,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.34/0.54 ( ( ( minus_minus_rat @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_rat @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less_eq
% 0.34/0.54 thf(fact_1587_diff__eq__diff__less__eq,axiom,
% 0.34/0.54 ! [A2: int,B3: int,C: int,D: int] :
% 0.34/0.54 ( ( ( minus_minus_int @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_int @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less_eq
% 0.34/0.54 thf(fact_1588_diff__right__mono,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_mono
% 0.34/0.54 thf(fact_1589_diff__right__mono,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_mono
% 0.34/0.54 thf(fact_1590_diff__right__mono,axiom,
% 0.34/0.54 ! [A2: int,B3: int,C: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_right_mono
% 0.34/0.54 thf(fact_1591_diff__left__mono,axiom,
% 0.34/0.54 ! [B3: real,A2: real,C: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_left_mono
% 0.34/0.54 thf(fact_1592_diff__left__mono,axiom,
% 0.34/0.54 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_left_mono
% 0.34/0.54 thf(fact_1593_diff__left__mono,axiom,
% 0.34/0.54 ! [B3: int,A2: int,C: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_left_mono
% 0.34/0.54 thf(fact_1594_diff__mono,axiom,
% 0.34/0.54 ! [A2: real,B3: real,D: real,C: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_real @ D @ C )
% 0.34/0.54 => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_mono
% 0.34/0.54 thf(fact_1595_diff__mono,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,D: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ D @ C )
% 0.34/0.54 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_mono
% 0.34/0.54 thf(fact_1596_diff__mono,axiom,
% 0.34/0.54 ! [A2: int,B3: int,D: int,C: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_int @ D @ C )
% 0.34/0.54 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_mono
% 0.34/0.54 thf(fact_1597_eq__iff__diff__eq__0,axiom,
% 0.34/0.54 ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
% 0.34/0.54 = ( ^ [A4: real,B4: real] :
% 0.34/0.54 ( ( minus_minus_real @ A4 @ B4 )
% 0.34/0.54 = zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_iff_diff_eq_0
% 0.34/0.54 thf(fact_1598_eq__iff__diff__eq__0,axiom,
% 0.34/0.54 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 0.34/0.54 = ( ^ [A4: rat,B4: rat] :
% 0.34/0.54 ( ( minus_minus_rat @ A4 @ B4 )
% 0.34/0.54 = zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_iff_diff_eq_0
% 0.34/0.54 thf(fact_1599_eq__iff__diff__eq__0,axiom,
% 0.34/0.54 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 0.34/0.54 = ( ^ [A4: int,B4: int] :
% 0.34/0.54 ( ( minus_minus_int @ A4 @ B4 )
% 0.34/0.54 = zero_zero_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_iff_diff_eq_0
% 0.34/0.54 thf(fact_1600_diff__strict__mono,axiom,
% 0.34/0.54 ! [A2: real,B3: real,D: real,C: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_real @ D @ C )
% 0.34/0.54 => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_mono
% 0.34/0.54 thf(fact_1601_diff__strict__mono,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,D: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_rat @ D @ C )
% 0.34/0.54 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_mono
% 0.34/0.54 thf(fact_1602_diff__strict__mono,axiom,
% 0.34/0.54 ! [A2: int,B3: int,D: int,C: int] :
% 0.34/0.54 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_int @ D @ C )
% 0.34/0.54 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_mono
% 0.34/0.54 thf(fact_1603_diff__eq__diff__less,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real,D: real] :
% 0.34/0.54 ( ( ( minus_minus_real @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_real @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_real @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less
% 0.34/0.54 thf(fact_1604_diff__eq__diff__less,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.34/0.54 ( ( ( minus_minus_rat @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_rat @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_rat @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less
% 0.34/0.54 thf(fact_1605_diff__eq__diff__less,axiom,
% 0.34/0.54 ! [A2: int,B3: int,C: int,D: int] :
% 0.34/0.54 ( ( ( minus_minus_int @ A2 @ B3 )
% 0.34/0.54 = ( minus_minus_int @ C @ D ) )
% 0.34/0.54 => ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.54 = ( ord_less_int @ C @ D ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_eq_diff_less
% 0.34/0.54 thf(fact_1606_diff__strict__left__mono,axiom,
% 0.34/0.54 ! [B3: real,A2: real,C: real] :
% 0.34/0.54 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_left_mono
% 0.34/0.54 thf(fact_1607_diff__strict__left__mono,axiom,
% 0.34/0.54 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_left_mono
% 0.34/0.54 thf(fact_1608_diff__strict__left__mono,axiom,
% 0.34/0.54 ! [B3: int,A2: int,C: int] :
% 0.34/0.54 ( ( ord_less_int @ B3 @ A2 )
% 0.34/0.54 => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_left_mono
% 0.34/0.54 thf(fact_1609_diff__strict__right__mono,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_right_mono
% 0.34/0.54 thf(fact_1610_diff__strict__right__mono,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_right_mono
% 0.34/0.54 thf(fact_1611_diff__strict__right__mono,axiom,
% 0.34/0.54 ! [A2: int,B3: int,C: int] :
% 0.34/0.54 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_strict_right_mono
% 0.34/0.54 thf(fact_1612_le__imp__neg__le,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_imp_neg_le
% 0.34/0.54 thf(fact_1613_le__imp__neg__le,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 0.34/0.54 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_imp_neg_le
% 0.34/0.54 thf(fact_1614_le__imp__neg__le,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_imp_neg_le
% 0.34/0.54 thf(fact_1615_le__imp__neg__le,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_imp_neg_le
% 0.34/0.54 thf(fact_1616_minus__le__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_le_iff
% 0.34/0.54 thf(fact_1617_minus__le__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_le_iff
% 0.34/0.54 thf(fact_1618_minus__le__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_le_iff
% 0.34/0.54 thf(fact_1619_minus__le__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_le_iff
% 0.34/0.54 thf(fact_1620_le__minus__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 0.34/0.54 = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_iff
% 0.34/0.54 thf(fact_1621_le__minus__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 0.34/0.54 = ( ord_le3102999989581377725nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_iff
% 0.34/0.54 thf(fact_1622_le__minus__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 0.34/0.54 = ( ord_less_eq_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_iff
% 0.34/0.54 thf(fact_1623_le__minus__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 0.34/0.54 = ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_iff
% 0.34/0.54 thf(fact_1624_verit__negate__coefficient_I2_J,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % verit_negate_coefficient(2)
% 0.34/0.54 thf(fact_1625_verit__negate__coefficient_I2_J,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % verit_negate_coefficient(2)
% 0.34/0.54 thf(fact_1626_verit__negate__coefficient_I2_J,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.54 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % verit_negate_coefficient(2)
% 0.34/0.54 thf(fact_1627_verit__negate__coefficient_I2_J,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
% 0.34/0.54 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % verit_negate_coefficient(2)
% 0.34/0.54 thf(fact_1628_less__minus__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 0.34/0.54 = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_iff
% 0.34/0.54 thf(fact_1629_less__minus__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 0.34/0.54 = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_iff
% 0.34/0.54 thf(fact_1630_less__minus__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 0.34/0.54 = ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_iff
% 0.34/0.54 thf(fact_1631_less__minus__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 0.34/0.54 = ( ord_le6747313008572928689nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_iff
% 0.34/0.54 thf(fact_1632_minus__less__iff,axiom,
% 0.34/0.54 ! [A2: int,B3: int] :
% 0.34/0.54 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_less_iff
% 0.34/0.54 thf(fact_1633_minus__less__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_less_iff
% 0.34/0.54 thf(fact_1634_minus__less__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_less_iff
% 0.34/0.54 thf(fact_1635_minus__less__iff,axiom,
% 0.34/0.54 ! [A2: code_integer,B3: code_integer] :
% 0.34/0.54 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 0.34/0.54 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % minus_less_iff
% 0.34/0.54 thf(fact_1636_zero__induct__lemma,axiom,
% 0.34/0.54 ! [P: nat > $o,K: nat,I: nat] :
% 0.34/0.54 ( ( P @ K )
% 0.34/0.54 => ( ! [N3: nat] :
% 0.34/0.54 ( ( P @ ( suc @ N3 ) )
% 0.34/0.54 => ( P @ N3 ) )
% 0.34/0.54 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_induct_lemma
% 0.34/0.54 thf(fact_1637_diffs0__imp__equal,axiom,
% 0.34/0.54 ! [M: nat,N: nat] :
% 0.34/0.54 ( ( ( minus_minus_nat @ M @ N )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 => ( ( ( minus_minus_nat @ N @ M )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 => ( M = N ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diffs0_imp_equal
% 0.34/0.54 thf(fact_1638_minus__nat_Odiff__0,axiom,
% 0.34/0.54 ! [M: nat] :
% 0.34/0.54 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.34/0.54 = M ) ).
% 0.34/0.54
% 0.34/0.54 % minus_nat.diff_0
% 0.34/0.54 thf(fact_1639_diff__less__mono2,axiom,
% 0.34/0.54 ! [M: nat,N: nat,L: nat] :
% 0.34/0.54 ( ( ord_less_nat @ M @ N )
% 0.34/0.54 => ( ( ord_less_nat @ M @ L )
% 0.34/0.54 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_less_mono2
% 0.34/0.54 thf(fact_1640_less__imp__diff__less,axiom,
% 0.34/0.54 ! [J: nat,K: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_nat @ J @ K )
% 0.34/0.54 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_imp_diff_less
% 0.34/0.54 thf(fact_1641_diff__le__mono2,axiom,
% 0.34/0.54 ! [M: nat,N: nat,L: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_le_mono2
% 0.34/0.54 thf(fact_1642_le__diff__iff_H,axiom,
% 0.34/0.54 ! [A2: nat,C: nat,B3: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ A2 @ C )
% 0.34/0.54 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
% 0.34/0.54 = ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_diff_iff'
% 0.34/0.54 thf(fact_1643_diff__le__self,axiom,
% 0.34/0.54 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 0.34/0.54
% 0.34/0.54 % diff_le_self
% 0.34/0.54 thf(fact_1644_diff__le__mono,axiom,
% 0.34/0.54 ! [M: nat,N: nat,L: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_le_mono
% 0.34/0.54 thf(fact_1645_Nat_Odiff__diff__eq,axiom,
% 0.34/0.54 ! [K: nat,M: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.34/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.34/0.54 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.34/0.54 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Nat.diff_diff_eq
% 0.34/0.54 thf(fact_1646_le__diff__iff,axiom,
% 0.34/0.54 ! [K: nat,M: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.34/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.34/0.54 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_diff_iff
% 0.34/0.54 thf(fact_1647_eq__diff__iff,axiom,
% 0.34/0.54 ! [K: nat,M: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.34/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.34/0.54 => ( ( ( minus_minus_nat @ M @ K )
% 0.34/0.54 = ( minus_minus_nat @ N @ K ) )
% 0.34/0.54 = ( M = N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_diff_iff
% 0.34/0.54 thf(fact_1648_vebt__delete_Osimps_I4_J,axiom,
% 0.34/0.54 ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 0.34/0.54 ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
% 0.34/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_delete.simps(4)
% 0.34/0.54 thf(fact_1649_vebt__member_Osimps_I2_J,axiom,
% 0.34/0.54 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 0.34/0.54 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_member.simps(2)
% 0.34/0.54 thf(fact_1650_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.34/0.54 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.minNull.simps(4)
% 0.34/0.54 thf(fact_1651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.34/0.54 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.34/0.54 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.minNull.simps(5)
% 0.34/0.54 thf(fact_1652_verit__less__mono__div__int2,axiom,
% 0.34/0.54 ! [A3: int,B2: int,N: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 0.34/0.54 => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % verit_less_mono_div_int2
% 0.34/0.54 thf(fact_1653_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_list_nat,N: nat] :
% 0.34/0.54 ( ~ ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ? [B8: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B8 )
% 0.34/0.54 & ( ( finite_card_list_nat @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_le6045566169113846134st_nat @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1654_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_set_nat,N: nat] :
% 0.34/0.54 ( ~ ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ? [B8: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B8 )
% 0.34/0.54 & ( ( finite_card_set_nat @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_le6893508408891458716et_nat @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1655_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_nat,N: nat] :
% 0.34/0.54 ( ~ ( finite_finite_nat @ A3 )
% 0.34/0.54 => ? [B8: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B8 )
% 0.34/0.54 & ( ( finite_card_nat @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1656_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_complex,N: nat] :
% 0.34/0.54 ( ~ ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ? [B8: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B8 )
% 0.34/0.54 & ( ( finite_card_complex @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_le211207098394363844omplex @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1657_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 0.34/0.54 ( ~ ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ? [B8: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B8 )
% 0.34/0.54 & ( ( finite711546835091564841at_nat @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_le3146513528884898305at_nat @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1658_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat,N: nat] :
% 0.34/0.54 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ? [B8: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B8 )
% 0.34/0.54 & ( ( finite121521170596916366d_enat @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_le7203529160286727270d_enat @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1659_infinite__arbitrarily__large,axiom,
% 0.34/0.54 ! [A3: set_int,N: nat] :
% 0.34/0.54 ( ~ ( finite_finite_int @ A3 )
% 0.34/0.54 => ? [B8: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B8 )
% 0.34/0.54 & ( ( finite_card_int @ B8 )
% 0.34/0.54 = N )
% 0.34/0.54 & ( ord_less_eq_set_int @ B8 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % infinite_arbitrarily_large
% 0.34/0.54 thf(fact_1660_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.54 = ( finite_card_list_nat @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1661_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.54 = ( finite_card_set_nat @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1662_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite_card_nat @ A3 )
% 0.34/0.54 = ( finite_card_nat @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1663_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite_card_complex @ A3 )
% 0.34/0.54 = ( finite_card_complex @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1664_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.54 = ( finite711546835091564841at_nat @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1665_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.54 = ( finite121521170596916366d_enat @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1666_card__subset__eq,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ( ( finite_card_int @ A3 )
% 0.34/0.54 = ( finite_card_int @ B2 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_subset_eq
% 0.34/0.54 thf(fact_1667_le__iff__diff__le__0,axiom,
% 0.34/0.54 ( ord_less_eq_real
% 0.34/0.54 = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_iff_diff_le_0
% 0.34/0.54 thf(fact_1668_le__iff__diff__le__0,axiom,
% 0.34/0.54 ( ord_less_eq_rat
% 0.34/0.54 = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_iff_diff_le_0
% 0.34/0.54 thf(fact_1669_le__iff__diff__le__0,axiom,
% 0.34/0.54 ( ord_less_eq_int
% 0.34/0.54 = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_iff_diff_le_0
% 0.34/0.54 thf(fact_1670_less__iff__diff__less__0,axiom,
% 0.34/0.54 ( ord_less_real
% 0.34/0.54 = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_iff_diff_less_0
% 0.34/0.54 thf(fact_1671_less__iff__diff__less__0,axiom,
% 0.34/0.54 ( ord_less_rat
% 0.34/0.54 = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_iff_diff_less_0
% 0.34/0.54 thf(fact_1672_less__iff__diff__less__0,axiom,
% 0.34/0.54 ( ord_less_int
% 0.34/0.54 = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_iff_diff_less_0
% 0.34/0.54 thf(fact_1673_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_real,A3: set_real,R2: real > real > $o] :
% 0.34/0.54 ( ( finite_finite_real @ B2 )
% 0.34/0.54 => ( ! [A: real] :
% 0.34/0.54 ( ( member_real @ A @ A3 )
% 0.34/0.54 => ? [B9: real] :
% 0.34/0.54 ( ( member_real @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: real,A22: real,B: real] :
% 0.34/0.54 ( ( member_real @ A1 @ A3 )
% 0.34/0.54 => ( ( member_real @ A22 @ A3 )
% 0.34/0.54 => ( ( member_real @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1674_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_o,A3: set_real,R2: real > $o > $o] :
% 0.34/0.54 ( ( finite_finite_o @ B2 )
% 0.34/0.54 => ( ! [A: real] :
% 0.34/0.54 ( ( member_real @ A @ A3 )
% 0.34/0.54 => ? [B9: $o] :
% 0.34/0.54 ( ( member_o @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: real,A22: real,B: $o] :
% 0.34/0.54 ( ( member_real @ A1 @ A3 )
% 0.34/0.54 => ( ( member_real @ A22 @ A3 )
% 0.34/0.54 => ( ( member_o @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1675_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_real,A3: set_o,R2: $o > real > $o] :
% 0.34/0.54 ( ( finite_finite_real @ B2 )
% 0.34/0.54 => ( ! [A: $o] :
% 0.34/0.54 ( ( member_o @ A @ A3 )
% 0.34/0.54 => ? [B9: real] :
% 0.34/0.54 ( ( member_real @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: $o,A22: $o,B: real] :
% 0.34/0.54 ( ( member_o @ A1 @ A3 )
% 0.34/0.54 => ( ( member_o @ A22 @ A3 )
% 0.34/0.54 => ( ( member_real @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1676_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_o,A3: set_o,R2: $o > $o > $o] :
% 0.34/0.54 ( ( finite_finite_o @ B2 )
% 0.34/0.54 => ( ! [A: $o] :
% 0.34/0.54 ( ( member_o @ A @ A3 )
% 0.34/0.54 => ? [B9: $o] :
% 0.34/0.54 ( ( member_o @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: $o,A22: $o,B: $o] :
% 0.34/0.54 ( ( member_o @ A1 @ A3 )
% 0.34/0.54 => ( ( member_o @ A22 @ A3 )
% 0.34/0.54 => ( ( member_o @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1677_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_real,A3: set_complex,R2: complex > real > $o] :
% 0.34/0.54 ( ( finite_finite_real @ B2 )
% 0.34/0.54 => ( ! [A: complex] :
% 0.34/0.54 ( ( member_complex @ A @ A3 )
% 0.34/0.54 => ? [B9: real] :
% 0.34/0.54 ( ( member_real @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: complex,A22: complex,B: real] :
% 0.34/0.54 ( ( member_complex @ A1 @ A3 )
% 0.34/0.54 => ( ( member_complex @ A22 @ A3 )
% 0.34/0.54 => ( ( member_real @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1678_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_o,A3: set_complex,R2: complex > $o > $o] :
% 0.34/0.54 ( ( finite_finite_o @ B2 )
% 0.34/0.54 => ( ! [A: complex] :
% 0.34/0.54 ( ( member_complex @ A @ A3 )
% 0.34/0.54 => ? [B9: $o] :
% 0.34/0.54 ( ( member_o @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: complex,A22: complex,B: $o] :
% 0.34/0.54 ( ( member_complex @ A1 @ A3 )
% 0.34/0.54 => ( ( member_complex @ A22 @ A3 )
% 0.34/0.54 => ( ( member_o @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1679_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_real,A3: set_nat,R2: nat > real > $o] :
% 0.34/0.54 ( ( finite_finite_real @ B2 )
% 0.34/0.54 => ( ! [A: nat] :
% 0.34/0.54 ( ( member_nat @ A @ A3 )
% 0.34/0.54 => ? [B9: real] :
% 0.34/0.54 ( ( member_real @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: nat,A22: nat,B: real] :
% 0.34/0.54 ( ( member_nat @ A1 @ A3 )
% 0.34/0.54 => ( ( member_nat @ A22 @ A3 )
% 0.34/0.54 => ( ( member_real @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1680_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_o,A3: set_nat,R2: nat > $o > $o] :
% 0.34/0.54 ( ( finite_finite_o @ B2 )
% 0.34/0.54 => ( ! [A: nat] :
% 0.34/0.54 ( ( member_nat @ A @ A3 )
% 0.34/0.54 => ? [B9: $o] :
% 0.34/0.54 ( ( member_o @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: nat,A22: nat,B: $o] :
% 0.34/0.54 ( ( member_nat @ A1 @ A3 )
% 0.34/0.54 => ( ( member_nat @ A22 @ A3 )
% 0.34/0.54 => ( ( member_o @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1681_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_real,A3: set_int,R2: int > real > $o] :
% 0.34/0.54 ( ( finite_finite_real @ B2 )
% 0.34/0.54 => ( ! [A: int] :
% 0.34/0.54 ( ( member_int @ A @ A3 )
% 0.34/0.54 => ? [B9: real] :
% 0.34/0.54 ( ( member_real @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: int,A22: int,B: real] :
% 0.34/0.54 ( ( member_int @ A1 @ A3 )
% 0.34/0.54 => ( ( member_int @ A22 @ A3 )
% 0.34/0.54 => ( ( member_real @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1682_card__le__if__inj__on__rel,axiom,
% 0.34/0.54 ! [B2: set_o,A3: set_int,R2: int > $o > $o] :
% 0.34/0.54 ( ( finite_finite_o @ B2 )
% 0.34/0.54 => ( ! [A: int] :
% 0.34/0.54 ( ( member_int @ A @ A3 )
% 0.34/0.54 => ? [B9: $o] :
% 0.34/0.54 ( ( member_o @ B9 @ B2 )
% 0.34/0.54 & ( R2 @ A @ B9 ) ) )
% 0.34/0.54 => ( ! [A1: int,A22: int,B: $o] :
% 0.34/0.54 ( ( member_int @ A1 @ A3 )
% 0.34/0.54 => ( ( member_int @ A22 @ A3 )
% 0.34/0.54 => ( ( member_o @ B @ B2 )
% 0.34/0.54 => ( ( R2 @ A1 @ B )
% 0.34/0.54 => ( ( R2 @ A22 @ B )
% 0.34/0.54 => ( A1 = A22 ) ) ) ) ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_if_inj_on_rel
% 0.34/0.54 thf(fact_1683_le__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(4)
% 0.34/0.54 thf(fact_1684_le__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(4)
% 0.34/0.54 thf(fact_1685_le__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(4)
% 0.34/0.54 thf(fact_1686_le__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(4)
% 0.34/0.54 thf(fact_1687_le__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(2)
% 0.34/0.54 thf(fact_1688_le__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(2)
% 0.34/0.54 thf(fact_1689_le__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(2)
% 0.34/0.54 thf(fact_1690_le__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(2)
% 0.34/0.54 thf(fact_1691_zero__neq__neg__one,axiom,
% 0.34/0.54 ( zero_zero_int
% 0.34/0.54 != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_neq_neg_one
% 0.34/0.54 thf(fact_1692_zero__neq__neg__one,axiom,
% 0.34/0.54 ( zero_zero_real
% 0.34/0.54 != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_neq_neg_one
% 0.34/0.54 thf(fact_1693_zero__neq__neg__one,axiom,
% 0.34/0.54 ( zero_zero_rat
% 0.34/0.54 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_neq_neg_one
% 0.34/0.54 thf(fact_1694_zero__neq__neg__one,axiom,
% 0.34/0.54 ( zero_z3403309356797280102nteger
% 0.34/0.54 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_neq_neg_one
% 0.34/0.54 thf(fact_1695_zero__neq__neg__one,axiom,
% 0.34/0.54 ( zero_zero_complex
% 0.34/0.54 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_neq_neg_one
% 0.34/0.54 thf(fact_1696_less__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(2)
% 0.34/0.54 thf(fact_1697_less__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(2)
% 0.34/0.54 thf(fact_1698_less__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(2)
% 0.34/0.54 thf(fact_1699_less__minus__one__simps_I2_J,axiom,
% 0.34/0.54 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(2)
% 0.34/0.54 thf(fact_1700_less__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(4)
% 0.34/0.54 thf(fact_1701_less__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(4)
% 0.34/0.54 thf(fact_1702_less__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(4)
% 0.34/0.54 thf(fact_1703_less__minus__one__simps_I4_J,axiom,
% 0.34/0.54 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(4)
% 0.34/0.54 thf(fact_1704_diff__less__Suc,axiom,
% 0.34/0.54 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_less_Suc
% 0.34/0.54 thf(fact_1705_Suc__diff__Suc,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_nat @ N @ M )
% 0.34/0.54 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 0.34/0.54 = ( minus_minus_nat @ M @ N ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Suc_diff_Suc
% 0.34/0.54 thf(fact_1706_diff__less,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.54 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.34/0.54 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_less
% 0.34/0.54 thf(fact_1707_Suc__diff__le,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.54 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.34/0.54 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Suc_diff_le
% 0.34/0.54 thf(fact_1708_diff__less__mono,axiom,
% 0.34/0.54 ! [A2: nat,B3: nat,C: nat] :
% 0.34/0.54 ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ C @ A2 )
% 0.34/0.54 => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_less_mono
% 0.34/0.54 thf(fact_1709_less__diff__iff,axiom,
% 0.34/0.54 ! [K: nat,M: nat,N: nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ K @ M )
% 0.34/0.54 => ( ( ord_less_eq_nat @ K @ N )
% 0.34/0.54 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 0.34/0.54 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_diff_iff
% 0.34/0.54 thf(fact_1710_diff__Suc__eq__diff__pred,axiom,
% 0.34/0.54 ! [M: nat,N: nat] :
% 0.34/0.54 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.34/0.54 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_Suc_eq_diff_pred
% 0.34/0.54 thf(fact_1711_not__int__zless__negative,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % not_int_zless_negative
% 0.34/0.54 thf(fact_1712_vebt__member_Osimps_I3_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 0.34/0.54 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_member.simps(3)
% 0.34/0.54 thf(fact_1713_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.34/0.54 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.34/0.54 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.membermima.simps(2)
% 0.34/0.54 thf(fact_1714_vebt__mint_Osimps_I2_J,axiom,
% 0.34/0.54 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.34/0.54 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_mint.simps(2)
% 0.34/0.54 thf(fact_1715_vebt__maxt_Osimps_I2_J,axiom,
% 0.34/0.54 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.34/0.54 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_maxt.simps(2)
% 0.34/0.54 thf(fact_1716_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.34/0.54 ! [X2: vEBT_VEBT] :
% 0.34/0.54 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 0.34/0.54 => ( ! [Uv2: $o] :
% 0.34/0.54 ( X2
% 0.34/0.54 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.34/0.54 => ( ! [Uu2: $o] :
% 0.34/0.54 ( X2
% 0.34/0.54 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.34/0.54 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.34/0.54 ( X2
% 0.34/0.54 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.minNull.elims(3)
% 0.34/0.54 thf(fact_1717_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.34/0.54 ! [X2: vEBT_VEBT] :
% 0.34/0.54 ( ( vEBT_VEBT_minNull @ X2 )
% 0.34/0.54 => ( ( X2
% 0.34/0.54 != ( vEBT_Leaf @ $false @ $false ) )
% 0.34/0.54 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.34/0.54 ( X2
% 0.34/0.54 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.minNull.elims(2)
% 0.34/0.54 thf(fact_1718_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.34/0.54 ! [X2: vEBT_VEBT,Y3: $o] :
% 0.34/0.54 ( ( ( vEBT_VEBT_minNull @ X2 )
% 0.34/0.54 = Y3 )
% 0.34/0.54 => ( ( ( X2
% 0.34/0.54 = ( vEBT_Leaf @ $false @ $false ) )
% 0.34/0.54 => ~ Y3 )
% 0.34/0.54 => ( ( ? [Uv2: $o] :
% 0.34/0.54 ( X2
% 0.34/0.54 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.34/0.54 => Y3 )
% 0.34/0.54 => ( ( ? [Uu2: $o] :
% 0.34/0.54 ( X2
% 0.34/0.54 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.34/0.54 => Y3 )
% 0.34/0.54 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.34/0.54 ( X2
% 0.34/0.54 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.34/0.54 => ~ Y3 )
% 0.34/0.54 => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.34/0.54 ( X2
% 0.34/0.54 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.34/0.54 => Y3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.minNull.elims(1)
% 0.34/0.54 thf(fact_1719_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.34/0.54 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.34/0.54 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT_internal.naive_member.simps(2)
% 0.34/0.54 thf(fact_1720_vebt__pred_Osimps_I4_J,axiom,
% 0.34/0.54 ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 0.34/0.54 ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_pred.simps(4)
% 0.34/0.54 thf(fact_1721_vebt__succ_Osimps_I3_J,axiom,
% 0.34/0.54 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 0.34/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_succ.simps(3)
% 0.34/0.54 thf(fact_1722_frac__ge__0,axiom,
% 0.34/0.54 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % frac_ge_0
% 0.34/0.54 thf(fact_1723_frac__ge__0,axiom,
% 0.34/0.54 ! [X2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % frac_ge_0
% 0.34/0.54 thf(fact_1724_frac__lt__1,axiom,
% 0.34/0.54 ! [X2: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X2 ) @ one_one_real ) ).
% 0.34/0.54
% 0.34/0.54 % frac_lt_1
% 0.34/0.54 thf(fact_1725_frac__lt__1,axiom,
% 0.34/0.54 ! [X2: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X2 ) @ one_one_rat ) ).
% 0.34/0.54
% 0.34/0.54 % frac_lt_1
% 0.34/0.54 thf(fact_1726_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_list_nat )
% 0.34/0.54 | ~ ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1727_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_set_nat )
% 0.34/0.54 | ~ ( finite1152437895449049373et_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1728_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ( ( finite_card_complex @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_complex )
% 0.34/0.54 | ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1729_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bo2099793752762293965at_nat )
% 0.34/0.54 | ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1730_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bo7653980558646680370d_enat )
% 0.34/0.54 | ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1731_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( ( finite_card_real @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_real )
% 0.34/0.54 | ~ ( finite_finite_real @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1732_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( ( finite_card_o @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_o )
% 0.34/0.54 | ~ ( finite_finite_o @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1733_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( ( finite_card_nat @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_nat )
% 0.34/0.54 | ~ ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1734_card__eq__0__iff,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( ( finite_card_int @ A3 )
% 0.34/0.54 = zero_zero_nat )
% 0.34/0.54 = ( ( A3 = bot_bot_set_int )
% 0.34/0.54 | ~ ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_eq_0_iff
% 0.34/0.54 thf(fact_1735_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
% 0.34/0.54 => ( finite8100373058378681591st_nat @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1736_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
% 0.34/0.54 => ( finite1152437895449049373et_nat @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1737_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 0.34/0.54 => ( finite_finite_nat @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1738_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 0.34/0.54 => ( finite_finite_int @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1739_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 0.34/0.54 => ( finite3207457112153483333omplex @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1740_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
% 0.34/0.54 => ( finite6177210948735845034at_nat @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1741_card__ge__0__finite,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
% 0.34/0.54 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_ge_0_finite
% 0.34/0.54 thf(fact_1742_card__mono,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1743_card__mono,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1744_card__mono,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1745_card__mono,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1746_card__mono,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1747_card__mono,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1748_card__mono,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_mono
% 0.34/0.54 thf(fact_1749_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B2 ) @ ( finite_card_list_nat @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1750_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1751_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1752_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1753_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B2 ) @ ( finite711546835091564841at_nat @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1754_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1755_card__seteq,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A3 ) )
% 0.34/0.54 => ( A3 = B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_seteq
% 0.34/0.54 thf(fact_1756_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_list_nat,C2: nat] :
% 0.34/0.54 ( ! [G: set_list_nat] :
% 0.34/0.54 ( ( ord_le6045566169113846134st_nat @ G @ F2 )
% 0.34/0.54 => ( ( finite8100373058378681591st_nat @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite8100373058378681591st_nat @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1757_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_set_nat,C2: nat] :
% 0.34/0.54 ( ! [G: set_set_nat] :
% 0.34/0.54 ( ( ord_le6893508408891458716et_nat @ G @ F2 )
% 0.34/0.54 => ( ( finite1152437895449049373et_nat @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite1152437895449049373et_nat @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1758_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_nat,C2: nat] :
% 0.34/0.54 ( ! [G: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ G @ F2 )
% 0.34/0.54 => ( ( finite_finite_nat @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite_finite_nat @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1759_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_complex,C2: nat] :
% 0.34/0.54 ( ! [G: set_complex] :
% 0.34/0.54 ( ( ord_le211207098394363844omplex @ G @ F2 )
% 0.34/0.54 => ( ( finite3207457112153483333omplex @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite3207457112153483333omplex @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1760_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
% 0.34/0.54 ( ! [G: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_le3146513528884898305at_nat @ G @ F2 )
% 0.34/0.54 => ( ( finite6177210948735845034at_nat @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite6177210948735845034at_nat @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1761_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_Extended_enat,C2: nat] :
% 0.34/0.54 ( ! [G: set_Extended_enat] :
% 0.34/0.54 ( ( ord_le7203529160286727270d_enat @ G @ F2 )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1762_finite__if__finite__subsets__card__bdd,axiom,
% 0.34/0.54 ! [F2: set_int,C2: nat] :
% 0.34/0.54 ( ! [G: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ G @ F2 )
% 0.34/0.54 => ( ( finite_finite_int @ G )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C2 ) ) )
% 0.34/0.54 => ( ( finite_finite_int @ F2 )
% 0.34/0.54 & ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_if_finite_subsets_card_bdd
% 0.34/0.54 thf(fact_1763_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_list_nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S ) )
% 0.34/0.54 => ~ ! [T4: set_list_nat] :
% 0.34/0.54 ( ( ord_le6045566169113846134st_nat @ T4 @ S )
% 0.34/0.54 => ( ( ( finite_card_list_nat @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1764_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_set_nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
% 0.34/0.54 => ~ ! [T4: set_set_nat] :
% 0.34/0.54 ( ( ord_le6893508408891458716et_nat @ T4 @ S )
% 0.34/0.54 => ( ( ( finite_card_set_nat @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite1152437895449049373et_nat @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1765_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ~ ! [T4: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ T4 @ S )
% 0.34/0.54 => ( ( ( finite_card_nat @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1766_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_complex] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S ) )
% 0.34/0.54 => ~ ! [T4: set_complex] :
% 0.34/0.54 ( ( ord_le211207098394363844omplex @ T4 @ S )
% 0.34/0.54 => ( ( ( finite_card_complex @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1767_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S ) )
% 0.34/0.54 => ~ ! [T4: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_le3146513528884898305at_nat @ T4 @ S )
% 0.34/0.54 => ( ( ( finite711546835091564841at_nat @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1768_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_Extended_enat] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ~ ! [T4: set_Extended_enat] :
% 0.34/0.54 ( ( ord_le7203529160286727270d_enat @ T4 @ S )
% 0.34/0.54 => ( ( ( finite121521170596916366d_enat @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite4001608067531595151d_enat @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1769_obtain__subset__with__card__n,axiom,
% 0.34/0.54 ! [N: nat,S: set_int] :
% 0.34/0.54 ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
% 0.34/0.54 => ~ ! [T4: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ T4 @ S )
% 0.34/0.54 => ( ( ( finite_card_int @ T4 )
% 0.34/0.54 = N )
% 0.34/0.54 => ~ ( finite_finite_int @ T4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % obtain_subset_with_card_n
% 0.34/0.54 thf(fact_1770_le__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(3)
% 0.34/0.54 thf(fact_1771_le__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(3)
% 0.34/0.54 thf(fact_1772_le__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(3)
% 0.34/0.54 thf(fact_1773_le__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(3)
% 0.34/0.54 thf(fact_1774_le__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(1)
% 0.34/0.54 thf(fact_1775_le__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(1)
% 0.34/0.54 thf(fact_1776_le__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(1)
% 0.34/0.54 thf(fact_1777_le__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.34/0.54
% 0.34/0.54 % le_minus_one_simps(1)
% 0.34/0.54 thf(fact_1778_less__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(1)
% 0.34/0.54 thf(fact_1779_less__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(1)
% 0.34/0.54 thf(fact_1780_less__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(1)
% 0.34/0.54 thf(fact_1781_less__minus__one__simps_I1_J,axiom,
% 0.34/0.54 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(1)
% 0.34/0.54 thf(fact_1782_less__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(3)
% 0.34/0.54 thf(fact_1783_less__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(3)
% 0.34/0.54 thf(fact_1784_less__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(3)
% 0.34/0.54 thf(fact_1785_less__minus__one__simps_I3_J,axiom,
% 0.34/0.54 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_minus_one_simps(3)
% 0.34/0.54 thf(fact_1786_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1787_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_set_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1788_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1789_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1790_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_less_set_complex @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1791_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1792_psubset__card__mono,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_card_mono
% 0.34/0.54 thf(fact_1793_diff__Suc__less,axiom,
% 0.34/0.54 ! [N: nat,I: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.54 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_Suc_less
% 0.34/0.54 thf(fact_1794_finite__enumerate__in__set,axiom,
% 0.34/0.54 ! [S: set_Extended_enat,N: nat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_in_set
% 0.34/0.54 thf(fact_1795_finite__enumerate__in__set,axiom,
% 0.34/0.54 ! [S: set_nat,N: nat] :
% 0.34/0.54 ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_in_set
% 0.34/0.54 thf(fact_1796_finite__enumerate__Ex,axiom,
% 0.34/0.54 ! [S: set_Extended_enat,S2: extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ( ( member_Extended_enat @ S2 @ S )
% 0.34/0.54 => ? [N3: nat] :
% 0.34/0.54 ( ( ord_less_nat @ N3 @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 & ( ( infini7641415182203889163d_enat @ S @ N3 )
% 0.34/0.54 = S2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_Ex
% 0.34/0.54 thf(fact_1797_finite__enumerate__Ex,axiom,
% 0.34/0.54 ! [S: set_nat,S2: nat] :
% 0.34/0.54 ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( member_nat @ S2 @ S )
% 0.34/0.54 => ? [N3: nat] :
% 0.34/0.54 ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
% 0.34/0.54 & ( ( infini8530281810654367211te_nat @ S @ N3 )
% 0.34/0.54 = S2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_Ex
% 0.34/0.54 thf(fact_1798_finite__enum__ext,axiom,
% 0.34/0.54 ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
% 0.34/0.54 ( ! [I2: nat] :
% 0.34/0.54 ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
% 0.34/0.54 => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
% 0.34/0.54 = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ X6 )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ Y7 )
% 0.34/0.54 => ( ( ( finite121521170596916366d_enat @ X6 )
% 0.34/0.54 = ( finite121521170596916366d_enat @ Y7 ) )
% 0.34/0.54 => ( X6 = Y7 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enum_ext
% 0.34/0.54 thf(fact_1799_finite__enum__ext,axiom,
% 0.34/0.54 ! [X6: set_nat,Y7: set_nat] :
% 0.34/0.54 ( ! [I2: nat] :
% 0.34/0.54 ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
% 0.34/0.54 => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
% 0.34/0.54 = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
% 0.34/0.54 => ( ( finite_finite_nat @ X6 )
% 0.34/0.54 => ( ( finite_finite_nat @ Y7 )
% 0.34/0.54 => ( ( ( finite_card_nat @ X6 )
% 0.34/0.54 = ( finite_card_nat @ Y7 ) )
% 0.34/0.54 => ( X6 = Y7 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enum_ext
% 0.34/0.54 thf(fact_1800_int__cases4,axiom,
% 0.34/0.54 ! [M: int] :
% 0.34/0.54 ( ! [N3: nat] :
% 0.34/0.54 ( M
% 0.34/0.54 != ( semiri1314217659103216013at_int @ N3 ) )
% 0.34/0.54 => ~ ! [N3: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.34/0.54 => ( M
% 0.34/0.54 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_cases4
% 0.34/0.54 thf(fact_1801_vebt__member_Osimps_I4_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.34/0.54 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_member.simps(4)
% 0.34/0.54 thf(fact_1802_int__zle__neg,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.34/0.54 = ( ( N = zero_zero_nat )
% 0.34/0.54 & ( M = zero_zero_nat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_zle_neg
% 0.34/0.54 thf(fact_1803_negative__zle__0,axiom,
% 0.34/0.54 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 0.34/0.54
% 0.34/0.54 % negative_zle_0
% 0.34/0.54 thf(fact_1804_nonpos__int__cases,axiom,
% 0.34/0.54 ! [K: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.34/0.54 => ~ ! [N3: nat] :
% 0.34/0.54 ( K
% 0.34/0.54 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % nonpos_int_cases
% 0.34/0.54 thf(fact_1805_vebt__pred_Osimps_I5_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 0.34/0.54 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_pred.simps(5)
% 0.34/0.54 thf(fact_1806_vebt__succ_Osimps_I4_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 0.34/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_succ.simps(4)
% 0.34/0.54 thf(fact_1807_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_list_nat )
% 0.34/0.54 & ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1808_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_set_nat )
% 0.34/0.54 & ( finite1152437895449049373et_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1809_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_complex )
% 0.34/0.54 & ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1810_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bo2099793752762293965at_nat )
% 0.34/0.54 & ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1811_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.34/0.54 & ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1812_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_real )
% 0.34/0.54 & ( finite_finite_real @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1813_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_o )
% 0.34/0.54 & ( finite_finite_o @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1814_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_nat )
% 0.34/0.54 & ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1815_card__gt__0__iff,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 0.34/0.54 = ( ( A3 != bot_bot_set_int )
% 0.34/0.54 & ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_gt_0_iff
% 0.34/0.54 thf(fact_1816_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: list_nat] :
% 0.34/0.54 ( ( member_list_nat @ X @ A3 )
% 0.34/0.54 => ! [Y: list_nat] :
% 0.34/0.54 ( ( member_list_nat @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1817_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: set_nat] :
% 0.34/0.54 ( ( member_set_nat @ X @ A3 )
% 0.34/0.54 => ! [Y: set_nat] :
% 0.34/0.54 ( ( member_set_nat @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1818_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: nat] :
% 0.34/0.54 ( ( member_nat @ X @ A3 )
% 0.34/0.54 => ! [Y: nat] :
% 0.34/0.54 ( ( member_nat @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1819_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: int] :
% 0.34/0.54 ( ( member_int @ X @ A3 )
% 0.34/0.54 => ! [Y: int] :
% 0.34/0.54 ( ( member_int @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1820_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: complex] :
% 0.34/0.54 ( ( member_complex @ X @ A3 )
% 0.34/0.54 => ! [Y: complex] :
% 0.34/0.54 ( ( member_complex @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1821_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: product_prod_nat_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ X @ A3 )
% 0.34/0.54 => ! [Y: product_prod_nat_nat] :
% 0.34/0.54 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1822_card__le__Suc0__iff__eq,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 0.34/0.54 = ( ! [X: extended_enat] :
% 0.34/0.54 ( ( member_Extended_enat @ X @ A3 )
% 0.34/0.54 => ! [Y: extended_enat] :
% 0.34/0.54 ( ( member_Extended_enat @ Y @ A3 )
% 0.34/0.54 => ( X = Y ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_Suc0_iff_eq
% 0.34/0.54 thf(fact_1823_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 0.34/0.54 => ( ord_le1190675801316882794st_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1824_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1825_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1826_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 0.34/0.54 => ( ord_less_set_complex @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1827_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 0.34/0.54 => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1828_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 0.34/0.54 => ( ord_le2529575680413868914d_enat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1829_card__psubset,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 0.34/0.54 => ( ord_less_set_int @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_psubset
% 0.34/0.54 thf(fact_1830_finite__enumerate__mono,axiom,
% 0.34/0.54 ! [M: nat,N: nat,S: set_Extended_enat] :
% 0.34/0.54 ( ( ord_less_nat @ M @ N )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_mono
% 0.34/0.54 thf(fact_1831_finite__enumerate__mono,axiom,
% 0.34/0.54 ! [M: nat,N: nat,S: set_nat] :
% 0.34/0.54 ( ( ord_less_nat @ M @ N )
% 0.34/0.54 => ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_mono
% 0.34/0.54 thf(fact_1832_Suc__diff__eq__diff__pred,axiom,
% 0.34/0.54 ! [N: nat,M: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.54 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 0.34/0.54 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Suc_diff_eq_diff_pred
% 0.34/0.54 thf(fact_1833_Suc__pred_H,axiom,
% 0.34/0.54 ! [N: nat] :
% 0.34/0.54 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.54 => ( N
% 0.34/0.54 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Suc_pred'
% 0.34/0.54 thf(fact_1834_int__cases3,axiom,
% 0.34/0.54 ! [K: int] :
% 0.34/0.54 ( ( K != zero_zero_int )
% 0.34/0.54 => ( ! [N3: nat] :
% 0.34/0.54 ( ( K
% 0.34/0.54 = ( semiri1314217659103216013at_int @ N3 ) )
% 0.34/0.54 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 0.34/0.54 => ~ ! [N3: nat] :
% 0.34/0.54 ( ( K
% 0.34/0.54 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.34/0.54 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_cases3
% 0.34/0.54 thf(fact_1835_not__zle__0__negative,axiom,
% 0.34/0.54 ! [N: nat] :
% 0.34/0.54 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % not_zle_0_negative
% 0.34/0.54 thf(fact_1836_negD,axiom,
% 0.34/0.54 ! [X2: int] :
% 0.34/0.54 ( ( ord_less_int @ X2 @ zero_zero_int )
% 0.34/0.54 => ? [N3: nat] :
% 0.34/0.54 ( X2
% 0.34/0.54 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % negD
% 0.34/0.54 thf(fact_1837_negative__zless__0,axiom,
% 0.34/0.54 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 0.34/0.54
% 0.34/0.54 % negative_zless_0
% 0.34/0.54 thf(fact_1838_vebt__pred_Osimps_I6_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 0.34/0.54 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_pred.simps(6)
% 0.34/0.54 thf(fact_1839_vebt__succ_Osimps_I5_J,axiom,
% 0.34/0.54 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 0.34/0.54 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 0.34/0.54 = none_nat ) ).
% 0.34/0.54
% 0.34/0.54 % vebt_succ.simps(5)
% 0.34/0.54 thf(fact_1840_finite__le__enumerate,axiom,
% 0.34/0.54 ! [S: set_nat,N: nat] :
% 0.34/0.54 ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_le_enumerate
% 0.34/0.54 thf(fact_1841_nat__approx__posE,axiom,
% 0.34/0.54 ! [E2: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.34/0.54 => ~ ! [N3: nat] :
% 0.34/0.54 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % nat_approx_posE
% 0.34/0.54 thf(fact_1842_nat__approx__posE,axiom,
% 0.34/0.54 ! [E2: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.34/0.54 => ~ ! [N3: nat] :
% 0.34/0.54 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % nat_approx_posE
% 0.34/0.54 thf(fact_1843_finite__enumerate__step,axiom,
% 0.34/0.54 ! [S: set_Extended_enat,N: nat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ S ) )
% 0.34/0.54 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_step
% 0.34/0.54 thf(fact_1844_finite__enumerate__step,axiom,
% 0.34/0.54 ! [S: set_nat,N: nat] :
% 0.34/0.54 ( ( finite_finite_nat @ S )
% 0.34/0.54 => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
% 0.34/0.54 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_enumerate_step
% 0.34/0.54 thf(fact_1845_le__divide__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.54 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_divide_eq_1_pos
% 0.34/0.54 thf(fact_1846_le__divide__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_divide_eq_1_pos
% 0.34/0.54 thf(fact_1847_le__divide__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_divide_eq_1_neg
% 0.34/0.54 thf(fact_1848_le__divide__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % le_divide_eq_1_neg
% 0.34/0.54 thf(fact_1849_divide__le__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.54 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_eq_1_pos
% 0.34/0.54 thf(fact_1850_divide__le__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.54 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_eq_1_pos
% 0.34/0.54 thf(fact_1851_divide__le__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.54 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_eq_1_neg
% 0.34/0.54 thf(fact_1852_divide__le__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.54 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_eq_1_neg
% 0.34/0.54 thf(fact_1853_zero__less__divide__1__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
% 0.34/0.54 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_less_divide_1_iff
% 0.34/0.54 thf(fact_1854_zero__less__divide__1__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
% 0.34/0.54 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_less_divide_1_iff
% 0.34/0.54 thf(fact_1855_less__divide__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.54 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_divide_eq_1_pos
% 0.34/0.54 thf(fact_1856_less__divide__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.54 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_divide_eq_1_pos
% 0.34/0.54 thf(fact_1857_less__divide__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_divide_eq_1_neg
% 0.34/0.54 thf(fact_1858_less__divide__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.54 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % less_divide_eq_1_neg
% 0.34/0.54 thf(fact_1859_divide__less__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.54 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.54 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_eq_1_pos
% 0.34/0.54 thf(fact_1860_divide__less__eq__1__pos,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.54 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.54 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_eq_1_pos
% 0.34/0.54 thf(fact_1861_divide__less__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.54 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_eq_1_neg
% 0.34/0.54 thf(fact_1862_divide__less__eq__1__neg,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.54 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_eq_1_neg
% 0.34/0.54 thf(fact_1863_divide__less__0__1__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.54 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_0_1_iff
% 0.34/0.54 thf(fact_1864_divide__less__0__1__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
% 0.34/0.54 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_less_0_1_iff
% 0.34/0.54 thf(fact_1865_Diff__empty,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
% 0.34/0.54 = A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_empty
% 0.34/0.54 thf(fact_1866_Diff__empty,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
% 0.34/0.54 = A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_empty
% 0.34/0.54 thf(fact_1867_Diff__empty,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
% 0.34/0.54 = A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_empty
% 0.34/0.54 thf(fact_1868_Diff__empty,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
% 0.34/0.54 = A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_empty
% 0.34/0.54 thf(fact_1869_empty__Diff,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
% 0.34/0.54 = bot_bot_set_real ) ).
% 0.34/0.54
% 0.34/0.54 % empty_Diff
% 0.34/0.54 thf(fact_1870_empty__Diff,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % empty_Diff
% 0.34/0.54 thf(fact_1871_empty__Diff,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % empty_Diff
% 0.34/0.54 thf(fact_1872_empty__Diff,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % empty_Diff
% 0.34/0.54 thf(fact_1873_Diff__cancel,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( minus_minus_set_real @ A3 @ A3 )
% 0.34/0.54 = bot_bot_set_real ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_cancel
% 0.34/0.54 thf(fact_1874_Diff__cancel,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( minus_minus_set_o @ A3 @ A3 )
% 0.34/0.54 = bot_bot_set_o ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_cancel
% 0.34/0.54 thf(fact_1875_Diff__cancel,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( minus_minus_set_int @ A3 @ A3 )
% 0.34/0.54 = bot_bot_set_int ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_cancel
% 0.34/0.54 thf(fact_1876_Diff__cancel,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( minus_minus_set_nat @ A3 @ A3 )
% 0.34/0.54 = bot_bot_set_nat ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_cancel
% 0.34/0.54 thf(fact_1877_finite__Diff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ A3 )
% 0.34/0.54 => ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff
% 0.34/0.54 thf(fact_1878_finite__Diff,axiom,
% 0.34/0.54 ! [A3: set_complex,B2: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff
% 0.34/0.54 thf(fact_1879_finite__Diff,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff
% 0.34/0.54 thf(fact_1880_finite__Diff,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff
% 0.34/0.54 thf(fact_1881_finite__Diff,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff
% 0.34/0.54 thf(fact_1882_finite__Diff2,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.54 = ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff2
% 0.34/0.54 thf(fact_1883_finite__Diff2,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.34/0.54 = ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff2
% 0.34/0.54 thf(fact_1884_finite__Diff2,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff2
% 0.34/0.54 thf(fact_1885_finite__Diff2,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.34/0.54 = ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff2
% 0.34/0.54 thf(fact_1886_finite__Diff2,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % finite_Diff2
% 0.34/0.54 thf(fact_1887_Compl__anti__mono,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Compl_anti_mono
% 0.34/0.54 thf(fact_1888_Compl__subset__Compl__iff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 0.34/0.54 = ( ord_less_eq_set_int @ B2 @ A3 ) ) ).
% 0.34/0.54
% 0.34/0.54 % Compl_subset_Compl_iff
% 0.34/0.54 thf(fact_1889_divide__eq__0__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ A2 @ B3 )
% 0.34/0.54 = zero_zero_rat )
% 0.34/0.54 = ( ( A2 = zero_zero_rat )
% 0.34/0.54 | ( B3 = zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_0_iff
% 0.34/0.54 thf(fact_1890_divide__eq__0__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ A2 @ B3 )
% 0.34/0.54 = zero_zero_real )
% 0.34/0.54 = ( ( A2 = zero_zero_real )
% 0.34/0.54 | ( B3 = zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_0_iff
% 0.34/0.54 thf(fact_1891_divide__eq__0__iff,axiom,
% 0.34/0.54 ! [A2: complex,B3: complex] :
% 0.34/0.54 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 0.34/0.54 = zero_zero_complex )
% 0.34/0.54 = ( ( A2 = zero_zero_complex )
% 0.34/0.54 | ( B3 = zero_zero_complex ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_0_iff
% 0.34/0.54 thf(fact_1892_divide__cancel__left,axiom,
% 0.34/0.54 ! [C: rat,A2: rat,B3: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ C @ A2 )
% 0.34/0.54 = ( divide_divide_rat @ C @ B3 ) )
% 0.34/0.54 = ( ( C = zero_zero_rat )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_left
% 0.34/0.54 thf(fact_1893_divide__cancel__left,axiom,
% 0.34/0.54 ! [C: real,A2: real,B3: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ C @ A2 )
% 0.34/0.54 = ( divide_divide_real @ C @ B3 ) )
% 0.34/0.54 = ( ( C = zero_zero_real )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_left
% 0.34/0.54 thf(fact_1894_divide__cancel__left,axiom,
% 0.34/0.54 ! [C: complex,A2: complex,B3: complex] :
% 0.34/0.54 ( ( ( divide1717551699836669952omplex @ C @ A2 )
% 0.34/0.54 = ( divide1717551699836669952omplex @ C @ B3 ) )
% 0.34/0.54 = ( ( C = zero_zero_complex )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_left
% 0.34/0.54 thf(fact_1895_divide__cancel__right,axiom,
% 0.34/0.54 ! [A2: rat,C: rat,B3: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ A2 @ C )
% 0.34/0.54 = ( divide_divide_rat @ B3 @ C ) )
% 0.34/0.54 = ( ( C = zero_zero_rat )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_right
% 0.34/0.54 thf(fact_1896_divide__cancel__right,axiom,
% 0.34/0.54 ! [A2: real,C: real,B3: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ A2 @ C )
% 0.34/0.54 = ( divide_divide_real @ B3 @ C ) )
% 0.34/0.54 = ( ( C = zero_zero_real )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_right
% 0.34/0.54 thf(fact_1897_divide__cancel__right,axiom,
% 0.34/0.54 ! [A2: complex,C: complex,B3: complex] :
% 0.34/0.54 ( ( ( divide1717551699836669952omplex @ A2 @ C )
% 0.34/0.54 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.34/0.54 = ( ( C = zero_zero_complex )
% 0.34/0.54 | ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_cancel_right
% 0.34/0.54 thf(fact_1898_division__ring__divide__zero,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( divide_divide_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 = zero_zero_rat ) ).
% 0.34/0.54
% 0.34/0.54 % division_ring_divide_zero
% 0.34/0.54 thf(fact_1899_division__ring__divide__zero,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( divide_divide_real @ A2 @ zero_zero_real )
% 0.34/0.54 = zero_zero_real ) ).
% 0.34/0.54
% 0.34/0.54 % division_ring_divide_zero
% 0.34/0.54 thf(fact_1900_division__ring__divide__zero,axiom,
% 0.34/0.54 ! [A2: complex] :
% 0.34/0.54 ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
% 0.34/0.54 = zero_zero_complex ) ).
% 0.34/0.54
% 0.34/0.54 % division_ring_divide_zero
% 0.34/0.54 thf(fact_1901_Diff__eq__empty__iff,axiom,
% 0.34/0.54 ! [A3: set_real,B2: set_real] :
% 0.34/0.54 ( ( ( minus_minus_set_real @ A3 @ B2 )
% 0.34/0.54 = bot_bot_set_real )
% 0.34/0.54 = ( ord_less_eq_set_real @ A3 @ B2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_eq_empty_iff
% 0.34/0.54 thf(fact_1902_Diff__eq__empty__iff,axiom,
% 0.34/0.54 ! [A3: set_o,B2: set_o] :
% 0.34/0.54 ( ( ( minus_minus_set_o @ A3 @ B2 )
% 0.34/0.54 = bot_bot_set_o )
% 0.34/0.54 = ( ord_less_eq_set_o @ A3 @ B2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_eq_empty_iff
% 0.34/0.54 thf(fact_1903_Diff__eq__empty__iff,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( ( minus_minus_set_nat @ A3 @ B2 )
% 0.34/0.54 = bot_bot_set_nat )
% 0.34/0.54 = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_eq_empty_iff
% 0.34/0.54 thf(fact_1904_Diff__eq__empty__iff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( ( minus_minus_set_int @ A3 @ B2 )
% 0.34/0.54 = bot_bot_set_int )
% 0.34/0.54 = ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_eq_empty_iff
% 0.34/0.54 thf(fact_1905_divide__eq__1__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ A2 @ B3 )
% 0.34/0.54 = one_one_rat )
% 0.34/0.54 = ( ( B3 != zero_zero_rat )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_1_iff
% 0.34/0.54 thf(fact_1906_divide__eq__1__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ A2 @ B3 )
% 0.34/0.54 = one_one_real )
% 0.34/0.54 = ( ( B3 != zero_zero_real )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_1_iff
% 0.34/0.54 thf(fact_1907_divide__eq__1__iff,axiom,
% 0.34/0.54 ! [A2: complex,B3: complex] :
% 0.34/0.54 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 0.34/0.54 = one_one_complex )
% 0.34/0.54 = ( ( B3 != zero_zero_complex )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_1_iff
% 0.34/0.54 thf(fact_1908_one__eq__divide__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( one_one_rat
% 0.34/0.54 = ( divide_divide_rat @ A2 @ B3 ) )
% 0.34/0.54 = ( ( B3 != zero_zero_rat )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % one_eq_divide_iff
% 0.34/0.54 thf(fact_1909_one__eq__divide__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( one_one_real
% 0.34/0.54 = ( divide_divide_real @ A2 @ B3 ) )
% 0.34/0.54 = ( ( B3 != zero_zero_real )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % one_eq_divide_iff
% 0.34/0.54 thf(fact_1910_one__eq__divide__iff,axiom,
% 0.34/0.54 ! [A2: complex,B3: complex] :
% 0.34/0.54 ( ( one_one_complex
% 0.34/0.54 = ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 0.34/0.54 = ( ( B3 != zero_zero_complex )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % one_eq_divide_iff
% 0.34/0.54 thf(fact_1911_divide__self,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( A2 != zero_zero_rat )
% 0.34/0.54 => ( ( divide_divide_rat @ A2 @ A2 )
% 0.34/0.54 = one_one_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self
% 0.34/0.54 thf(fact_1912_divide__self,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( A2 != zero_zero_real )
% 0.34/0.54 => ( ( divide_divide_real @ A2 @ A2 )
% 0.34/0.54 = one_one_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self
% 0.34/0.54 thf(fact_1913_divide__self,axiom,
% 0.34/0.54 ! [A2: complex] :
% 0.34/0.54 ( ( A2 != zero_zero_complex )
% 0.34/0.54 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 0.34/0.54 = one_one_complex ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self
% 0.34/0.54 thf(fact_1914_divide__self__if,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ( A2 = zero_zero_rat )
% 0.34/0.54 => ( ( divide_divide_rat @ A2 @ A2 )
% 0.34/0.54 = zero_zero_rat ) )
% 0.34/0.54 & ( ( A2 != zero_zero_rat )
% 0.34/0.54 => ( ( divide_divide_rat @ A2 @ A2 )
% 0.34/0.54 = one_one_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self_if
% 0.34/0.54 thf(fact_1915_divide__self__if,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ( A2 = zero_zero_real )
% 0.34/0.54 => ( ( divide_divide_real @ A2 @ A2 )
% 0.34/0.54 = zero_zero_real ) )
% 0.34/0.54 & ( ( A2 != zero_zero_real )
% 0.34/0.54 => ( ( divide_divide_real @ A2 @ A2 )
% 0.34/0.54 = one_one_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self_if
% 0.34/0.54 thf(fact_1916_divide__self__if,axiom,
% 0.34/0.54 ! [A2: complex] :
% 0.34/0.54 ( ( ( A2 = zero_zero_complex )
% 0.34/0.54 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 0.34/0.54 = zero_zero_complex ) )
% 0.34/0.54 & ( ( A2 != zero_zero_complex )
% 0.34/0.54 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 0.34/0.54 = one_one_complex ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_self_if
% 0.34/0.54 thf(fact_1917_divide__eq__eq__1,axiom,
% 0.34/0.54 ! [B3: rat,A2: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ B3 @ A2 )
% 0.34/0.54 = one_one_rat )
% 0.34/0.54 = ( ( A2 != zero_zero_rat )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_eq_1
% 0.34/0.54 thf(fact_1918_divide__eq__eq__1,axiom,
% 0.34/0.54 ! [B3: real,A2: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ B3 @ A2 )
% 0.34/0.54 = one_one_real )
% 0.34/0.54 = ( ( A2 != zero_zero_real )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_eq_eq_1
% 0.34/0.54 thf(fact_1919_eq__divide__eq__1,axiom,
% 0.34/0.54 ! [B3: rat,A2: rat] :
% 0.34/0.54 ( ( one_one_rat
% 0.34/0.54 = ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.54 = ( ( A2 != zero_zero_rat )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_divide_eq_1
% 0.34/0.54 thf(fact_1920_eq__divide__eq__1,axiom,
% 0.34/0.54 ! [B3: real,A2: real] :
% 0.34/0.54 ( ( one_one_real
% 0.34/0.54 = ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.54 = ( ( A2 != zero_zero_real )
% 0.34/0.54 & ( A2 = B3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % eq_divide_eq_1
% 0.34/0.54 thf(fact_1921_one__divide__eq__0__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ( divide_divide_rat @ one_one_rat @ A2 )
% 0.34/0.54 = zero_zero_rat )
% 0.34/0.54 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % one_divide_eq_0_iff
% 0.34/0.54 thf(fact_1922_one__divide__eq__0__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ( divide_divide_real @ one_one_real @ A2 )
% 0.34/0.54 = zero_zero_real )
% 0.34/0.54 = ( A2 = zero_zero_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % one_divide_eq_0_iff
% 0.34/0.54 thf(fact_1923_zero__eq__1__divide__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( zero_zero_rat
% 0.34/0.54 = ( divide_divide_rat @ one_one_rat @ A2 ) )
% 0.34/0.54 = ( A2 = zero_zero_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_eq_1_divide_iff
% 0.34/0.54 thf(fact_1924_zero__eq__1__divide__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( zero_zero_real
% 0.34/0.54 = ( divide_divide_real @ one_one_real @ A2 ) )
% 0.34/0.54 = ( A2 = zero_zero_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_eq_1_divide_iff
% 0.34/0.54 thf(fact_1925_zle__diff1__eq,axiom,
% 0.34/0.54 ! [W2: int,Z: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
% 0.34/0.54 = ( ord_less_int @ W2 @ Z ) ) ).
% 0.34/0.54
% 0.34/0.54 % zle_diff1_eq
% 0.34/0.54 thf(fact_1926_zero__le__divide__1__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_le_divide_1_iff
% 0.34/0.54 thf(fact_1927_zero__le__divide__1__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
% 0.34/0.54 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_le_divide_1_iff
% 0.34/0.54 thf(fact_1928_divide__le__0__1__iff,axiom,
% 0.34/0.54 ! [A2: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
% 0.34/0.54 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_0_1_iff
% 0.34/0.54 thf(fact_1929_divide__le__0__1__iff,axiom,
% 0.34/0.54 ! [A2: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
% 0.34/0.54 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_le_0_1_iff
% 0.34/0.54 thf(fact_1930_Diff__infinite__finite,axiom,
% 0.34/0.54 ! [T2: set_int,S: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ T2 )
% 0.34/0.54 => ( ~ ( finite_finite_int @ S )
% 0.34/0.54 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_infinite_finite
% 0.34/0.54 thf(fact_1931_Diff__infinite__finite,axiom,
% 0.34/0.54 ! [T2: set_complex,S: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ T2 )
% 0.34/0.54 => ( ~ ( finite3207457112153483333omplex @ S )
% 0.34/0.54 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_infinite_finite
% 0.34/0.54 thf(fact_1932_Diff__infinite__finite,axiom,
% 0.34/0.54 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ T2 )
% 0.34/0.54 => ( ~ ( finite6177210948735845034at_nat @ S )
% 0.34/0.54 => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ T2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_infinite_finite
% 0.34/0.54 thf(fact_1933_Diff__infinite__finite,axiom,
% 0.34/0.54 ! [T2: set_Extended_enat,S: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ T2 )
% 0.34/0.54 => ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.54 => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ T2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_infinite_finite
% 0.34/0.54 thf(fact_1934_Diff__infinite__finite,axiom,
% 0.34/0.54 ! [T2: set_nat,S: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ T2 )
% 0.34/0.54 => ( ~ ( finite_finite_nat @ S )
% 0.34/0.54 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_infinite_finite
% 0.34/0.54 thf(fact_1935_Diff__mono,axiom,
% 0.34/0.54 ! [A3: set_nat,C2: set_nat,D4: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ D4 @ B2 )
% 0.34/0.54 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_mono
% 0.34/0.54 thf(fact_1936_Diff__mono,axiom,
% 0.34/0.54 ! [A3: set_int,C2: set_int,D4: set_int,B2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ D4 @ B2 )
% 0.34/0.54 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ C2 @ D4 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_mono
% 0.34/0.54 thf(fact_1937_Diff__subset,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_subset
% 0.34/0.54 thf(fact_1938_Diff__subset,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ A3 ) ).
% 0.34/0.54
% 0.34/0.54 % Diff_subset
% 0.34/0.54 thf(fact_1939_double__diff,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 0.34/0.54 => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A3 ) )
% 0.34/0.54 = A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % double_diff
% 0.34/0.54 thf(fact_1940_double__diff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.34/0.54 => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A3 ) )
% 0.34/0.54 = A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % double_diff
% 0.34/0.54 thf(fact_1941_psubset__imp__ex__mem,axiom,
% 0.34/0.54 ! [A3: set_real,B2: set_real] :
% 0.34/0.54 ( ( ord_less_set_real @ A3 @ B2 )
% 0.34/0.54 => ? [B: real] : ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_imp_ex_mem
% 0.34/0.54 thf(fact_1942_psubset__imp__ex__mem,axiom,
% 0.34/0.54 ! [A3: set_o,B2: set_o] :
% 0.34/0.54 ( ( ord_less_set_o @ A3 @ B2 )
% 0.34/0.54 => ? [B: $o] : ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_imp_ex_mem
% 0.34/0.54 thf(fact_1943_psubset__imp__ex__mem,axiom,
% 0.34/0.54 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.54 ( ( ord_less_set_set_nat @ A3 @ B2 )
% 0.34/0.54 => ? [B: set_nat] : ( member_set_nat @ B @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_imp_ex_mem
% 0.34/0.54 thf(fact_1944_psubset__imp__ex__mem,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( ord_less_set_int @ A3 @ B2 )
% 0.34/0.54 => ? [B: int] : ( member_int @ B @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_imp_ex_mem
% 0.34/0.54 thf(fact_1945_psubset__imp__ex__mem,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( ord_less_set_nat @ A3 @ B2 )
% 0.34/0.54 => ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % psubset_imp_ex_mem
% 0.34/0.54 thf(fact_1946_subset__Compl__self__eq,axiom,
% 0.34/0.54 ! [A3: set_real] :
% 0.34/0.54 ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
% 0.34/0.54 = ( A3 = bot_bot_set_real ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_Compl_self_eq
% 0.34/0.54 thf(fact_1947_subset__Compl__self__eq,axiom,
% 0.34/0.54 ! [A3: set_o] :
% 0.34/0.54 ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
% 0.34/0.54 = ( A3 = bot_bot_set_o ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_Compl_self_eq
% 0.34/0.54 thf(fact_1948_subset__Compl__self__eq,axiom,
% 0.34/0.54 ! [A3: set_nat] :
% 0.34/0.54 ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
% 0.34/0.54 = ( A3 = bot_bot_set_nat ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_Compl_self_eq
% 0.34/0.54 thf(fact_1949_subset__Compl__self__eq,axiom,
% 0.34/0.54 ! [A3: set_int] :
% 0.34/0.54 ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
% 0.34/0.54 = ( A3 = bot_bot_set_int ) ) ).
% 0.34/0.54
% 0.34/0.54 % subset_Compl_self_eq
% 0.34/0.54 thf(fact_1950_int__le__induct,axiom,
% 0.34/0.54 ! [I: int,K: int,P: int > $o] :
% 0.34/0.54 ( ( ord_less_eq_int @ I @ K )
% 0.34/0.54 => ( ( P @ K )
% 0.34/0.54 => ( ! [I2: int] :
% 0.34/0.54 ( ( ord_less_eq_int @ I2 @ K )
% 0.34/0.54 => ( ( P @ I2 )
% 0.34/0.54 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.34/0.54 => ( P @ I ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_le_induct
% 0.34/0.54 thf(fact_1951_int__less__induct,axiom,
% 0.34/0.54 ! [I: int,K: int,P: int > $o] :
% 0.34/0.54 ( ( ord_less_int @ I @ K )
% 0.34/0.54 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.34/0.54 => ( ! [I2: int] :
% 0.34/0.54 ( ( ord_less_int @ I2 @ K )
% 0.34/0.54 => ( ( P @ I2 )
% 0.34/0.54 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.34/0.54 => ( P @ I ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_less_induct
% 0.34/0.54 thf(fact_1952_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_list_nat,B2: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1953_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1954_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ A3 )
% 0.34/0.54 => ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1955_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_complex,B2: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1956_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1957_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1958_card__less__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_less_sym_Diff
% 0.34/0.54 thf(fact_1959_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_list_nat,B2: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.54 => ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1960_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.54 => ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1961_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_int,B2: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ A3 )
% 0.34/0.54 => ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1962_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_complex,B2: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.54 => ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1963_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.54 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1964_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.54 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1965_card__le__sym__Diff,axiom,
% 0.34/0.54 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ A3 )
% 0.34/0.54 => ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 0.34/0.54 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_le_sym_Diff
% 0.34/0.54 thf(fact_1966_linordered__field__no__ub,axiom,
% 0.34/0.54 ! [X4: real] :
% 0.34/0.54 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 0.34/0.54
% 0.34/0.54 % linordered_field_no_ub
% 0.34/0.54 thf(fact_1967_linordered__field__no__ub,axiom,
% 0.34/0.54 ! [X4: rat] :
% 0.34/0.54 ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% 0.34/0.54
% 0.34/0.54 % linordered_field_no_ub
% 0.34/0.54 thf(fact_1968_linordered__field__no__lb,axiom,
% 0.34/0.54 ! [X4: real] :
% 0.34/0.54 ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).
% 0.34/0.54
% 0.34/0.54 % linordered_field_no_lb
% 0.34/0.54 thf(fact_1969_linordered__field__no__lb,axiom,
% 0.34/0.54 ! [X4: rat] :
% 0.34/0.54 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).
% 0.34/0.54
% 0.34/0.54 % linordered_field_no_lb
% 0.34/0.54 thf(fact_1970_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
% 0.34/0.54 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1971_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
% 0.34/0.54 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1972_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.34/0.54 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1973_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 0.34/0.54 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1974_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.34/0.54 => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1975_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.34/0.54 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1976_card__Diff__subset,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.34/0.54 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.54 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % card_Diff_subset
% 0.34/0.54 thf(fact_1977_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_list_nat,A3: set_list_nat] :
% 0.34/0.54 ( ( finite8100373058378681591st_nat @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1978_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.54 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1979_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_int,A3: set_int] :
% 0.34/0.54 ( ( finite_finite_int @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1980_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_complex,A3: set_complex] :
% 0.34/0.54 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1981_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.54 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1982_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 0.34/0.54 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1983_diff__card__le__card__Diff,axiom,
% 0.34/0.54 ! [B2: set_nat,A3: set_nat] :
% 0.34/0.54 ( ( finite_finite_nat @ B2 )
% 0.34/0.54 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % diff_card_le_card_Diff
% 0.34/0.54 thf(fact_1984_int__ops_I6_J,axiom,
% 0.34/0.54 ! [A2: nat,B3: nat] :
% 0.34/0.54 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
% 0.34/0.54 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.34/0.54 = zero_zero_int ) )
% 0.34/0.54 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
% 0.34/0.54 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.34/0.54 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % int_ops(6)
% 0.34/0.54 thf(fact_1985_VEBT_Osize_I4_J,axiom,
% 0.34/0.54 ! [X21: $o,X222: $o] :
% 0.34/0.54 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.34/0.54 = zero_zero_nat ) ).
% 0.34/0.54
% 0.34/0.54 % VEBT.size(4)
% 0.34/0.54 thf(fact_1986_zdiff__int__split,axiom,
% 0.34/0.54 ! [P: int > $o,X2: nat,Y3: nat] :
% 0.34/0.54 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
% 0.34/0.54 = ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.34/0.54 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 0.34/0.54 & ( ( ord_less_nat @ X2 @ Y3 )
% 0.34/0.54 => ( P @ zero_zero_int ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % zdiff_int_split
% 0.34/0.54 thf(fact_1987_divide__right__mono__neg,axiom,
% 0.34/0.54 ! [A2: real,B3: real,C: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.34/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_right_mono_neg
% 0.34/0.54 thf(fact_1988_divide__right__mono__neg,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.34/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( divide_divide_rat @ A2 @ C ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_right_mono_neg
% 0.34/0.54 thf(fact_1989_divide__nonpos__nonpos,axiom,
% 0.34/0.54 ! [X2: real,Y3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.34/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonpos_nonpos
% 0.34/0.54 thf(fact_1990_divide__nonpos__nonpos,axiom,
% 0.34/0.54 ! [X2: rat,Y3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.34/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonpos_nonpos
% 0.34/0.54 thf(fact_1991_divide__nonpos__nonneg,axiom,
% 0.34/0.54 ! [X2: real,Y3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.34/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.34/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonpos_nonneg
% 0.34/0.54 thf(fact_1992_divide__nonpos__nonneg,axiom,
% 0.34/0.54 ! [X2: rat,Y3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.34/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.34/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonpos_nonneg
% 0.34/0.54 thf(fact_1993_divide__nonneg__nonpos,axiom,
% 0.34/0.54 ! [X2: real,Y3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.54 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.34/0.54 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonneg_nonpos
% 0.34/0.54 thf(fact_1994_divide__nonneg__nonpos,axiom,
% 0.34/0.54 ! [X2: rat,Y3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.34/0.54 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonneg_nonpos
% 0.34/0.54 thf(fact_1995_divide__nonneg__nonneg,axiom,
% 0.34/0.54 ! [X2: real,Y3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.54 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.34/0.54 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonneg_nonneg
% 0.34/0.54 thf(fact_1996_divide__nonneg__nonneg,axiom,
% 0.34/0.54 ! [X2: rat,Y3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.54 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.34/0.54 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % divide_nonneg_nonneg
% 0.34/0.54 thf(fact_1997_zero__le__divide__iff,axiom,
% 0.34/0.54 ! [A2: real,B3: real] :
% 0.34/0.54 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.34/0.54 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.34/0.54 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 0.34/0.54 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.34/0.54 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.34/0.54
% 0.34/0.54 % zero_le_divide_iff
% 0.34/0.54 thf(fact_1998_zero__le__divide__iff,axiom,
% 0.34/0.54 ! [A2: rat,B3: rat] :
% 0.34/0.54 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 0.34/0.54 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.54 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 0.34/0.54 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.54 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zero_le_divide_iff
% 0.34/0.55 thf(fact_1999_divide__right__mono,axiom,
% 0.34/0.55 ! [A2: real,B3: real,C: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.34/0.55 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.34/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_right_mono
% 0.34/0.55 thf(fact_2000_divide__right__mono,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.34/0.55 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.34/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_right_mono
% 0.34/0.55 thf(fact_2001_divide__le__0__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
% 0.34/0.55 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 0.34/0.55 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_0_iff
% 0.34/0.55 thf(fact_2002_divide__le__0__iff,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
% 0.34/0.55 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 0.34/0.55 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_0_iff
% 0.34/0.55 thf(fact_2003_divide__neg__neg,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.34/0.55 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_neg_neg
% 0.34/0.55 thf(fact_2004_divide__neg__neg,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.34/0.55 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.34/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_neg_neg
% 0.34/0.55 thf(fact_2005_divide__neg__pos,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_neg_pos
% 0.34/0.55 thf(fact_2006_divide__neg__pos,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_neg_pos
% 0.34/0.55 thf(fact_2007_divide__pos__neg,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_pos_neg
% 0.34/0.55 thf(fact_2008_divide__pos__neg,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_pos_neg
% 0.34/0.55 thf(fact_2009_divide__pos__pos,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.34/0.55 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_pos_pos
% 0.34/0.55 thf(fact_2010_divide__pos__pos,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.34/0.55 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_pos_pos
% 0.34/0.55 thf(fact_2011_divide__less__0__iff,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_rat @ B3 @ zero_zero_rat ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_0_iff
% 0.34/0.55 thf(fact_2012_divide__less__0__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real] :
% 0.34/0.55 ( ( ord_less_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_real @ B3 @ zero_zero_real ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_0_iff
% 0.34/0.55 thf(fact_2013_divide__less__cancel,axiom,
% 0.34/0.55 ! [A2: rat,C: rat,B3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.55 => ( ord_less_rat @ A2 @ B3 ) )
% 0.34/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_rat @ B3 @ A2 ) )
% 0.34/0.55 & ( C != zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_cancel
% 0.34/0.55 thf(fact_2014_divide__less__cancel,axiom,
% 0.34/0.55 ! [A2: real,C: real,B3: real] :
% 0.34/0.55 ( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.34/0.55 => ( ord_less_real @ A2 @ B3 ) )
% 0.34/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.34/0.55 => ( ord_less_real @ B3 @ A2 ) )
% 0.34/0.55 & ( C != zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_cancel
% 0.34/0.55 thf(fact_2015_zero__less__divide__iff,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat] :
% 0.34/0.55 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_rat @ zero_zero_rat @ B3 ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zero_less_divide_iff
% 0.34/0.55 thf(fact_2016_zero__less__divide__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real] :
% 0.34/0.55 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_real @ zero_zero_real @ B3 ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zero_less_divide_iff
% 0.34/0.55 thf(fact_2017_divide__strict__right__mono,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat,C: rat] :
% 0.34/0.55 ( ( ord_less_rat @ A2 @ B3 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_strict_right_mono
% 0.34/0.55 thf(fact_2018_divide__strict__right__mono,axiom,
% 0.34/0.55 ! [A2: real,B3: real,C: real] :
% 0.34/0.55 ( ( ord_less_real @ A2 @ B3 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_strict_right_mono
% 0.34/0.55 thf(fact_2019_divide__strict__right__mono__neg,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat,C: rat] :
% 0.34/0.55 ( ( ord_less_rat @ B3 @ A2 )
% 0.34/0.55 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_strict_right_mono_neg
% 0.34/0.55 thf(fact_2020_divide__strict__right__mono__neg,axiom,
% 0.34/0.55 ! [B3: real,A2: real,C: real] :
% 0.34/0.55 ( ( ord_less_real @ B3 @ A2 )
% 0.34/0.55 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_strict_right_mono_neg
% 0.34/0.55 thf(fact_2021_right__inverse__eq,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( B3 != zero_zero_rat )
% 0.34/0.55 => ( ( ( divide_divide_rat @ A2 @ B3 )
% 0.34/0.55 = one_one_rat )
% 0.34/0.55 = ( A2 = B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % right_inverse_eq
% 0.34/0.55 thf(fact_2022_right__inverse__eq,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( B3 != zero_zero_real )
% 0.34/0.55 => ( ( ( divide_divide_real @ A2 @ B3 )
% 0.34/0.55 = one_one_real )
% 0.34/0.55 = ( A2 = B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % right_inverse_eq
% 0.34/0.55 thf(fact_2023_right__inverse__eq,axiom,
% 0.34/0.55 ! [B3: complex,A2: complex] :
% 0.34/0.55 ( ( B3 != zero_zero_complex )
% 0.34/0.55 => ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 0.34/0.55 = one_one_complex )
% 0.34/0.55 = ( A2 = B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % right_inverse_eq
% 0.34/0.55 thf(fact_2024_nonzero__minus__divide__divide,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( B3 != zero_zero_real )
% 0.34/0.55 => ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) )
% 0.34/0.55 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_divide
% 0.34/0.55 thf(fact_2025_nonzero__minus__divide__divide,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( B3 != zero_zero_rat )
% 0.34/0.55 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) )
% 0.34/0.55 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_divide
% 0.34/0.55 thf(fact_2026_nonzero__minus__divide__divide,axiom,
% 0.34/0.55 ! [B3: complex,A2: complex] :
% 0.34/0.55 ( ( B3 != zero_zero_complex )
% 0.34/0.55 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) )
% 0.34/0.55 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_divide
% 0.34/0.55 thf(fact_2027_nonzero__minus__divide__right,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( B3 != zero_zero_real )
% 0.34/0.55 => ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.34/0.55 = ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_right
% 0.34/0.55 thf(fact_2028_nonzero__minus__divide__right,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( B3 != zero_zero_rat )
% 0.34/0.55 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 0.34/0.55 = ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_right
% 0.34/0.55 thf(fact_2029_nonzero__minus__divide__right,axiom,
% 0.34/0.55 ! [B3: complex,A2: complex] :
% 0.34/0.55 ( ( B3 != zero_zero_complex )
% 0.34/0.55 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 0.34/0.55 = ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonzero_minus_divide_right
% 0.34/0.55 thf(fact_2030_frac__le,axiom,
% 0.34/0.55 ! [Y3: real,X2: real,W2: real,Z: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.34/0.55 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.34/0.55 => ( ( ord_less_eq_real @ W2 @ Z )
% 0.34/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_le
% 0.34/0.55 thf(fact_2031_frac__le,axiom,
% 0.34/0.55 ! [Y3: rat,X2: rat,W2: rat,Z: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.34/0.55 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.34/0.55 => ( ( ord_less_eq_rat @ W2 @ Z )
% 0.34/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_le
% 0.34/0.55 thf(fact_2032_frac__less,axiom,
% 0.34/0.55 ! [X2: real,Y3: real,W2: real,Z: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_real @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.34/0.55 => ( ( ord_less_eq_real @ W2 @ Z )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_less
% 0.34/0.55 thf(fact_2033_frac__less,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_rat @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.34/0.55 => ( ( ord_less_eq_rat @ W2 @ Z )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_less
% 0.34/0.55 thf(fact_2034_frac__less2,axiom,
% 0.34/0.55 ! [X2: real,Y3: real,W2: real,Z: real] :
% 0.34/0.55 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.34/0.55 => ( ( ord_less_real @ W2 @ Z )
% 0.34/0.55 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_less2
% 0.34/0.55 thf(fact_2035_frac__less2,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
% 0.34/0.55 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.34/0.55 => ( ( ord_less_rat @ W2 @ Z )
% 0.34/0.55 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % frac_less2
% 0.34/0.55 thf(fact_2036_divide__le__cancel,axiom,
% 0.34/0.55 ! [A2: real,C: real,B3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.34/0.55 => ( ord_less_eq_real @ A2 @ B3 ) )
% 0.34/0.55 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.34/0.55 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_cancel
% 0.34/0.55 thf(fact_2037_divide__le__cancel,axiom,
% 0.34/0.55 ! [A2: rat,C: rat,B3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.34/0.55 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.34/0.55 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_cancel
% 0.34/0.55 thf(fact_2038_divide__nonneg__neg,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.34/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonneg_neg
% 0.34/0.55 thf(fact_2039_divide__nonneg__neg,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonneg_neg
% 0.34/0.55 thf(fact_2040_divide__nonneg__pos,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.34/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonneg_pos
% 0.34/0.55 thf(fact_2041_divide__nonneg__pos,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.34/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonneg_pos
% 0.34/0.55 thf(fact_2042_divide__nonpos__neg,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.34/0.55 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.34/0.55 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonpos_neg
% 0.34/0.55 thf(fact_2043_divide__nonpos__neg,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.34/0.55 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.34/0.55 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonpos_neg
% 0.34/0.55 thf(fact_2044_divide__nonpos__pos,axiom,
% 0.34/0.55 ! [X2: real,Y3: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.34/0.55 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.34/0.55 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonpos_pos
% 0.34/0.55 thf(fact_2045_divide__nonpos__pos,axiom,
% 0.34/0.55 ! [X2: rat,Y3: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.34/0.55 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.34/0.55 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_nonpos_pos
% 0.34/0.55 thf(fact_2046_divide__less__eq__1,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_rat @ B3 @ A2 ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_rat @ A2 @ B3 ) )
% 0.34/0.55 | ( A2 = zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_eq_1
% 0.34/0.55 thf(fact_2047_divide__less__eq__1,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_real @ B3 @ A2 ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_real @ A2 @ B3 ) )
% 0.34/0.55 | ( A2 = zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_less_eq_1
% 0.34/0.55 thf(fact_2048_less__divide__eq__1,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_rat @ A2 @ B3 ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % less_divide_eq_1
% 0.34/0.55 thf(fact_2049_less__divide__eq__1,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_real @ A2 @ B3 ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % less_divide_eq_1
% 0.34/0.55 thf(fact_2050_divide__eq__minus__1__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real] :
% 0.34/0.55 ( ( ( divide_divide_real @ A2 @ B3 )
% 0.34/0.55 = ( uminus_uminus_real @ one_one_real ) )
% 0.34/0.55 = ( ( B3 != zero_zero_real )
% 0.34/0.55 & ( A2
% 0.34/0.55 = ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_eq_minus_1_iff
% 0.34/0.55 thf(fact_2051_divide__eq__minus__1__iff,axiom,
% 0.34/0.55 ! [A2: rat,B3: rat] :
% 0.34/0.55 ( ( ( divide_divide_rat @ A2 @ B3 )
% 0.34/0.55 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.34/0.55 = ( ( B3 != zero_zero_rat )
% 0.34/0.55 & ( A2
% 0.34/0.55 = ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_eq_minus_1_iff
% 0.34/0.55 thf(fact_2052_divide__eq__minus__1__iff,axiom,
% 0.34/0.55 ! [A2: complex,B3: complex] :
% 0.34/0.55 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 0.34/0.55 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.34/0.55 = ( ( B3 != zero_zero_complex )
% 0.34/0.55 & ( A2
% 0.34/0.55 = ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_eq_minus_1_iff
% 0.34/0.55 thf(fact_2053_divide__le__eq__1,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_eq_real @ B3 @ A2 ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_eq_real @ A2 @ B3 ) )
% 0.34/0.55 | ( A2 = zero_zero_real ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_eq_1
% 0.34/0.55 thf(fact_2054_divide__le__eq__1,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_eq_rat @ B3 @ A2 ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.34/0.55 | ( A2 = zero_zero_rat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % divide_le_eq_1
% 0.34/0.55 thf(fact_2055_le__divide__eq__1,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 0.34/0.55 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.34/0.55 & ( ord_less_eq_real @ A2 @ B3 ) )
% 0.34/0.55 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.34/0.55 & ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % le_divide_eq_1
% 0.34/0.55 thf(fact_2056_le__divide__eq__1,axiom,
% 0.34/0.55 ! [B3: rat,A2: rat] :
% 0.34/0.55 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 0.34/0.55 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.34/0.55 & ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.34/0.55 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.34/0.55 & ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % le_divide_eq_1
% 0.34/0.55 thf(fact_2057_div__pos__pos__trivial,axiom,
% 0.34/0.55 ! [K: int,L: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.34/0.55 => ( ( ord_less_int @ K @ L )
% 0.34/0.55 => ( ( divide_divide_int @ K @ L )
% 0.34/0.55 = zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_pos_pos_trivial
% 0.34/0.55 thf(fact_2058_div__neg__neg__trivial,axiom,
% 0.34/0.55 ! [K: int,L: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_int @ L @ K )
% 0.34/0.55 => ( ( divide_divide_int @ K @ L )
% 0.34/0.55 = zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_neg_neg_trivial
% 0.34/0.55 thf(fact_2059_div__eq__minus1,axiom,
% 0.34/0.55 ! [B3: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
% 0.34/0.55 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_eq_minus1
% 0.34/0.55 thf(fact_2060_geqmaxNone,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.34/0.55 => ( ( ord_less_eq_nat @ Ma @ X2 )
% 0.34/0.55 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.34/0.55 = none_nat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % geqmaxNone
% 0.34/0.55 thf(fact_2061_le__div__geq,axiom,
% 0.34/0.55 ! [N: nat,M: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.55 => ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.34/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % le_div_geq
% 0.34/0.55 thf(fact_2062_div__less,axiom,
% 0.34/0.55 ! [M: nat,N: nat] :
% 0.34/0.55 ( ( ord_less_nat @ M @ N )
% 0.34/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.34/0.55 = zero_zero_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_less
% 0.34/0.55 thf(fact_2063_div__by__Suc__0,axiom,
% 0.34/0.55 ! [M: nat] :
% 0.34/0.55 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.34/0.55 = M ) ).
% 0.34/0.55
% 0.34/0.55 % div_by_Suc_0
% 0.34/0.55 thf(fact_2064_compl__le__compl__iff,axiom,
% 0.34/0.55 ! [X2: set_int,Y3: set_int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 0.34/0.55 = ( ord_less_eq_set_int @ Y3 @ X2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % compl_le_compl_iff
% 0.34/0.55 thf(fact_2065_bits__div__0,axiom,
% 0.34/0.55 ! [A2: int] :
% 0.34/0.55 ( ( divide_divide_int @ zero_zero_int @ A2 )
% 0.34/0.55 = zero_zero_int ) ).
% 0.34/0.55
% 0.34/0.55 % bits_div_0
% 0.34/0.55 thf(fact_2066_bits__div__0,axiom,
% 0.34/0.55 ! [A2: nat] :
% 0.34/0.55 ( ( divide_divide_nat @ zero_zero_nat @ A2 )
% 0.34/0.55 = zero_zero_nat ) ).
% 0.34/0.55
% 0.34/0.55 % bits_div_0
% 0.34/0.55 thf(fact_2067_DiffI,axiom,
% 0.34/0.55 ! [C: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ C @ A3 )
% 0.34/0.55 => ( ~ ( member_real @ C @ B2 )
% 0.34/0.55 => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffI
% 0.34/0.55 thf(fact_2068_DiffI,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ C @ A3 )
% 0.34/0.55 => ( ~ ( member_o @ C @ B2 )
% 0.34/0.55 => ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffI
% 0.34/0.55 thf(fact_2069_DiffI,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ A3 )
% 0.34/0.55 => ( ~ ( member_set_nat @ C @ B2 )
% 0.34/0.55 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffI
% 0.34/0.55 thf(fact_2070_DiffI,axiom,
% 0.34/0.55 ! [C: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ C @ A3 )
% 0.34/0.55 => ( ~ ( member_int @ C @ B2 )
% 0.34/0.55 => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffI
% 0.34/0.55 thf(fact_2071_DiffI,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ A3 )
% 0.34/0.55 => ( ~ ( member_nat @ C @ B2 )
% 0.34/0.55 => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffI
% 0.34/0.55 thf(fact_2072_Diff__iff,axiom,
% 0.34/0.55 ! [C: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 0.34/0.55 = ( ( member_real @ C @ A3 )
% 0.34/0.55 & ~ ( member_real @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_iff
% 0.34/0.55 thf(fact_2073_Diff__iff,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 0.34/0.55 = ( ( member_o @ C @ A3 )
% 0.34/0.55 & ~ ( member_o @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_iff
% 0.34/0.55 thf(fact_2074_Diff__iff,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.34/0.55 = ( ( member_set_nat @ C @ A3 )
% 0.34/0.55 & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_iff
% 0.34/0.55 thf(fact_2075_Diff__iff,axiom,
% 0.34/0.55 ! [C: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.55 = ( ( member_int @ C @ A3 )
% 0.34/0.55 & ~ ( member_int @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_iff
% 0.34/0.55 thf(fact_2076_Diff__iff,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.55 = ( ( member_nat @ C @ A3 )
% 0.34/0.55 & ~ ( member_nat @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_iff
% 0.34/0.55 thf(fact_2077_Diff__idemp,axiom,
% 0.34/0.55 ! [A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_nat @ A3 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_idemp
% 0.34/0.55 thf(fact_2078_ComplI,axiom,
% 0.34/0.55 ! [C: real,A3: set_real] :
% 0.34/0.55 ( ~ ( member_real @ C @ A3 )
% 0.34/0.55 => ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplI
% 0.34/0.55 thf(fact_2079_ComplI,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o] :
% 0.34/0.55 ( ~ ( member_o @ C @ A3 )
% 0.34/0.55 => ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplI
% 0.34/0.55 thf(fact_2080_ComplI,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ~ ( member_set_nat @ C @ A3 )
% 0.34/0.55 => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplI
% 0.34/0.55 thf(fact_2081_ComplI,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat] :
% 0.34/0.55 ( ~ ( member_nat @ C @ A3 )
% 0.34/0.55 => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplI
% 0.34/0.55 thf(fact_2082_ComplI,axiom,
% 0.34/0.55 ! [C: int,A3: set_int] :
% 0.34/0.55 ( ~ ( member_int @ C @ A3 )
% 0.34/0.55 => ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplI
% 0.34/0.55 thf(fact_2083_Compl__iff,axiom,
% 0.34/0.55 ! [C: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
% 0.34/0.55 = ( ~ ( member_real @ C @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Compl_iff
% 0.34/0.55 thf(fact_2084_Compl__iff,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
% 0.34/0.55 = ( ~ ( member_o @ C @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Compl_iff
% 0.34/0.55 thf(fact_2085_Compl__iff,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
% 0.34/0.55 = ( ~ ( member_set_nat @ C @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Compl_iff
% 0.34/0.55 thf(fact_2086_Compl__iff,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
% 0.34/0.55 = ( ~ ( member_nat @ C @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Compl_iff
% 0.34/0.55 thf(fact_2087_Compl__iff,axiom,
% 0.34/0.55 ! [C: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
% 0.34/0.55 = ( ~ ( member_int @ C @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Compl_iff
% 0.34/0.55 thf(fact_2088_bits__div__by__0,axiom,
% 0.34/0.55 ! [A2: int] :
% 0.34/0.55 ( ( divide_divide_int @ A2 @ zero_zero_int )
% 0.34/0.55 = zero_zero_int ) ).
% 0.34/0.55
% 0.34/0.55 % bits_div_by_0
% 0.34/0.55 thf(fact_2089_bits__div__by__0,axiom,
% 0.34/0.55 ! [A2: nat] :
% 0.34/0.55 ( ( divide_divide_nat @ A2 @ zero_zero_nat )
% 0.34/0.55 = zero_zero_nat ) ).
% 0.34/0.55
% 0.34/0.55 % bits_div_by_0
% 0.34/0.55 thf(fact_2090_real__of__nat__div3,axiom,
% 0.34/0.55 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% 0.34/0.55
% 0.34/0.55 % real_of_nat_div3
% 0.34/0.55 thf(fact_2091_DiffE,axiom,
% 0.34/0.55 ! [C: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( ( member_real @ C @ A3 )
% 0.34/0.55 => ( member_real @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffE
% 0.34/0.55 thf(fact_2092_DiffE,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( ( member_o @ C @ A3 )
% 0.34/0.55 => ( member_o @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffE
% 0.34/0.55 thf(fact_2093_DiffE,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( ( member_set_nat @ C @ A3 )
% 0.34/0.55 => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffE
% 0.34/0.55 thf(fact_2094_DiffE,axiom,
% 0.34/0.55 ! [C: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( ( member_int @ C @ A3 )
% 0.34/0.55 => ( member_int @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffE
% 0.34/0.55 thf(fact_2095_DiffE,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( ( member_nat @ C @ A3 )
% 0.34/0.55 => ( member_nat @ C @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffE
% 0.34/0.55 thf(fact_2096_ComplD,axiom,
% 0.34/0.55 ! [C: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
% 0.34/0.55 => ~ ( member_real @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplD
% 0.34/0.55 thf(fact_2097_ComplD,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
% 0.34/0.55 => ~ ( member_o @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplD
% 0.34/0.55 thf(fact_2098_ComplD,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
% 0.34/0.55 => ~ ( member_set_nat @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplD
% 0.34/0.55 thf(fact_2099_ComplD,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
% 0.34/0.55 => ~ ( member_nat @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplD
% 0.34/0.55 thf(fact_2100_ComplD,axiom,
% 0.34/0.55 ! [C: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
% 0.34/0.55 => ~ ( member_int @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % ComplD
% 0.34/0.55 thf(fact_2101_DiffD1,axiom,
% 0.34/0.55 ! [C: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 0.34/0.55 => ( member_real @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD1
% 0.34/0.55 thf(fact_2102_DiffD1,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 0.34/0.55 => ( member_o @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD1
% 0.34/0.55 thf(fact_2103_DiffD1,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.34/0.55 => ( member_set_nat @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD1
% 0.34/0.55 thf(fact_2104_DiffD1,axiom,
% 0.34/0.55 ! [C: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.55 => ( member_int @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD1
% 0.34/0.55 thf(fact_2105_DiffD1,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.55 => ( member_nat @ C @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD1
% 0.34/0.55 thf(fact_2106_DiffD2,axiom,
% 0.34/0.55 ! [C: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( member_real @ C @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD2
% 0.34/0.55 thf(fact_2107_DiffD2,axiom,
% 0.34/0.55 ! [C: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( member_o @ C @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD2
% 0.34/0.55 thf(fact_2108_DiffD2,axiom,
% 0.34/0.55 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( member_set_nat @ C @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD2
% 0.34/0.55 thf(fact_2109_DiffD2,axiom,
% 0.34/0.55 ! [C: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( member_int @ C @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD2
% 0.34/0.55 thf(fact_2110_DiffD2,axiom,
% 0.34/0.55 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.34/0.55 => ~ ( member_nat @ C @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % DiffD2
% 0.34/0.55 thf(fact_2111_real__of__nat__div4,axiom,
% 0.34/0.55 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % real_of_nat_div4
% 0.34/0.55 thf(fact_2112_real__of__nat__div2,axiom,
% 0.34/0.55 ! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % real_of_nat_div2
% 0.34/0.55 thf(fact_2113_subrelI,axiom,
% 0.34/0.55 ! [R2: set_Pr6588086440996610945on_nat,S2: set_Pr6588086440996610945on_nat] :
% 0.34/0.55 ( ! [X5: option_nat,Y4: option_nat] :
% 0.34/0.55 ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ R2 )
% 0.34/0.55 => ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ S2 ) )
% 0.34/0.55 => ( ord_le6406482658798684961on_nat @ R2 @ S2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subrelI
% 0.34/0.55 thf(fact_2114_subrelI,axiom,
% 0.34/0.55 ! [R2: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
% 0.34/0.55 ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
% 0.34/0.55 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ R2 )
% 0.34/0.55 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ S2 ) )
% 0.34/0.55 => ( ord_le5997549366648089703at_nat @ R2 @ S2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subrelI
% 0.34/0.55 thf(fact_2115_subrelI,axiom,
% 0.34/0.55 ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
% 0.34/0.55 ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ R2 )
% 0.34/0.55 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ S2 ) )
% 0.34/0.55 => ( ord_le1268244103169919719at_nat @ R2 @ S2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subrelI
% 0.34/0.55 thf(fact_2116_subrelI,axiom,
% 0.34/0.55 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ! [X5: nat,Y4: nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
% 0.34/0.55 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S2 ) )
% 0.34/0.55 => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subrelI
% 0.34/0.55 thf(fact_2117_subrelI,axiom,
% 0.34/0.55 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 0.34/0.55 ( ! [X5: int,Y4: int] :
% 0.34/0.55 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
% 0.34/0.55 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S2 ) )
% 0.34/0.55 => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subrelI
% 0.34/0.55 thf(fact_2118_div__le__dividend,axiom,
% 0.34/0.55 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 0.34/0.55
% 0.34/0.55 % div_le_dividend
% 0.34/0.55 thf(fact_2119_div__le__mono,axiom,
% 0.34/0.55 ! [M: nat,N: nat,K: nat] :
% 0.34/0.55 ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.55 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_le_mono
% 0.34/0.55 thf(fact_2120_vebt__delete_Osimps_I5_J,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] :
% 0.34/0.55 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X2 )
% 0.34/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_delete.simps(5)
% 0.34/0.55 thf(fact_2121_vebt__mint_Ocases,axiom,
% 0.34/0.55 ! [X2: vEBT_VEBT] :
% 0.34/0.55 ( ! [A: $o,B: $o] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( vEBT_Leaf @ A @ B ) )
% 0.34/0.55 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.34/0.55 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_mint.cases
% 0.34/0.55 thf(fact_2122_vebt__mint_Osimps_I3_J,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.34/0.55 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.34/0.55 = ( some_nat @ Mi ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_mint.simps(3)
% 0.34/0.55 thf(fact_2123_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
% 0.34/0.55 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
% 0.34/0.55 = ( ( X2 = Mi )
% 0.34/0.55 | ( X2 = Ma ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.membermima.simps(3)
% 0.34/0.55 thf(fact_2124_vebt__maxt_Osimps_I3_J,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.34/0.55 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.34/0.55 = ( some_nat @ Ma ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_maxt.simps(3)
% 0.34/0.55 thf(fact_2125_vebt__delete_Osimps_I6_J,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] :
% 0.34/0.55 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X2 )
% 0.34/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_delete.simps(6)
% 0.34/0.55 thf(fact_2126_compl__le__swap2,axiom,
% 0.34/0.55 ! [Y3: set_int,X2: set_int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X2 )
% 0.34/0.55 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % compl_le_swap2
% 0.34/0.55 thf(fact_2127_compl__le__swap1,axiom,
% 0.34/0.55 ! [Y3: set_int,X2: set_int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) )
% 0.34/0.55 => ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % compl_le_swap1
% 0.34/0.55 thf(fact_2128_compl__mono,axiom,
% 0.34/0.55 ! [X2: set_int,Y3: set_int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.34/0.55 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % compl_mono
% 0.34/0.55 thf(fact_2129_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.34/0.55 ! [M: nat,N: nat] :
% 0.34/0.55 ( ( ( divide_divide_nat @ M @ N )
% 0.34/0.55 = zero_zero_nat )
% 0.34/0.55 = ( ( ord_less_nat @ M @ N )
% 0.34/0.55 | ( N = zero_zero_nat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Euclidean_Division.div_eq_0_iff
% 0.34/0.55 thf(fact_2130_Suc__div__le__mono,axiom,
% 0.34/0.55 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 0.34/0.55
% 0.34/0.55 % Suc_div_le_mono
% 0.34/0.55 thf(fact_2131_vebt__mint_Oelims,axiom,
% 0.34/0.55 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.34/0.55 ( ( ( vEBT_vebt_mint @ X2 )
% 0.34/0.55 = Y3 )
% 0.34/0.55 => ( ! [A: $o,B: $o] :
% 0.34/0.55 ( ( X2
% 0.34/0.55 = ( vEBT_Leaf @ A @ B ) )
% 0.34/0.55 => ~ ( ( A
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.55 & ( ~ A
% 0.34/0.55 => ( ( B
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.55 & ( ~ B
% 0.34/0.55 => ( Y3 = none_nat ) ) ) ) ) )
% 0.34/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.34/0.55 => ( Y3 != none_nat ) )
% 0.34/0.55 => ~ ! [Mi2: nat] :
% 0.34/0.55 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.34/0.55 => ( Y3
% 0.34/0.55 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_mint.elims
% 0.34/0.55 thf(fact_2132_vebt__maxt_Oelims,axiom,
% 0.34/0.55 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.34/0.55 ( ( ( vEBT_vebt_maxt @ X2 )
% 0.34/0.55 = Y3 )
% 0.34/0.55 => ( ! [A: $o,B: $o] :
% 0.34/0.55 ( ( X2
% 0.34/0.55 = ( vEBT_Leaf @ A @ B ) )
% 0.34/0.55 => ~ ( ( B
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( some_nat @ one_one_nat ) ) )
% 0.34/0.55 & ( ~ B
% 0.34/0.55 => ( ( A
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( some_nat @ zero_zero_nat ) ) )
% 0.34/0.55 & ( ~ A
% 0.34/0.55 => ( Y3 = none_nat ) ) ) ) ) )
% 0.34/0.55 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.34/0.55 => ( Y3 != none_nat ) )
% 0.34/0.55 => ~ ! [Mi2: nat,Ma2: nat] :
% 0.34/0.55 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.34/0.55 ( X2
% 0.34/0.55 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.34/0.55 => ( Y3
% 0.34/0.55 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % vebt_maxt.elims
% 0.34/0.55 thf(fact_2133_diff__shunt__var,axiom,
% 0.34/0.55 ! [X2: set_real,Y3: set_real] :
% 0.34/0.55 ( ( ( minus_minus_set_real @ X2 @ Y3 )
% 0.34/0.55 = bot_bot_set_real )
% 0.34/0.55 = ( ord_less_eq_set_real @ X2 @ Y3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % diff_shunt_var
% 0.34/0.55 thf(fact_2134_diff__shunt__var,axiom,
% 0.34/0.55 ! [X2: set_o,Y3: set_o] :
% 0.34/0.55 ( ( ( minus_minus_set_o @ X2 @ Y3 )
% 0.34/0.55 = bot_bot_set_o )
% 0.34/0.55 = ( ord_less_eq_set_o @ X2 @ Y3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % diff_shunt_var
% 0.34/0.55 thf(fact_2135_diff__shunt__var,axiom,
% 0.34/0.55 ! [X2: set_nat,Y3: set_nat] :
% 0.34/0.55 ( ( ( minus_minus_set_nat @ X2 @ Y3 )
% 0.34/0.55 = bot_bot_set_nat )
% 0.34/0.55 = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % diff_shunt_var
% 0.34/0.55 thf(fact_2136_diff__shunt__var,axiom,
% 0.34/0.55 ! [X2: set_int,Y3: set_int] :
% 0.34/0.55 ( ( ( minus_minus_set_int @ X2 @ Y3 )
% 0.34/0.55 = bot_bot_set_int )
% 0.34/0.55 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % diff_shunt_var
% 0.34/0.55 thf(fact_2137_div__greater__zero__iff,axiom,
% 0.34/0.55 ! [M: nat,N: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 0.34/0.55 = ( ( ord_less_eq_nat @ N @ M )
% 0.34/0.55 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_greater_zero_iff
% 0.34/0.55 thf(fact_2138_div__le__mono2,axiom,
% 0.34/0.55 ! [M: nat,N: nat,K: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.34/0.55 => ( ( ord_less_eq_nat @ M @ N )
% 0.34/0.55 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_le_mono2
% 0.34/0.55 thf(fact_2139_div__eq__dividend__iff,axiom,
% 0.34/0.55 ! [M: nat,N: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.34/0.55 => ( ( ( divide_divide_nat @ M @ N )
% 0.34/0.55 = M )
% 0.34/0.55 = ( N = one_one_nat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_eq_dividend_iff
% 0.34/0.55 thf(fact_2140_div__less__dividend,axiom,
% 0.34/0.55 ! [N: nat,M: nat] :
% 0.34/0.55 ( ( ord_less_nat @ one_one_nat @ N )
% 0.34/0.55 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.34/0.55 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_less_dividend
% 0.34/0.55 thf(fact_2141_pos__imp__zdiv__neg__iff,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
% 0.34/0.55 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % pos_imp_zdiv_neg_iff
% 0.34/0.55 thf(fact_2142_neg__imp__zdiv__neg__iff,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
% 0.34/0.55 = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % neg_imp_zdiv_neg_iff
% 0.34/0.55 thf(fact_2143_div__neg__pos__less0,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_neg_pos_less0
% 0.34/0.55 thf(fact_2144_div__positive,axiom,
% 0.34/0.55 ! [B3: nat,A2: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.34/0.55 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.34/0.55 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_positive
% 0.34/0.55 thf(fact_2145_div__positive,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.55 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_positive
% 0.34/0.55 thf(fact_2146_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.34/0.55 ! [A2: nat,B3: nat] :
% 0.34/0.55 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.34/0.55 => ( ( ord_less_nat @ A2 @ B3 )
% 0.34/0.55 => ( ( divide_divide_nat @ A2 @ B3 )
% 0.34/0.55 = zero_zero_nat ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % unique_euclidean_semiring_numeral_class.div_less
% 0.34/0.55 thf(fact_2147_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.55 => ( ( ord_less_int @ A2 @ B3 )
% 0.34/0.55 => ( ( divide_divide_int @ A2 @ B3 )
% 0.34/0.55 = zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % unique_euclidean_semiring_numeral_class.div_less
% 0.34/0.55 thf(fact_2148_div__geq,axiom,
% 0.34/0.55 ! [N: nat,M: nat] :
% 0.34/0.55 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.34/0.55 => ( ~ ( ord_less_nat @ M @ N )
% 0.34/0.55 => ( ( divide_divide_nat @ M @ N )
% 0.34/0.55 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_geq
% 0.34/0.55 thf(fact_2149_div__if,axiom,
% 0.34/0.55 ( divide_divide_nat
% 0.34/0.55 = ( ^ [M2: nat,N2: nat] :
% 0.34/0.55 ( if_nat
% 0.34/0.55 @ ( ( ord_less_nat @ M2 @ N2 )
% 0.34/0.55 | ( N2 = zero_zero_nat ) )
% 0.34/0.55 @ zero_zero_nat
% 0.34/0.55 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_if
% 0.34/0.55 thf(fact_2150_zdiv__mono1,axiom,
% 0.34/0.55 ! [A2: int,A7: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ A2 @ A7 )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A7 @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zdiv_mono1
% 0.34/0.55 thf(fact_2151_zdiv__mono2,axiom,
% 0.34/0.55 ! [A2: int,B7: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.34/0.55 => ( ( ord_less_eq_int @ B7 @ B3 )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zdiv_mono2
% 0.34/0.55 thf(fact_2152_zdiv__eq__0__iff,axiom,
% 0.34/0.55 ! [I: int,K: int] :
% 0.34/0.55 ( ( ( divide_divide_int @ I @ K )
% 0.34/0.55 = zero_zero_int )
% 0.34/0.55 = ( ( K = zero_zero_int )
% 0.34/0.55 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.34/0.55 & ( ord_less_int @ I @ K ) )
% 0.34/0.55 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.34/0.55 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zdiv_eq_0_iff
% 0.34/0.55 thf(fact_2153_zdiv__mono1__neg,axiom,
% 0.34/0.55 ! [A2: int,A7: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ A2 @ A7 )
% 0.34/0.55 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B3 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zdiv_mono1_neg
% 0.34/0.55 thf(fact_2154_zdiv__mono2__neg,axiom,
% 0.34/0.55 ! [A2: int,B7: int,B3: int] :
% 0.34/0.55 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.34/0.55 => ( ( ord_less_eq_int @ B7 @ B3 )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % zdiv_mono2_neg
% 0.34/0.55 thf(fact_2155_div__int__pos__iff,axiom,
% 0.34/0.55 ! [K: int,L: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.34/0.55 = ( ( K = zero_zero_int )
% 0.34/0.55 | ( L = zero_zero_int )
% 0.34/0.55 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.34/0.55 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.34/0.55 | ( ( ord_less_int @ K @ zero_zero_int )
% 0.34/0.55 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_int_pos_iff
% 0.34/0.55 thf(fact_2156_div__positive__int,axiom,
% 0.34/0.55 ! [L: int,K: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ L @ K )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ L )
% 0.34/0.55 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_positive_int
% 0.34/0.55 thf(fact_2157_div__nonneg__neg__le0,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.55 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_nonneg_neg_le0
% 0.34/0.55 thf(fact_2158_div__nonpos__pos__le0,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % div_nonpos_pos_le0
% 0.34/0.55 thf(fact_2159_pos__imp__zdiv__pos__iff,axiom,
% 0.34/0.55 ! [K: int,I: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ K )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.34/0.55 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % pos_imp_zdiv_pos_iff
% 0.34/0.55 thf(fact_2160_neg__imp__zdiv__nonneg__iff,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.34/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 0.34/0.55 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % neg_imp_zdiv_nonneg_iff
% 0.34/0.55 thf(fact_2161_pos__imp__zdiv__nonneg__iff,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.34/0.55 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 0.34/0.55 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % pos_imp_zdiv_nonneg_iff
% 0.34/0.55 thf(fact_2162_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.34/0.55 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 0.34/0.55 = ( ( ord_less_eq_int @ B3 @ A2 )
% 0.34/0.55 & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % nonneg1_imp_zdiv_pos_iff
% 0.34/0.55 thf(fact_2163_int__div__less__self,axiom,
% 0.34/0.55 ! [X2: int,K: int] :
% 0.34/0.55 ( ( ord_less_int @ zero_zero_int @ X2 )
% 0.34/0.55 => ( ( ord_less_int @ one_one_int @ K )
% 0.34/0.55 => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % int_div_less_self
% 0.34/0.55 thf(fact_2164_delete__correct,axiom,
% 0.34/0.55 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.55 => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
% 0.34/0.55 = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % delete_correct
% 0.34/0.55 thf(fact_2165_mi__eq__ma__no__ch,axiom,
% 0.34/0.55 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.34/0.55 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.34/0.55 => ( ( Mi = Ma )
% 0.34/0.55 => ( ! [X4: vEBT_VEBT] :
% 0.34/0.55 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.34/0.55 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 0.34/0.55 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mi_eq_ma_no_ch
% 0.34/0.55 thf(fact_2166_divides__aux__eq,axiom,
% 0.34/0.55 ! [Q3: nat,R2: nat] :
% 0.34/0.55 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.34/0.55 = ( R2 = zero_zero_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % divides_aux_eq
% 0.34/0.55 thf(fact_2167_divides__aux__eq,axiom,
% 0.34/0.55 ! [Q3: int,R2: int] :
% 0.34/0.55 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.34/0.55 = ( R2 = zero_zero_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % divides_aux_eq
% 0.34/0.55 thf(fact_2168_delete__correct_H,axiom,
% 0.34/0.55 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.34/0.55 ( ( vEBT_invar_vebt @ T @ N )
% 0.34/0.55 => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
% 0.34/0.55 = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % delete_correct'
% 0.34/0.55 thf(fact_2169_dbl__dec__simps_I2_J,axiom,
% 0.34/0.55 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.34/0.55 = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % dbl_dec_simps(2)
% 0.34/0.55 thf(fact_2170_dbl__dec__simps_I2_J,axiom,
% 0.34/0.55 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.34/0.55 = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.34/0.55
% 0.34/0.55 % dbl_dec_simps(2)
% 0.34/0.55 thf(fact_2171_dbl__dec__simps_I2_J,axiom,
% 0.34/0.55 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.34/0.55 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.34/0.55
% 0.34/0.55 % dbl_dec_simps(2)
% 0.34/0.55 thf(fact_2172_dbl__dec__simps_I2_J,axiom,
% 0.34/0.55 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.34/0.55 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.34/0.55
% 0.34/0.55 % dbl_dec_simps(2)
% 0.34/0.55 thf(fact_2173_dbl__dec__simps_I2_J,axiom,
% 0.34/0.55 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.34/0.55 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.34/0.55
% 0.34/0.55 % dbl_dec_simps(2)
% 0.34/0.55 thf(fact_2174_prod__decode__aux_Osimps,axiom,
% 0.34/0.55 ( nat_prod_decode_aux
% 0.34/0.55 = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_decode_aux.simps
% 0.34/0.55 thf(fact_2175_prod__decode__aux_Oelims,axiom,
% 0.34/0.55 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 0.34/0.55 = Y3 )
% 0.34/0.55 => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 0.34/0.55 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.34/0.55 => ( Y3
% 0.34/0.55 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % prod_decode_aux.elims
% 0.34/0.55 thf(fact_2176_insert__absorb2,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb2
% 0.34/0.55 thf(fact_2177_insert__absorb2,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real] :
% 0.34/0.55 ( ( insert_real @ X2 @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.55 = ( insert_real @ X2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb2
% 0.34/0.55 thf(fact_2178_insert__absorb2,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o] :
% 0.34/0.55 ( ( insert_o @ X2 @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.55 = ( insert_o @ X2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb2
% 0.34/0.55 thf(fact_2179_insert__absorb2,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat] :
% 0.34/0.55 ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( insert_nat @ X2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb2
% 0.34/0.55 thf(fact_2180_insert__absorb2,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int] :
% 0.34/0.55 ( ( insert_int @ X2 @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.55 = ( insert_int @ X2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb2
% 0.34/0.55 thf(fact_2181_insert__iff,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2182_insert__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member_real @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2183_insert__iff,axiom,
% 0.34/0.55 ! [A2: $o,B3: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member_o @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2184_insert__iff,axiom,
% 0.34/0.55 ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member_set_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2185_insert__iff,axiom,
% 0.34/0.55 ! [A2: nat,B3: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2186_insert__iff,axiom,
% 0.34/0.55 ! [A2: int,B3: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 | ( member_int @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_iff
% 0.34/0.55 thf(fact_2187_insertCI,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2188_insertCI,axiom,
% 0.34/0.55 ! [A2: real,B2: set_real,B3: real] :
% 0.34/0.55 ( ( ~ ( member_real @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2189_insertCI,axiom,
% 0.34/0.55 ! [A2: $o,B2: set_o,B3: $o] :
% 0.34/0.55 ( ( ~ ( member_o @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2190_insertCI,axiom,
% 0.34/0.55 ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
% 0.34/0.55 ( ( ~ ( member_set_nat @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2191_insertCI,axiom,
% 0.34/0.55 ! [A2: nat,B2: set_nat,B3: nat] :
% 0.34/0.55 ( ( ~ ( member_nat @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2192_insertCI,axiom,
% 0.34/0.55 ! [A2: int,B2: set_int,B3: int] :
% 0.34/0.55 ( ( ~ ( member_int @ A2 @ B2 )
% 0.34/0.55 => ( A2 = B3 ) )
% 0.34/0.55 => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertCI
% 0.34/0.55 thf(fact_2193_singletonI,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2194_singletonI,axiom,
% 0.34/0.55 ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2195_singletonI,axiom,
% 0.34/0.55 ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2196_singletonI,axiom,
% 0.34/0.55 ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2197_singletonI,axiom,
% 0.34/0.55 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2198_singletonI,axiom,
% 0.34/0.55 ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonI
% 0.34/0.55 thf(fact_2199_finite__insert,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ ( insert_real @ A2 @ A3 ) )
% 0.34/0.55 = ( finite_finite_real @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2200_finite__insert,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ ( insert_o @ A2 @ A3 ) )
% 0.34/0.55 = ( finite_finite_o @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2201_finite__insert,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) )
% 0.34/0.55 = ( finite_finite_nat @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2202_finite__insert,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ ( insert_int @ A2 @ A3 ) )
% 0.34/0.55 = ( finite_finite_int @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2203_finite__insert,axiom,
% 0.34/0.55 ! [A2: complex,A3: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) )
% 0.34/0.55 = ( finite3207457112153483333omplex @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2204_finite__insert,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 0.34/0.55 = ( finite6177210948735845034at_nat @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2205_finite__insert,axiom,
% 0.34/0.55 ! [A2: extended_enat,A3: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) )
% 0.34/0.55 = ( finite4001608067531595151d_enat @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_insert
% 0.34/0.55 thf(fact_2206_insert__subset,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.55 & ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2207_insert__subset,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member_real @ X2 @ B2 )
% 0.34/0.55 & ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2208_insert__subset,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member_o @ X2 @ B2 )
% 0.34/0.55 & ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2209_insert__subset,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member_set_nat @ X2 @ B2 )
% 0.34/0.55 & ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2210_insert__subset,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member_nat @ X2 @ B2 )
% 0.34/0.55 & ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2211_insert__subset,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( ( member_int @ X2 @ B2 )
% 0.34/0.55 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subset
% 0.34/0.55 thf(fact_2212_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2213_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: real,B2: set_real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2214_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: $o,B2: set_o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2215_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2216_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: int,B2: set_int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2217_insert__Diff1,axiom,
% 0.34/0.55 ! [X2: nat,B2: set_nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff1
% 0.34/0.55 thf(fact_2218_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2219_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2220_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2221_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2222_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2223_Diff__insert0,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Diff_insert0
% 0.34/0.55 thf(fact_2224_singleton__insert__inj__eq,axiom,
% 0.34/0.55 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq
% 0.34/0.55 thf(fact_2225_singleton__insert__inj__eq,axiom,
% 0.34/0.55 ! [B3: real,A2: real,A3: set_real] :
% 0.34/0.55 ( ( ( insert_real @ B3 @ bot_bot_set_real )
% 0.34/0.55 = ( insert_real @ A2 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq
% 0.34/0.55 thf(fact_2226_singleton__insert__inj__eq,axiom,
% 0.34/0.55 ! [B3: $o,A2: $o,A3: set_o] :
% 0.34/0.55 ( ( ( insert_o @ B3 @ bot_bot_set_o )
% 0.34/0.55 = ( insert_o @ A2 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq
% 0.34/0.55 thf(fact_2227_singleton__insert__inj__eq,axiom,
% 0.34/0.55 ! [B3: nat,A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( ( insert_nat @ B3 @ bot_bot_set_nat )
% 0.34/0.55 = ( insert_nat @ A2 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq
% 0.34/0.55 thf(fact_2228_singleton__insert__inj__eq,axiom,
% 0.34/0.55 ! [B3: int,A2: int,A3: set_int] :
% 0.34/0.55 ( ( ( insert_int @ B3 @ bot_bot_set_int )
% 0.34/0.55 = ( insert_int @ A2 @ A3 ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq
% 0.34/0.55 thf(fact_2229_singleton__insert__inj__eq_H,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq'
% 0.34/0.55 thf(fact_2230_singleton__insert__inj__eq_H,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real,B3: real] :
% 0.34/0.55 ( ( ( insert_real @ A2 @ A3 )
% 0.34/0.55 = ( insert_real @ B3 @ bot_bot_set_real ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq'
% 0.34/0.55 thf(fact_2231_singleton__insert__inj__eq_H,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o,B3: $o] :
% 0.34/0.55 ( ( ( insert_o @ A2 @ A3 )
% 0.34/0.55 = ( insert_o @ B3 @ bot_bot_set_o ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq'
% 0.34/0.55 thf(fact_2232_singleton__insert__inj__eq_H,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat,B3: nat] :
% 0.34/0.55 ( ( ( insert_nat @ A2 @ A3 )
% 0.34/0.55 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq'
% 0.34/0.55 thf(fact_2233_singleton__insert__inj__eq_H,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int,B3: int] :
% 0.34/0.55 ( ( ( insert_int @ A2 @ A3 )
% 0.34/0.55 = ( insert_int @ B3 @ bot_bot_set_int ) )
% 0.34/0.55 = ( ( A2 = B3 )
% 0.34/0.55 & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_insert_inj_eq'
% 0.34/0.55 thf(fact_2234_insert__Diff__single,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_single
% 0.34/0.55 thf(fact_2235_insert__Diff__single,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real] :
% 0.34/0.55 ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.34/0.55 = ( insert_real @ A2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_single
% 0.34/0.55 thf(fact_2236_insert__Diff__single,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o] :
% 0.34/0.55 ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( insert_o @ A2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_single
% 0.34/0.55 thf(fact_2237_insert__Diff__single,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int] :
% 0.34/0.55 ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( insert_int @ A2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_single
% 0.34/0.55 thf(fact_2238_insert__Diff__single,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( insert_nat @ A2 @ A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_single
% 0.34/0.55 thf(fact_2239_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_real,A2: real,B2: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite_finite_real @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2240_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_o,A2: $o,B2: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite_finite_o @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2241_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_int,A2: int,B2: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2242_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_complex,A2: complex,B2: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2243_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2244_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_Extended_enat,A2: extended_enat,B2: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2245_finite__Diff__insert,axiom,
% 0.34/0.55 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_Diff_insert
% 0.34/0.55 thf(fact_2246_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_real,X2: real] :
% 0.34/0.55 ( ( finite_finite_real @ A3 )
% 0.34/0.55 => ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2247_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_o,X2: $o] :
% 0.34/0.55 ( ( finite_finite_o @ A3 )
% 0.34/0.55 => ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2248_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.55 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.55 => ( ~ ( member_list_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2249_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.55 => ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2250_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_nat,X2: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A3 )
% 0.34/0.55 => ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2251_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_int,X2: int] :
% 0.34/0.55 ( ( finite_finite_int @ A3 )
% 0.34/0.55 => ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2252_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_complex,X2: complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.55 => ( ~ ( member_complex @ X2 @ A3 )
% 0.34/0.55 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2253_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2254_card__insert__disjoint,axiom,
% 0.34/0.55 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.55 => ( ~ ( member_Extended_enat @ X2 @ A3 )
% 0.34/0.55 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.34/0.55 = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_insert_disjoint
% 0.34/0.55 thf(fact_2255_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) )
% 0.34/0.55 = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2256_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_set_nat,B3: set_nat] :
% 0.34/0.55 ( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B3 @ bot_bot_set_set_nat ) ) )
% 0.34/0.55 = ( ~ ( member_set_nat @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2257_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_real,B3: real] :
% 0.34/0.55 ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ ( insert_real @ B3 @ bot_bot_set_real ) ) )
% 0.34/0.55 = ( ~ ( member_real @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2258_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_o,B3: $o] :
% 0.34/0.55 ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ ( insert_o @ B3 @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( ~ ( member_o @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2259_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_nat,B3: nat] :
% 0.34/0.55 ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( ~ ( member_nat @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2260_subset__Compl__singleton,axiom,
% 0.34/0.55 ! [A3: set_int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( insert_int @ B3 @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( ~ ( member_int @ B3 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_Compl_singleton
% 0.34/0.55 thf(fact_2261_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.34/0.55 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2262_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ( member_real @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_real @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2263_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ( member_o @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_o @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2264_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: complex,A3: set_complex,B2: set_complex] :
% 0.34/0.55 ( ( member_complex @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_complex @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2265_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: list_nat,A3: set_list_nat,B2: set_list_nat] :
% 0.34/0.55 ( ( member_list_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_list_nat @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2266_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_set_nat @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2267_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ( member_int @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_int @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2268_card__Diff__insert,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_nat @ A2 @ B2 )
% 0.34/0.55 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
% 0.34/0.55 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % card_Diff_insert
% 0.34/0.55 thf(fact_2269_less__eq__real__def,axiom,
% 0.34/0.55 ( ord_less_eq_real
% 0.34/0.55 = ( ^ [X: real,Y: real] :
% 0.34/0.55 ( ( ord_less_real @ X @ Y )
% 0.34/0.55 | ( X = Y ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % less_eq_real_def
% 0.34/0.55 thf(fact_2270_complete__real,axiom,
% 0.34/0.55 ! [S: set_real] :
% 0.34/0.55 ( ? [X4: real] : ( member_real @ X4 @ S )
% 0.34/0.55 => ( ? [Z5: real] :
% 0.34/0.55 ! [X5: real] :
% 0.34/0.55 ( ( member_real @ X5 @ S )
% 0.34/0.55 => ( ord_less_eq_real @ X5 @ Z5 ) )
% 0.34/0.55 => ? [Y4: real] :
% 0.34/0.55 ( ! [X4: real] :
% 0.34/0.55 ( ( member_real @ X4 @ S )
% 0.34/0.55 => ( ord_less_eq_real @ X4 @ Y4 ) )
% 0.34/0.55 & ! [Z5: real] :
% 0.34/0.55 ( ! [X5: real] :
% 0.34/0.55 ( ( member_real @ X5 @ S )
% 0.34/0.55 => ( ord_less_eq_real @ X5 @ Z5 ) )
% 0.34/0.55 => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % complete_real
% 0.34/0.55 thf(fact_2271_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert8211810215607154385at_nat @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member8440522571783428010at_nat @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2272_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_real] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_real @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member_real @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2273_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_o] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_o @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member_o @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2274_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_set_nat @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member_set_nat @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2275_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_nat @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member_nat @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2276_mk__disjoint__insert,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ A2 @ A3 )
% 0.34/0.55 => ? [B8: set_int] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_int @ A2 @ B8 ) )
% 0.34/0.55 & ~ ( member_int @ A2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % mk_disjoint_insert
% 0.34/0.55 thf(fact_2277_insert__commute,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y3 @ A3 ) )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_commute
% 0.34/0.55 thf(fact_2278_insert__commute,axiom,
% 0.34/0.55 ! [X2: real,Y3: real,A3: set_real] :
% 0.34/0.55 ( ( insert_real @ X2 @ ( insert_real @ Y3 @ A3 ) )
% 0.34/0.55 = ( insert_real @ Y3 @ ( insert_real @ X2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_commute
% 0.34/0.55 thf(fact_2279_insert__commute,axiom,
% 0.34/0.55 ! [X2: $o,Y3: $o,A3: set_o] :
% 0.34/0.55 ( ( insert_o @ X2 @ ( insert_o @ Y3 @ A3 ) )
% 0.34/0.55 = ( insert_o @ Y3 @ ( insert_o @ X2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_commute
% 0.34/0.55 thf(fact_2280_insert__commute,axiom,
% 0.34/0.55 ! [X2: nat,Y3: nat,A3: set_nat] :
% 0.34/0.55 ( ( insert_nat @ X2 @ ( insert_nat @ Y3 @ A3 ) )
% 0.34/0.55 = ( insert_nat @ Y3 @ ( insert_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_commute
% 0.34/0.55 thf(fact_2281_insert__commute,axiom,
% 0.34/0.55 ! [X2: int,Y3: int,A3: set_int] :
% 0.34/0.55 ( ( insert_int @ X2 @ ( insert_int @ Y3 @ A3 ) )
% 0.34/0.55 = ( insert_int @ Y3 @ ( insert_int @ X2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_commute
% 0.34/0.55 thf(fact_2282_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 != B3 )
% 0.34/0.55 => ? [C4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert8211810215607154385at_nat @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member8440522571783428010at_nat @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert8211810215607154385at_nat @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member8440522571783428010at_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2283_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real,B3: real,B2: set_real] :
% 0.34/0.55 ( ~ ( member_real @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_real @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert_real @ A2 @ A3 )
% 0.34/0.55 = ( insert_real @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 != B3 )
% 0.34/0.55 => ? [C4: set_real] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_real @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member_real @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert_real @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2284_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o,B3: $o,B2: set_o] :
% 0.34/0.55 ( ~ ( member_o @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_o @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert_o @ A2 @ A3 )
% 0.34/0.55 = ( insert_o @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 = ( ~ B3 ) )
% 0.34/0.55 => ? [C4: set_o] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_o @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member_o @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert_o @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member_o @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2285_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: set_nat,A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ~ ( member_set_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_set_nat @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert_set_nat @ A2 @ A3 )
% 0.34/0.55 = ( insert_set_nat @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 != B3 )
% 0.34/0.55 => ? [C4: set_set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_set_nat @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member_set_nat @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert_set_nat @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member_set_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2286_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat,B3: nat,B2: set_nat] :
% 0.34/0.55 ( ~ ( member_nat @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_nat @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert_nat @ A2 @ A3 )
% 0.34/0.55 = ( insert_nat @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 != B3 )
% 0.34/0.55 => ? [C4: set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_nat @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member_nat @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert_nat @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2287_insert__eq__iff,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int,B3: int,B2: set_int] :
% 0.34/0.55 ( ~ ( member_int @ A2 @ A3 )
% 0.34/0.55 => ( ~ ( member_int @ B3 @ B2 )
% 0.34/0.55 => ( ( ( insert_int @ A2 @ A3 )
% 0.34/0.55 = ( insert_int @ B3 @ B2 ) )
% 0.34/0.55 = ( ( ( A2 = B3 )
% 0.34/0.55 => ( A3 = B2 ) )
% 0.34/0.55 & ( ( A2 != B3 )
% 0.34/0.55 => ? [C4: set_int] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_int @ B3 @ C4 ) )
% 0.34/0.55 & ~ ( member_int @ B3 @ C4 )
% 0.34/0.55 & ( B2
% 0.34/0.55 = ( insert_int @ A2 @ C4 ) )
% 0.34/0.55 & ~ ( member_int @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_eq_iff
% 0.34/0.55 thf(fact_2288_insert__absorb,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.34/0.55 => ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2289_insert__absorb,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ A2 @ A3 )
% 0.34/0.55 => ( ( insert_real @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2290_insert__absorb,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ A2 @ A3 )
% 0.34/0.55 => ( ( insert_o @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2291_insert__absorb,axiom,
% 0.34/0.55 ! [A2: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.55 => ( ( insert_set_nat @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2292_insert__absorb,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ A3 )
% 0.34/0.55 => ( ( insert_nat @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2293_insert__absorb,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ A2 @ A3 )
% 0.34/0.55 => ( ( insert_int @ A2 @ A3 )
% 0.34/0.55 = A3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_absorb
% 0.34/0.55 thf(fact_2294_insert__ident,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert8211810215607154385at_nat @ X2 @ A3 )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2295_insert__ident,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member_real @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert_real @ X2 @ A3 )
% 0.34/0.55 = ( insert_real @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2296_insert__ident,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member_o @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert_o @ X2 @ A3 )
% 0.34/0.55 = ( insert_o @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2297_insert__ident,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member_set_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert_set_nat @ X2 @ A3 )
% 0.34/0.55 = ( insert_set_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2298_insert__ident,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert_nat @ X2 @ A3 )
% 0.34/0.55 = ( insert_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2299_insert__ident,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.55 => ( ~ ( member_int @ X2 @ B2 )
% 0.34/0.55 => ( ( ( insert_int @ X2 @ A3 )
% 0.34/0.55 = ( insert_int @ X2 @ B2 ) )
% 0.34/0.55 = ( A3 = B2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_ident
% 0.34/0.55 thf(fact_2300_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert8211810215607154385at_nat @ X2 @ B8 ) )
% 0.34/0.55 => ( member8440522571783428010at_nat @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2301_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_real] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_real @ X2 @ B8 ) )
% 0.34/0.55 => ( member_real @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2302_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_o] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_o @ X2 @ B8 ) )
% 0.34/0.55 => ( member_o @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2303_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_set_nat @ X2 @ B8 ) )
% 0.34/0.55 => ( member_set_nat @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2304_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_nat] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_nat @ X2 @ B8 ) )
% 0.34/0.55 => ( member_nat @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2305_Set_Oset__insert,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ X2 @ A3 )
% 0.34/0.55 => ~ ! [B8: set_int] :
% 0.34/0.55 ( ( A3
% 0.34/0.55 = ( insert_int @ X2 @ B8 ) )
% 0.34/0.55 => ( member_int @ X2 @ B8 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % Set.set_insert
% 0.34/0.55 thf(fact_2306_insertI2,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.34/0.55 => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2307_insertI2,axiom,
% 0.34/0.55 ! [A2: real,B2: set_real,B3: real] :
% 0.34/0.55 ( ( member_real @ A2 @ B2 )
% 0.34/0.55 => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2308_insertI2,axiom,
% 0.34/0.55 ! [A2: $o,B2: set_o,B3: $o] :
% 0.34/0.55 ( ( member_o @ A2 @ B2 )
% 0.34/0.55 => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2309_insertI2,axiom,
% 0.34/0.55 ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ B2 )
% 0.34/0.55 => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2310_insertI2,axiom,
% 0.34/0.55 ! [A2: nat,B2: set_nat,B3: nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ B2 )
% 0.34/0.55 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2311_insertI2,axiom,
% 0.34/0.55 ! [A2: int,B2: set_int,B3: int] :
% 0.34/0.55 ( ( member_int @ A2 @ B2 )
% 0.34/0.55 => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI2
% 0.34/0.55 thf(fact_2312_insertI1,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2313_insertI1,axiom,
% 0.34/0.55 ! [A2: real,B2: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2314_insertI1,axiom,
% 0.34/0.55 ! [A2: $o,B2: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2315_insertI1,axiom,
% 0.34/0.55 ! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2316_insertI1,axiom,
% 0.34/0.55 ! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2317_insertI1,axiom,
% 0.34/0.55 ! [A2: int,B2: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertI1
% 0.34/0.55 thf(fact_2318_insertE,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 != B3 )
% 0.34/0.55 => ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2319_insertE,axiom,
% 0.34/0.55 ! [A2: real,B3: real,A3: set_real] :
% 0.34/0.55 ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 != B3 )
% 0.34/0.55 => ( member_real @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2320_insertE,axiom,
% 0.34/0.55 ! [A2: $o,B3: $o,A3: set_o] :
% 0.34/0.55 ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 = ( ~ B3 ) )
% 0.34/0.55 => ( member_o @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2321_insertE,axiom,
% 0.34/0.55 ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 != B3 )
% 0.34/0.55 => ( member_set_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2322_insertE,axiom,
% 0.34/0.55 ! [A2: nat,B3: nat,A3: set_nat] :
% 0.34/0.55 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 != B3 )
% 0.34/0.55 => ( member_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2323_insertE,axiom,
% 0.34/0.55 ! [A2: int,B3: int,A3: set_int] :
% 0.34/0.55 ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
% 0.34/0.55 => ( ( A2 != B3 )
% 0.34/0.55 => ( member_int @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insertE
% 0.34/0.55 thf(fact_2324_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc8306885398267862888on_nat] :
% 0.34/0.55 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.34/0.55 => ~ ! [F3: nat > nat > nat,A: nat,B: nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc8929957630744042906on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_shift.cases
% 0.34/0.55 thf(fact_2325_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc5542196010084753463at_nat] :
% 0.34/0.55 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.34/0.55 => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc2899441246263362727at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_shift.cases
% 0.34/0.55 thf(fact_2326_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc1193250871479095198on_num] :
% 0.34/0.55 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: num > num > num,V2: num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.34/0.55 => ~ ! [F3: num > num > num,A: num,B: num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc5778274026573060048on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_shift.cases
% 0.34/0.55 thf(fact_2327_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc2233624965454879586on_nat] :
% 0.34/0.55 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.34/0.55 => ~ ! [F3: nat > nat > $o,X5: nat,Y4: nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc4035269172776083154on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_comp_shift.cases
% 0.34/0.55 thf(fact_2328_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc5491161045314408544at_nat] :
% 0.34/0.55 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.34/0.55 => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3994169339658061776at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_comp_shift.cases
% 0.34/0.55 thf(fact_2329_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.34/0.55 ! [X2: produc7036089656553540234on_num] :
% 0.34/0.55 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.34/0.55 => ( ! [Uw2: num > num > $o,V2: num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.34/0.55 => ~ ! [F3: num > num > $o,X5: num,Y4: num] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3576312749637752826on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % VEBT_internal.option_comp_shift.cases
% 0.34/0.55 thf(fact_2330_singletonD,axiom,
% 0.34/0.55 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2331_singletonD,axiom,
% 0.34/0.55 ! [B3: set_nat,A2: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2332_singletonD,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2333_singletonD,axiom,
% 0.34/0.55 ! [B3: $o,A2: $o] :
% 0.34/0.55 ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2334_singletonD,axiom,
% 0.34/0.55 ! [B3: nat,A2: nat] :
% 0.34/0.55 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2335_singletonD,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 => ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singletonD
% 0.34/0.55 thf(fact_2336_singleton__iff,axiom,
% 0.34/0.55 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2337_singleton__iff,axiom,
% 0.34/0.55 ! [B3: set_nat,A2: set_nat] :
% 0.34/0.55 ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2338_singleton__iff,axiom,
% 0.34/0.55 ! [B3: real,A2: real] :
% 0.34/0.55 ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2339_singleton__iff,axiom,
% 0.34/0.55 ! [B3: $o,A2: $o] :
% 0.34/0.55 ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2340_singleton__iff,axiom,
% 0.34/0.55 ! [B3: nat,A2: nat] :
% 0.34/0.55 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2341_singleton__iff,axiom,
% 0.34/0.55 ! [B3: int,A2: int] :
% 0.34/0.55 ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.55 = ( B3 = A2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_iff
% 0.34/0.55 thf(fact_2342_doubleton__eq__iff,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 0.34/0.55 ( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 0.34/0.55 = ( ( ( A2 = C )
% 0.34/0.55 & ( B3 = D ) )
% 0.34/0.55 | ( ( A2 = D )
% 0.34/0.55 & ( B3 = C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % doubleton_eq_iff
% 0.34/0.55 thf(fact_2343_doubleton__eq__iff,axiom,
% 0.34/0.55 ! [A2: real,B3: real,C: real,D: real] :
% 0.34/0.55 ( ( ( insert_real @ A2 @ ( insert_real @ B3 @ bot_bot_set_real ) )
% 0.34/0.55 = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 0.34/0.55 = ( ( ( A2 = C )
% 0.34/0.55 & ( B3 = D ) )
% 0.34/0.55 | ( ( A2 = D )
% 0.34/0.55 & ( B3 = C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % doubleton_eq_iff
% 0.34/0.55 thf(fact_2344_doubleton__eq__iff,axiom,
% 0.34/0.55 ! [A2: $o,B3: $o,C: $o,D: $o] :
% 0.34/0.55 ( ( ( insert_o @ A2 @ ( insert_o @ B3 @ bot_bot_set_o ) )
% 0.34/0.55 = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
% 0.34/0.55 = ( ( ( A2 = C )
% 0.34/0.55 & ( B3 = D ) )
% 0.34/0.55 | ( ( A2 = D )
% 0.34/0.55 & ( B3 = C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % doubleton_eq_iff
% 0.34/0.55 thf(fact_2345_doubleton__eq__iff,axiom,
% 0.34/0.55 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.34/0.55 ( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.34/0.55 = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.34/0.55 = ( ( ( A2 = C )
% 0.34/0.55 & ( B3 = D ) )
% 0.34/0.55 | ( ( A2 = D )
% 0.34/0.55 & ( B3 = C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % doubleton_eq_iff
% 0.34/0.55 thf(fact_2346_doubleton__eq__iff,axiom,
% 0.34/0.55 ! [A2: int,B3: int,C: int,D: int] :
% 0.34/0.55 ( ( ( insert_int @ A2 @ ( insert_int @ B3 @ bot_bot_set_int ) )
% 0.34/0.55 = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 0.34/0.55 = ( ( ( A2 = C )
% 0.34/0.55 & ( B3 = D ) )
% 0.34/0.55 | ( ( A2 = D )
% 0.34/0.55 & ( B3 = C ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % doubleton_eq_iff
% 0.34/0.55 thf(fact_2347_insert__not__empty,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 0.34/0.55 != bot_bo2099793752762293965at_nat ) ).
% 0.34/0.55
% 0.34/0.55 % insert_not_empty
% 0.34/0.55 thf(fact_2348_insert__not__empty,axiom,
% 0.34/0.55 ! [A2: real,A3: set_real] :
% 0.34/0.55 ( ( insert_real @ A2 @ A3 )
% 0.34/0.55 != bot_bot_set_real ) ).
% 0.34/0.55
% 0.34/0.55 % insert_not_empty
% 0.34/0.55 thf(fact_2349_insert__not__empty,axiom,
% 0.34/0.55 ! [A2: $o,A3: set_o] :
% 0.34/0.55 ( ( insert_o @ A2 @ A3 )
% 0.34/0.55 != bot_bot_set_o ) ).
% 0.34/0.55
% 0.34/0.55 % insert_not_empty
% 0.34/0.55 thf(fact_2350_insert__not__empty,axiom,
% 0.34/0.55 ! [A2: nat,A3: set_nat] :
% 0.34/0.55 ( ( insert_nat @ A2 @ A3 )
% 0.34/0.55 != bot_bot_set_nat ) ).
% 0.34/0.55
% 0.34/0.55 % insert_not_empty
% 0.34/0.55 thf(fact_2351_insert__not__empty,axiom,
% 0.34/0.55 ! [A2: int,A3: set_int] :
% 0.34/0.55 ( ( insert_int @ A2 @ A3 )
% 0.34/0.55 != bot_bot_set_int ) ).
% 0.34/0.55
% 0.34/0.55 % insert_not_empty
% 0.34/0.55 thf(fact_2352_singleton__inject,axiom,
% 0.34/0.55 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 => ( A2 = B3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_inject
% 0.34/0.55 thf(fact_2353_singleton__inject,axiom,
% 0.34/0.55 ! [A2: real,B3: real] :
% 0.34/0.55 ( ( ( insert_real @ A2 @ bot_bot_set_real )
% 0.34/0.55 = ( insert_real @ B3 @ bot_bot_set_real ) )
% 0.34/0.55 => ( A2 = B3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_inject
% 0.34/0.55 thf(fact_2354_singleton__inject,axiom,
% 0.34/0.55 ! [A2: $o,B3: $o] :
% 0.34/0.55 ( ( ( insert_o @ A2 @ bot_bot_set_o )
% 0.34/0.55 = ( insert_o @ B3 @ bot_bot_set_o ) )
% 0.34/0.55 => ( A2 = B3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_inject
% 0.34/0.55 thf(fact_2355_singleton__inject,axiom,
% 0.34/0.55 ! [A2: nat,B3: nat] :
% 0.34/0.55 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 0.34/0.55 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 0.34/0.55 => ( A2 = B3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_inject
% 0.34/0.55 thf(fact_2356_singleton__inject,axiom,
% 0.34/0.55 ! [A2: int,B3: int] :
% 0.34/0.55 ( ( ( insert_int @ A2 @ bot_bot_set_int )
% 0.34/0.55 = ( insert_int @ B3 @ bot_bot_set_int ) )
% 0.34/0.55 => ( A2 = B3 ) ) ).
% 0.34/0.55
% 0.34/0.55 % singleton_inject
% 0.34/0.55 thf(fact_2357_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_real,A2: real] :
% 0.34/0.55 ( ( finite_finite_real @ A3 )
% 0.34/0.55 => ( finite_finite_real @ ( insert_real @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2358_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_o,A2: $o] :
% 0.34/0.55 ( ( finite_finite_o @ A3 )
% 0.34/0.55 => ( finite_finite_o @ ( insert_o @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2359_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_nat,A2: nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A3 )
% 0.34/0.55 => ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2360_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_int,A2: int] :
% 0.34/0.55 ( ( finite_finite_int @ A3 )
% 0.34/0.55 => ( finite_finite_int @ ( insert_int @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2361_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_complex,A2: complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.55 => ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2362_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.55 => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2363_finite_OinsertI,axiom,
% 0.34/0.55 ! [A3: set_Extended_enat,A2: extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.55 => ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.insertI
% 0.34/0.55 thf(fact_2364_insert__mono,axiom,
% 0.34/0.55 ! [C2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ C2 @ D4 )
% 0.34/0.55 => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A2 @ C2 ) @ ( insert8211810215607154385at_nat @ A2 @ D4 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_mono
% 0.34/0.55 thf(fact_2365_insert__mono,axiom,
% 0.34/0.55 ! [C2: set_real,D4: set_real,A2: real] :
% 0.34/0.55 ( ( ord_less_eq_set_real @ C2 @ D4 )
% 0.34/0.55 => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C2 ) @ ( insert_real @ A2 @ D4 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_mono
% 0.34/0.55 thf(fact_2366_insert__mono,axiom,
% 0.34/0.55 ! [C2: set_o,D4: set_o,A2: $o] :
% 0.34/0.55 ( ( ord_less_eq_set_o @ C2 @ D4 )
% 0.34/0.55 => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D4 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_mono
% 0.34/0.55 thf(fact_2367_insert__mono,axiom,
% 0.34/0.55 ! [C2: set_nat,D4: set_nat,A2: nat] :
% 0.34/0.55 ( ( ord_less_eq_set_nat @ C2 @ D4 )
% 0.34/0.55 => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D4 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_mono
% 0.34/0.55 thf(fact_2368_insert__mono,axiom,
% 0.34/0.55 ! [C2: set_int,D4: set_int,A2: int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ C2 @ D4 )
% 0.34/0.55 => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C2 ) @ ( insert_int @ A2 @ D4 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_mono
% 0.34/0.55 thf(fact_2369_subset__insert,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2370_subset__insert,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real,B2: set_real] :
% 0.34/0.55 ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2371_subset__insert,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o,B2: set_o] :
% 0.34/0.55 ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2372_subset__insert,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.34/0.55 ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2373_subset__insert,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 0.34/0.55 ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2374_subset__insert,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int,B2: set_int] :
% 0.34/0.55 ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 0.34/0.55 = ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insert
% 0.34/0.55 thf(fact_2375_subset__insertI,axiom,
% 0.34/0.55 ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI
% 0.34/0.55 thf(fact_2376_subset__insertI,axiom,
% 0.34/0.55 ! [B2: set_real,A2: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI
% 0.34/0.55 thf(fact_2377_subset__insertI,axiom,
% 0.34/0.55 ! [B2: set_o,A2: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI
% 0.34/0.55 thf(fact_2378_subset__insertI,axiom,
% 0.34/0.55 ! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI
% 0.34/0.55 thf(fact_2379_subset__insertI,axiom,
% 0.34/0.55 ! [B2: set_int,A2: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A2 @ B2 ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI
% 0.34/0.55 thf(fact_2380_subset__insertI2,axiom,
% 0.34/0.55 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.34/0.55 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.34/0.55 => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI2
% 0.34/0.55 thf(fact_2381_subset__insertI2,axiom,
% 0.34/0.55 ! [A3: set_real,B2: set_real,B3: real] :
% 0.34/0.55 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.34/0.55 => ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI2
% 0.34/0.55 thf(fact_2382_subset__insertI2,axiom,
% 0.34/0.55 ! [A3: set_o,B2: set_o,B3: $o] :
% 0.34/0.55 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.34/0.55 => ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI2
% 0.34/0.55 thf(fact_2383_subset__insertI2,axiom,
% 0.34/0.55 ! [A3: set_nat,B2: set_nat,B3: nat] :
% 0.34/0.55 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.34/0.55 => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI2
% 0.34/0.55 thf(fact_2384_subset__insertI2,axiom,
% 0.34/0.55 ! [A3: set_int,B2: set_int,B3: int] :
% 0.34/0.55 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.34/0.55 => ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % subset_insertI2
% 0.34/0.55 thf(fact_2385_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_le3146513528884898305at_nat @ X6 @ A3 )
% 0.34/0.55 => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2386_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: real,A3: set_real,X6: set_real] :
% 0.34/0.55 ( ( member_real @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_real @ X6 @ A3 )
% 0.34/0.55 => ( ord_less_eq_set_real @ ( insert_real @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2387_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: $o,A3: set_o,X6: set_o] :
% 0.34/0.55 ( ( member_o @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_o @ X6 @ A3 )
% 0.34/0.55 => ( ord_less_eq_set_o @ ( insert_o @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2388_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: set_nat,A3: set_set_nat,X6: set_set_nat] :
% 0.34/0.55 ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_le6893508408891458716et_nat @ X6 @ A3 )
% 0.34/0.55 => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2389_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: nat,A3: set_nat,X6: set_nat] :
% 0.34/0.55 ( ( member_nat @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_nat @ X6 @ A3 )
% 0.34/0.55 => ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2390_insert__subsetI,axiom,
% 0.34/0.55 ! [X2: int,A3: set_int,X6: set_int] :
% 0.34/0.55 ( ( member_int @ X2 @ A3 )
% 0.34/0.55 => ( ( ord_less_eq_set_int @ X6 @ A3 )
% 0.34/0.55 => ( ord_less_eq_set_int @ ( insert_int @ X2 @ X6 ) @ A3 ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_subsetI
% 0.34/0.55 thf(fact_2391_fold__atLeastAtMost__nat_Ocases,axiom,
% 0.34/0.55 ! [X2: produc4471711990508489141at_nat] :
% 0.34/0.55 ~ ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
% 0.34/0.55 ( X2
% 0.34/0.55 != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % fold_atLeastAtMost_nat.cases
% 0.34/0.55 thf(fact_2392_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert8211810215607154385at_nat @ X2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2393_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: real,B2: set_real,A3: set_real] :
% 0.34/0.55 ( ( ( member_real @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_real @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member_real @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert_real @ X2 @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2394_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: $o,B2: set_o,A3: set_o] :
% 0.34/0.55 ( ( ( member_o @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_o @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member_o @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert_o @ X2 @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2395_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 0.34/0.55 ( ( ( member_set_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member_set_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2396_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: int,B2: set_int,A3: set_int] :
% 0.34/0.55 ( ( ( member_int @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_int @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member_int @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert_int @ X2 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2397_insert__Diff__if,axiom,
% 0.34/0.55 ! [X2: nat,B2: set_nat,A3: set_nat] :
% 0.34/0.55 ( ( ( member_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( minus_minus_set_nat @ A3 @ B2 ) ) )
% 0.34/0.55 & ( ~ ( member_nat @ X2 @ B2 )
% 0.34/0.55 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 0.34/0.55 = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % insert_Diff_if
% 0.34/0.55 thf(fact_2398_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_complex )
% 0.34/0.55 => ~ ! [A5: set_complex] :
% 0.34/0.55 ( ? [A: complex] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_complex @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2399_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bo2099793752762293965at_nat )
% 0.34/0.55 => ~ ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ? [A: product_prod_nat_nat] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert8211810215607154385at_nat @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite6177210948735845034at_nat @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2400_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bo7653980558646680370d_enat )
% 0.34/0.55 => ~ ! [A5: set_Extended_enat] :
% 0.34/0.55 ( ? [A: extended_enat] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_Extended_enat @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite4001608067531595151d_enat @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2401_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_real )
% 0.34/0.55 => ~ ! [A5: set_real] :
% 0.34/0.55 ( ? [A: real] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_real @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite_finite_real @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2402_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_o )
% 0.34/0.55 => ~ ! [A5: set_o] :
% 0.34/0.55 ( ? [A: $o] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_o @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite_finite_o @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2403_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_nat )
% 0.34/0.55 => ~ ! [A5: set_nat] :
% 0.34/0.55 ( ? [A: nat] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_nat @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2404_finite_Ocases,axiom,
% 0.34/0.55 ! [A2: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ A2 )
% 0.34/0.55 => ( ( A2 != bot_bot_set_int )
% 0.34/0.55 => ~ ! [A5: set_int] :
% 0.34/0.55 ( ? [A: int] :
% 0.34/0.55 ( A2
% 0.34/0.55 = ( insert_int @ A @ A5 ) )
% 0.34/0.55 => ~ ( finite_finite_int @ A5 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.cases
% 0.34/0.55 thf(fact_2405_finite_Osimps,axiom,
% 0.34/0.55 ( finite3207457112153483333omplex
% 0.34/0.55 = ( ^ [A4: set_complex] :
% 0.34/0.55 ( ( A4 = bot_bot_set_complex )
% 0.34/0.55 | ? [A6: set_complex,B4: complex] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_complex @ B4 @ A6 ) )
% 0.34/0.55 & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2406_finite_Osimps,axiom,
% 0.34/0.55 ( finite6177210948735845034at_nat
% 0.34/0.55 = ( ^ [A4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( A4 = bot_bo2099793752762293965at_nat )
% 0.34/0.55 | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
% 0.34/0.55 & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2407_finite_Osimps,axiom,
% 0.34/0.55 ( finite4001608067531595151d_enat
% 0.34/0.55 = ( ^ [A4: set_Extended_enat] :
% 0.34/0.55 ( ( A4 = bot_bo7653980558646680370d_enat )
% 0.34/0.55 | ? [A6: set_Extended_enat,B4: extended_enat] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_Extended_enat @ B4 @ A6 ) )
% 0.34/0.55 & ( finite4001608067531595151d_enat @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2408_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_real
% 0.34/0.55 = ( ^ [A4: set_real] :
% 0.34/0.55 ( ( A4 = bot_bot_set_real )
% 0.34/0.55 | ? [A6: set_real,B4: real] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_real @ B4 @ A6 ) )
% 0.34/0.55 & ( finite_finite_real @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2409_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_o
% 0.34/0.55 = ( ^ [A4: set_o] :
% 0.34/0.55 ( ( A4 = bot_bot_set_o )
% 0.34/0.55 | ? [A6: set_o,B4: $o] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_o @ B4 @ A6 ) )
% 0.34/0.55 & ( finite_finite_o @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2410_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_nat
% 0.34/0.55 = ( ^ [A4: set_nat] :
% 0.34/0.55 ( ( A4 = bot_bot_set_nat )
% 0.34/0.55 | ? [A6: set_nat,B4: nat] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_nat @ B4 @ A6 ) )
% 0.34/0.55 & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2411_finite_Osimps,axiom,
% 0.34/0.55 ( finite_finite_int
% 0.34/0.55 = ( ^ [A4: set_int] :
% 0.34/0.55 ( ( A4 = bot_bot_set_int )
% 0.34/0.55 | ? [A6: set_int,B4: int] :
% 0.34/0.55 ( ( A4
% 0.34/0.55 = ( insert_int @ B4 @ A6 ) )
% 0.34/0.55 & ( finite_finite_int @ A6 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite.simps
% 0.34/0.55 thf(fact_2412_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_set_nat )
% 0.34/0.55 => ( ! [X5: set_nat,F4: set_set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ F4 )
% 0.34/0.55 => ( ~ ( member_set_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2413_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_complex,P: set_complex > $o] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_complex )
% 0.34/0.55 => ( ! [X5: complex,F4: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ F4 )
% 0.34/0.55 => ( ~ ( member_complex @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2414_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ F2 )
% 0.34/0.55 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.55 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ F4 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2415_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ F2 )
% 0.34/0.55 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.55 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ F4 )
% 0.34/0.55 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2416_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_real,P: set_real > $o] :
% 0.34/0.55 ( ( finite_finite_real @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_real )
% 0.34/0.55 => ( ! [X5: real,F4: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ F4 )
% 0.34/0.55 => ( ~ ( member_real @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2417_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.55 => ( ! [X5: $o,F4: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F4 )
% 0.34/0.55 => ( ~ ( member_o @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2418_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_nat )
% 0.34/0.55 => ( ! [X5: nat,F4: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F4 )
% 0.34/0.55 => ( ~ ( member_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2419_finite__induct,axiom,
% 0.34/0.55 ! [F2: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( P @ bot_bot_set_int )
% 0.34/0.55 => ( ! [X5: int,F4: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F4 )
% 0.34/0.55 => ( ~ ( member_int @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_induct
% 0.34/0.55 thf(fact_2420_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_set_nat )
% 0.34/0.55 => ( ! [X5: set_nat] : ( P @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
% 0.34/0.55 => ( ! [X5: set_nat,F4: set_set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_set_nat )
% 0.34/0.55 => ( ~ ( member_set_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2421_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_complex,P: set_complex > $o] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_complex )
% 0.34/0.55 => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
% 0.34/0.55 => ( ! [X5: complex,F4: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_complex )
% 0.34/0.55 => ( ~ ( member_complex @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2422_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bo2099793752762293965at_nat )
% 0.34/0.55 => ( ! [X5: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.55 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2423_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bo7653980558646680370d_enat )
% 0.34/0.55 => ( ! [X5: extended_enat] : ( P @ ( insert_Extended_enat @ X5 @ bot_bo7653980558646680370d_enat ) )
% 0.34/0.55 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bo7653980558646680370d_enat )
% 0.34/0.55 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2424_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_real,P: set_real > $o] :
% 0.34/0.55 ( ( finite_finite_real @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_real )
% 0.34/0.55 => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
% 0.34/0.55 => ( ! [X5: real,F4: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_real )
% 0.34/0.55 => ( ~ ( member_real @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2425_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_o,P: set_o > $o] :
% 0.34/0.55 ( ( finite_finite_o @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_o )
% 0.34/0.55 => ( ! [X5: $o] : ( P @ ( insert_o @ X5 @ bot_bot_set_o ) )
% 0.34/0.55 => ( ! [X5: $o,F4: set_o] :
% 0.34/0.55 ( ( finite_finite_o @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_o )
% 0.34/0.55 => ( ~ ( member_o @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2426_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_nat,P: set_nat > $o] :
% 0.34/0.55 ( ( finite_finite_nat @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_nat )
% 0.34/0.55 => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
% 0.34/0.55 => ( ! [X5: nat,F4: set_nat] :
% 0.34/0.55 ( ( finite_finite_nat @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_nat )
% 0.34/0.55 => ( ~ ( member_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2427_finite__ne__induct,axiom,
% 0.34/0.55 ! [F2: set_int,P: set_int > $o] :
% 0.34/0.55 ( ( finite_finite_int @ F2 )
% 0.34/0.55 => ( ( F2 != bot_bot_set_int )
% 0.34/0.55 => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
% 0.34/0.55 => ( ! [X5: int,F4: set_int] :
% 0.34/0.55 ( ( finite_finite_int @ F4 )
% 0.34/0.55 => ( ( F4 != bot_bot_set_int )
% 0.34/0.55 => ( ~ ( member_int @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) ) )
% 0.34/0.55 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % finite_ne_induct
% 0.34/0.55 thf(fact_2428_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_set_nat > $o,A3: set_set_nat] :
% 0.34/0.55 ( ! [A5: set_set_nat] :
% 0.34/0.55 ( ~ ( finite1152437895449049373et_nat @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_set_nat )
% 0.34/0.55 => ( ! [X5: set_nat,F4: set_set_nat] :
% 0.34/0.55 ( ( finite1152437895449049373et_nat @ F4 )
% 0.34/0.55 => ( ~ ( member_set_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ A3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_2429_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_complex > $o,A3: set_complex] :
% 0.34/0.55 ( ! [A5: set_complex] :
% 0.34/0.55 ( ~ ( finite3207457112153483333omplex @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_complex )
% 0.34/0.55 => ( ! [X5: complex,F4: set_complex] :
% 0.34/0.55 ( ( finite3207457112153483333omplex @ F4 )
% 0.34/0.55 => ( ~ ( member_complex @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ A3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_2430_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ~ ( finite6177210948735845034at_nat @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.55 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.34/0.55 ( ( finite6177210948735845034at_nat @ F4 )
% 0.34/0.55 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ A3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_2431_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_Extended_enat > $o,A3: set_Extended_enat] :
% 0.34/0.55 ( ! [A5: set_Extended_enat] :
% 0.34/0.55 ( ~ ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.55 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 0.34/0.55 ( ( finite4001608067531595151d_enat @ F4 )
% 0.34/0.55 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ A3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_2432_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_real > $o,A3: set_real] :
% 0.34/0.55 ( ! [A5: set_real] :
% 0.34/0.55 ( ~ ( finite_finite_real @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_real )
% 0.34/0.55 => ( ! [X5: real,F4: set_real] :
% 0.34/0.55 ( ( finite_finite_real @ F4 )
% 0.34/0.55 => ( ~ ( member_real @ X5 @ F4 )
% 0.34/0.55 => ( ( P @ F4 )
% 0.34/0.55 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
% 0.34/0.55 => ( P @ A3 ) ) ) ) ).
% 0.34/0.55
% 0.34/0.55 % infinite_finite_induct
% 0.34/0.55 thf(fact_2433_infinite__finite__induct,axiom,
% 0.34/0.55 ! [P: set_o > $o,A3: set_o] :
% 0.34/0.55 ( ! [A5: set_o] :
% 0.34/0.55 ( ~ ( finite_finite_o @ A5 )
% 0.34/0.55 => ( P @ A5 ) )
% 0.34/0.55 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [X5: $o,F4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ F4 )
% 0.34/0.56 => ( ~ ( member_o @ X5 @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_finite_induct
% 0.34/0.56 thf(fact_2434_infinite__finite__induct,axiom,
% 0.34/0.56 ! [P: set_nat > $o,A3: set_nat] :
% 0.34/0.56 ( ! [A5: set_nat] :
% 0.34/0.56 ( ~ ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( P @ A5 ) )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [X5: nat,F4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ F4 )
% 0.34/0.56 => ( ~ ( member_nat @ X5 @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_finite_induct
% 0.34/0.56 thf(fact_2435_infinite__finite__induct,axiom,
% 0.34/0.56 ! [P: set_int > $o,A3: set_int] :
% 0.34/0.56 ( ! [A5: set_int] :
% 0.34/0.56 ( ~ ( finite_finite_int @ A5 )
% 0.34/0.56 => ( P @ A5 ) )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [X5: int,F4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ F4 )
% 0.34/0.56 => ( ~ ( member_int @ X5 @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_finite_induct
% 0.34/0.56 thf(fact_2436_subset__singletonD,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.56 => ( ( A3 = bot_bo2099793752762293965at_nat )
% 0.34/0.56 | ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singletonD
% 0.34/0.56 thf(fact_2437_subset__singletonD,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] :
% 0.34/0.56 ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.34/0.56 => ( ( A3 = bot_bot_set_real )
% 0.34/0.56 | ( A3
% 0.34/0.56 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singletonD
% 0.34/0.56 thf(fact_2438_subset__singletonD,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.34/0.56 => ( ( A3 = bot_bot_set_o )
% 0.34/0.56 | ( A3
% 0.34/0.56 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singletonD
% 0.34/0.56 thf(fact_2439_subset__singletonD,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.34/0.56 => ( ( A3 = bot_bot_set_nat )
% 0.34/0.56 | ( A3
% 0.34/0.56 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singletonD
% 0.34/0.56 thf(fact_2440_subset__singletonD,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.34/0.56 => ( ( A3 = bot_bot_set_int )
% 0.34/0.56 | ( A3
% 0.34/0.56 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singletonD
% 0.34/0.56 thf(fact_2441_subset__singleton__iff,axiom,
% 0.34/0.56 ! [X6: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ X6 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.56 = ( ( X6 = bot_bo2099793752762293965at_nat )
% 0.34/0.56 | ( X6
% 0.34/0.56 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singleton_iff
% 0.34/0.56 thf(fact_2442_subset__singleton__iff,axiom,
% 0.34/0.56 ! [X6: set_real,A2: real] :
% 0.34/0.56 ( ( ord_less_eq_set_real @ X6 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 0.34/0.56 = ( ( X6 = bot_bot_set_real )
% 0.34/0.56 | ( X6
% 0.34/0.56 = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singleton_iff
% 0.34/0.56 thf(fact_2443_subset__singleton__iff,axiom,
% 0.34/0.56 ! [X6: set_o,A2: $o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 0.34/0.56 = ( ( X6 = bot_bot_set_o )
% 0.34/0.56 | ( X6
% 0.34/0.56 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singleton_iff
% 0.34/0.56 thf(fact_2444_subset__singleton__iff,axiom,
% 0.34/0.56 ! [X6: set_nat,A2: nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 0.34/0.56 = ( ( X6 = bot_bot_set_nat )
% 0.34/0.56 | ( X6
% 0.34/0.56 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singleton_iff
% 0.34/0.56 thf(fact_2445_subset__singleton__iff,axiom,
% 0.34/0.56 ! [X6: set_int,A2: int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 0.34/0.56 = ( ( X6 = bot_bot_set_int )
% 0.34/0.56 | ( X6
% 0.34/0.56 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_singleton_iff
% 0.34/0.56 thf(fact_2446_Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert
% 0.34/0.56 thf(fact_2447_Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_real,A2: real,B2: set_real] :
% 0.34/0.56 ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ B2 ) @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert
% 0.34/0.56 thf(fact_2448_Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_o,A2: $o,B2: set_o] :
% 0.34/0.56 ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert
% 0.34/0.56 thf(fact_2449_Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_int,A2: int,B2: set_int] :
% 0.34/0.56 ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert
% 0.34/0.56 thf(fact_2450_Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 0.34/0.56 ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert
% 0.34/0.56 thf(fact_2451_insert__Diff,axiom,
% 0.34/0.56 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.34/0.56 => ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2452_insert__Diff,axiom,
% 0.34/0.56 ! [A2: set_nat,A3: set_set_nat] :
% 0.34/0.56 ( ( member_set_nat @ A2 @ A3 )
% 0.34/0.56 => ( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2453_insert__Diff,axiom,
% 0.34/0.56 ! [A2: real,A3: set_real] :
% 0.34/0.56 ( ( member_real @ A2 @ A3 )
% 0.34/0.56 => ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2454_insert__Diff,axiom,
% 0.34/0.56 ! [A2: $o,A3: set_o] :
% 0.34/0.56 ( ( member_o @ A2 @ A3 )
% 0.34/0.56 => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2455_insert__Diff,axiom,
% 0.34/0.56 ! [A2: int,A3: set_int] :
% 0.34/0.56 ( ( member_int @ A2 @ A3 )
% 0.34/0.56 => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2456_insert__Diff,axiom,
% 0.34/0.56 ! [A2: nat,A3: set_nat] :
% 0.34/0.56 ( ( member_nat @ A2 @ A3 )
% 0.34/0.56 => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % insert_Diff
% 0.34/0.56 thf(fact_2457_Diff__insert2,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert2
% 0.34/0.56 thf(fact_2458_Diff__insert2,axiom,
% 0.34/0.56 ! [A3: set_real,A2: real,B2: set_real] :
% 0.34/0.56 ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) @ B2 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert2
% 0.34/0.56 thf(fact_2459_Diff__insert2,axiom,
% 0.34/0.56 ! [A3: set_o,A2: $o,B2: set_o] :
% 0.34/0.56 ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B2 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert2
% 0.34/0.56 thf(fact_2460_Diff__insert2,axiom,
% 0.34/0.56 ! [A3: set_int,A2: int,B2: set_int] :
% 0.34/0.56 ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B2 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert2
% 0.34/0.56 thf(fact_2461_Diff__insert2,axiom,
% 0.34/0.56 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 0.34/0.56 ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.34/0.56 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert2
% 0.34/0.56 thf(fact_2462_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2463_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: set_nat,A3: set_set_nat] :
% 0.34/0.56 ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2464_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: real,A3: set_real] :
% 0.34/0.56 ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2465_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: $o,A3: set_o] :
% 0.34/0.56 ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2466_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: int,A3: set_int] :
% 0.34/0.56 ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2467_Diff__insert__absorb,axiom,
% 0.34/0.56 ! [X2: nat,A3: set_nat] :
% 0.34/0.56 ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.34/0.56 = A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_insert_absorb
% 0.34/0.56 thf(fact_2468_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2469_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_real,B2: set_real,X2: real,C2: set_real] :
% 0.34/0.56 ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member_real @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2470_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_o,B2: set_o,X2: $o,C2: set_o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member_o @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2471_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat,C2: set_set_nat] :
% 0.34/0.56 ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member_set_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2472_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_nat,B2: set_nat,X2: nat,C2: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2473_subset__Diff__insert,axiom,
% 0.34/0.56 ! [A3: set_int,B2: set_int,X2: int,C2: set_int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X2 @ C2 ) ) )
% 0.34/0.56 = ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ C2 ) )
% 0.34/0.56 & ~ ( member_int @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_Diff_insert
% 0.34/0.56 thf(fact_2474_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2475_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ ( insert_real @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2476_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ ( insert_o @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2477_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2478_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2479_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2480_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2481_card__insert__le,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( insert_int @ X2 @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_le
% 0.34/0.56 thf(fact_2482_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_complex,P: set_complex > $o,F: complex > rat] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [X5: complex,S4: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ S4 )
% 0.34/0.56 => ( ! [Y5: complex] :
% 0.34/0.56 ( ( member_complex @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2483_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > rat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [X5: extended_enat,S4: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ S4 )
% 0.34/0.56 => ( ! [Y5: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2484_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_real,P: set_real > $o,F: real > rat] :
% 0.34/0.56 ( ( finite_finite_real @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [X5: real,S4: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ S4 )
% 0.34/0.56 => ( ! [Y5: real] :
% 0.34/0.56 ( ( member_real @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2485_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_o,P: set_o > $o,F: $o > rat] :
% 0.34/0.56 ( ( finite_finite_o @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [X5: $o,S4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ S4 )
% 0.34/0.56 => ( ! [Y5: $o] :
% 0.34/0.56 ( ( member_o @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2486_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_nat,P: set_nat > $o,F: nat > rat] :
% 0.34/0.56 ( ( finite_finite_nat @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [X5: nat,S4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ S4 )
% 0.34/0.56 => ( ! [Y5: nat] :
% 0.34/0.56 ( ( member_nat @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2487_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_int,P: set_int > $o,F: int > rat] :
% 0.34/0.56 ( ( finite_finite_int @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [X5: int,S4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ S4 )
% 0.34/0.56 => ( ! [Y5: int] :
% 0.34/0.56 ( ( member_int @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2488_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_complex,P: set_complex > $o,F: complex > num] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [X5: complex,S4: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ S4 )
% 0.34/0.56 => ( ! [Y5: complex] :
% 0.34/0.56 ( ( member_complex @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2489_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > num] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [X5: extended_enat,S4: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ S4 )
% 0.34/0.56 => ( ! [Y5: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2490_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_real,P: set_real > $o,F: real > num] :
% 0.34/0.56 ( ( finite_finite_real @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [X5: real,S4: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ S4 )
% 0.34/0.56 => ( ! [Y5: real] :
% 0.34/0.56 ( ( member_real @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2491_finite__ranking__induct,axiom,
% 0.34/0.56 ! [S: set_o,P: set_o > $o,F: $o > num] :
% 0.34/0.56 ( ( finite_finite_o @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [X5: $o,S4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ S4 )
% 0.34/0.56 => ( ! [Y5: $o] :
% 0.34/0.56 ( ( member_o @ Y5 @ S4 )
% 0.34/0.56 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 0.34/0.56 => ( ( P @ S4 )
% 0.34/0.56 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_ranking_induct
% 0.34/0.56 thf(fact_2492_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [B: extended_enat,A5: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.56 => ( ! [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ A5 )
% 0.34/0.56 => ( ord_le72135733267957522d_enat @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2493_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [B: $o,A5: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A5 )
% 0.34/0.56 => ( ! [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_o @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2494_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [B: real,A5: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ A5 )
% 0.34/0.56 => ( ! [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_real @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2495_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_rat,P: set_rat > $o] :
% 0.34/0.56 ( ( finite_finite_rat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_rat )
% 0.34/0.56 => ( ! [B: rat,A5: set_rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A5 )
% 0.34/0.56 => ( ! [X4: rat] :
% 0.34/0.56 ( ( member_rat @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_rat @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2496_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_num,P: set_num > $o] :
% 0.34/0.56 ( ( finite_finite_num @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_num )
% 0.34/0.56 => ( ! [B: num,A5: set_num] :
% 0.34/0.56 ( ( finite_finite_num @ A5 )
% 0.34/0.56 => ( ! [X4: num] :
% 0.34/0.56 ( ( member_num @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_num @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2497_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [B: nat,A5: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( ! [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_nat @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2498_finite__linorder__max__induct,axiom,
% 0.34/0.56 ! [A3: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [B: int,A5: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A5 )
% 0.34/0.56 => ( ! [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_int @ X4 @ B ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_max_induct
% 0.34/0.56 thf(fact_2499_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [B: extended_enat,A5: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.56 => ( ! [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ A5 )
% 0.34/0.56 => ( ord_le72135733267957522d_enat @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2500_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [B: $o,A5: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A5 )
% 0.34/0.56 => ( ! [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_o @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2501_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [B: real,A5: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ A5 )
% 0.34/0.56 => ( ! [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_real @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2502_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_rat,P: set_rat > $o] :
% 0.34/0.56 ( ( finite_finite_rat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_rat )
% 0.34/0.56 => ( ! [B: rat,A5: set_rat] :
% 0.34/0.56 ( ( finite_finite_rat @ A5 )
% 0.34/0.56 => ( ! [X4: rat] :
% 0.34/0.56 ( ( member_rat @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_rat @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2503_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_num,P: set_num > $o] :
% 0.34/0.56 ( ( finite_finite_num @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_num )
% 0.34/0.56 => ( ! [B: num,A5: set_num] :
% 0.34/0.56 ( ( finite_finite_num @ A5 )
% 0.34/0.56 => ( ! [X4: num] :
% 0.34/0.56 ( ( member_num @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_num @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2504_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [B: nat,A5: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( ! [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_nat @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2505_finite__linorder__min__induct,axiom,
% 0.34/0.56 ! [A3: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [B: int,A5: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A5 )
% 0.34/0.56 => ( ! [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ A5 )
% 0.34/0.56 => ( ord_less_int @ B @ X4 ) )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_linorder_min_induct
% 0.34/0.56 thf(fact_2506_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ F2 )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [A: set_nat,F4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ F4 )
% 0.34/0.56 => ( ( member_set_nat @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_set_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2507_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ F2 )
% 0.34/0.56 => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [A: complex,F4: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ F4 )
% 0.34/0.56 => ( ( member_complex @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_complex @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2508_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ F2 )
% 0.34/0.56 => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ F4 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ A @ A3 )
% 0.34/0.56 => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2509_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ F2 )
% 0.34/0.56 => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [A: extended_enat,F4: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ F4 )
% 0.34/0.56 => ( ( member_Extended_enat @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_Extended_enat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2510_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_real,A3: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_real @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [A: real,F4: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ F4 )
% 0.34/0.56 => ( ( member_real @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_real @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2511_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_o,A3: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [A: $o,F4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ F4 )
% 0.34/0.56 => ( ( member_o @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_o @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2512_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [A: nat,F4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ F4 )
% 0.34/0.56 => ( ( member_nat @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2513_finite__subset__induct,axiom,
% 0.34/0.56 ! [F2: set_int,A3: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [A: int,F4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ F4 )
% 0.34/0.56 => ( ( member_int @ A @ A3 )
% 0.34/0.56 => ( ~ ( member_int @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct
% 0.34/0.56 thf(fact_2514_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ F2 )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [A: set_nat,F4: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ F4 )
% 0.34/0.56 => ( ( member_set_nat @ A @ A3 )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_set_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2515_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ F2 )
% 0.34/0.56 => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [A: complex,F4: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ F4 )
% 0.34/0.56 => ( ( member_complex @ A @ A3 )
% 0.34/0.56 => ( ( ord_le211207098394363844omplex @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_complex @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2516_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ F2 )
% 0.34/0.56 => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ F4 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ A @ A3 )
% 0.34/0.56 => ( ( ord_le3146513528884898305at_nat @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2517_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ F2 )
% 0.34/0.56 => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [A: extended_enat,F4: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ F4 )
% 0.34/0.56 => ( ( member_Extended_enat @ A @ A3 )
% 0.34/0.56 => ( ( ord_le7203529160286727270d_enat @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_Extended_enat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2518_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_real,A3: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_real @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [A: real,F4: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ F4 )
% 0.34/0.56 => ( ( member_real @ A @ A3 )
% 0.34/0.56 => ( ( ord_less_eq_set_real @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_real @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2519_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_o,A3: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [A: $o,F4: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ F4 )
% 0.34/0.56 => ( ( member_o @ A @ A3 )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_o @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2520_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [A: nat,F4: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ F4 )
% 0.34/0.56 => ( ( member_nat @ A @ A3 )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_nat @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2521_finite__subset__induct_H,axiom,
% 0.34/0.56 ! [F2: set_int,A3: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ F2 )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ F2 @ A3 )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [A: int,F4: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ F4 )
% 0.34/0.56 => ( ( member_int @ A @ A3 )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ F4 @ A3 )
% 0.34/0.56 => ( ~ ( member_int @ A @ F4 )
% 0.34/0.56 => ( ( P @ F4 )
% 0.34/0.56 => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) ) )
% 0.34/0.56 => ( P @ F2 ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_subset_induct'
% 0.34/0.56 thf(fact_2522_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_real,K: nat] :
% 0.34/0.56 ( ( ( finite_card_real @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: real,B6: set_real] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_real @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_real @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_real @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite_finite_real @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2523_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_o,K: nat] :
% 0.34/0.56 ( ( ( finite_card_o @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: $o,B6: set_o] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_o @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_o @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_o @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite_finite_o @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2524_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_list_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: list_nat,B6: set_list_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_list_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_list_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_list_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2525_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: set_nat,B6: set_set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_set_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_set_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_set_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2526_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: nat,B6: set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite_finite_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2527_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_int,K: nat] :
% 0.34/0.56 ( ( ( finite_card_int @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: int,B6: set_int] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_int @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_int @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_int @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite_finite_int @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2528_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_complex,K: nat] :
% 0.34/0.56 ( ( ( finite_card_complex @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: complex,B6: set_complex] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_complex @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_complex @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_complex @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2529_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 0.34/0.56 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite711546835091564841at_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2530_card__Suc__eq__finite,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,K: nat] :
% 0.34/0.56 ( ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: extended_enat,B6: set_Extended_enat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_Extended_enat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_Extended_enat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite121521170596916366d_enat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq_finite
% 0.34/0.56 thf(fact_2531_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_real @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2532_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_o @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2533_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.56 => ( ( ( member_list_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_list_nat @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_list_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2534_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_set_nat @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2535_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_nat @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2536_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_int @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2537_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( ( member_complex @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 0.34/0.56 = ( finite_card_complex @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_complex @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2538_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.56 => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( finite711546835091564841at_nat @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2539_card__insert__if,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( ( member_Extended_enat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.34/0.56 = ( finite121521170596916366d_enat @ A3 ) ) )
% 0.34/0.56 & ( ~ ( member_Extended_enat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_insert_if
% 0.34/0.56 thf(fact_2540_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_complex,A2: complex] :
% 0.34/0.56 ( ~ ( finite3207457112153483333omplex @ S )
% 0.34/0.56 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2541_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 0.34/0.56 ( ~ ( finite6177210948735845034at_nat @ S )
% 0.34/0.56 => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2542_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_Extended_enat,A2: extended_enat] :
% 0.34/0.56 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.34/0.56 => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2543_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_real,A2: real] :
% 0.34/0.56 ( ~ ( finite_finite_real @ S )
% 0.34/0.56 => ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2544_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_o,A2: $o] :
% 0.34/0.56 ( ~ ( finite_finite_o @ S )
% 0.34/0.56 => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2545_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_int,A2: int] :
% 0.34/0.56 ( ~ ( finite_finite_int @ S )
% 0.34/0.56 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2546_infinite__remove,axiom,
% 0.34/0.56 ! [S: set_nat,A2: nat] :
% 0.34/0.56 ( ~ ( finite_finite_nat @ S )
% 0.34/0.56 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_remove
% 0.34/0.56 thf(fact_2547_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_complex > $o,A3: set_complex] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_complex] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: complex] :
% 0.34/0.56 ( ( member_complex @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) )
% 0.34/0.56 | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2548_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: product_prod_nat_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) )
% 0.34/0.56 | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2549_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_Extended_enat > $o,A3: set_Extended_enat] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_Extended_enat] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) )
% 0.34/0.56 | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2550_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_real > $o,A3: set_real] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_real] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
% 0.34/0.56 | ~ ( finite_finite_real @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite_finite_real @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2551_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_o > $o,A3: set_o] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_o] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
% 0.34/0.56 | ~ ( finite_finite_o @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite_finite_o @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2552_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_int > $o,A3: set_int] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_int] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
% 0.34/0.56 | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite_finite_int @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2553_infinite__coinduct,axiom,
% 0.34/0.56 ! [X6: set_nat > $o,A3: set_nat] :
% 0.34/0.56 ( ( X6 @ A3 )
% 0.34/0.56 => ( ! [A5: set_nat] :
% 0.34/0.56 ( ( X6 @ A5 )
% 0.34/0.56 => ? [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ A5 )
% 0.34/0.56 & ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
% 0.34/0.56 | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
% 0.34/0.56 => ~ ( finite_finite_nat @ A3 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % infinite_coinduct
% 0.34/0.56 thf(fact_2554_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: set_nat,A5: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A5 )
% 0.34/0.56 => ( ( member_set_nat @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2555_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_complex,P: set_complex > $o] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: complex,A5: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A5 )
% 0.34/0.56 => ( ( member_complex @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_complex ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2556_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A5 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2557_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: extended_enat,A5: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.56 => ( ( member_Extended_enat @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bo7653980558646680370d_enat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2558_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: real,A5: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ A5 )
% 0.34/0.56 => ( ( member_real @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2559_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: $o,A5: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A5 )
% 0.34/0.56 => ( ( member_o @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2560_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: int,A5: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A5 )
% 0.34/0.56 => ( ( member_int @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2561_finite__empty__induct,axiom,
% 0.34/0.56 ! [A3: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( P @ A3 )
% 0.34/0.56 => ( ! [A: nat,A5: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( ( member_nat @ A @ A5 )
% 0.34/0.56 => ( ( P @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
% 0.34/0.56 => ( P @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_empty_induct
% 0.34/0.56 thf(fact_2562_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2563_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
% 0.34/0.56 ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2564_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real,B2: set_real] :
% 0.34/0.56 ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2565_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o,B2: set_o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2566_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2567_subset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int,B2: set_int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % subset_insert_iff
% 0.34/0.56 thf(fact_2568_Diff__single__insert,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 0.34/0.56 => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_single_insert
% 0.34/0.56 thf(fact_2569_Diff__single__insert,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real,B2: set_real] :
% 0.34/0.56 ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 )
% 0.34/0.56 => ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_single_insert
% 0.34/0.56 thf(fact_2570_Diff__single__insert,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o,B2: set_o] :
% 0.34/0.56 ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 )
% 0.34/0.56 => ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_single_insert
% 0.34/0.56 thf(fact_2571_Diff__single__insert,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_single_insert
% 0.34/0.56 thf(fact_2572_Diff__single__insert,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int,B2: set_int] :
% 0.34/0.56 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 )
% 0.34/0.56 => ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Diff_single_insert
% 0.34/0.56 thf(fact_2573_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: product_prod_nat_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2574_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_complex] :
% 0.34/0.56 ( ( ( finite_card_complex @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: complex] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2575_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_list_nat] :
% 0.34/0.56 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: list_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_list_nat @ X5 @ bot_bot_set_list_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2576_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_set_nat] :
% 0.34/0.56 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: set_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2577_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_real] :
% 0.34/0.56 ( ( ( finite_card_real @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: real] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2578_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_o] :
% 0.34/0.56 ( ( ( finite_card_o @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: $o] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2579_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_nat] :
% 0.34/0.56 ( ( ( finite_card_nat @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2580_card__1__singletonE,axiom,
% 0.34/0.56 ! [A3: set_int] :
% 0.34/0.56 ( ( ( finite_card_int @ A3 )
% 0.34/0.56 = one_one_nat )
% 0.34/0.56 => ~ ! [X5: int] :
% 0.34/0.56 ( A3
% 0.34/0.56 != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singletonE
% 0.34/0.56 thf(fact_2581_Compl__insert,axiom,
% 0.34/0.56 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Compl_insert
% 0.34/0.56 thf(fact_2582_Compl__insert,axiom,
% 0.34/0.56 ! [X2: real,A3: set_real] :
% 0.34/0.56 ( ( uminus612125837232591019t_real @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.56 = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Compl_insert
% 0.34/0.56 thf(fact_2583_Compl__insert,axiom,
% 0.34/0.56 ! [X2: $o,A3: set_o] :
% 0.34/0.56 ( ( uminus_uminus_set_o @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.56 = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Compl_insert
% 0.34/0.56 thf(fact_2584_Compl__insert,axiom,
% 0.34/0.56 ! [X2: int,A3: set_int] :
% 0.34/0.56 ( ( uminus1532241313380277803et_int @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.56 = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Compl_insert
% 0.34/0.56 thf(fact_2585_Compl__insert,axiom,
% 0.34/0.56 ! [X2: nat,A3: set_nat] :
% 0.34/0.56 ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % Compl_insert
% 0.34/0.56 thf(fact_2586_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 0.34/0.56 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite711546835091564841at_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2587_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_complex,K: nat] :
% 0.34/0.56 ( ( ( finite_card_complex @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: complex,B6: set_complex] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_complex @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_complex @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_complex @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_complex ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2588_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_list_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: list_nat,B6: set_list_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_list_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_list_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_list_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_list_nat ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2589_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: set_nat,B6: set_set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_set_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_set_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_set_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_set_nat ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2590_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_real,K: nat] :
% 0.34/0.56 ( ( ( finite_card_real @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: real,B6: set_real] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_real @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_real @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_real @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_real ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2591_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_o,K: nat] :
% 0.34/0.56 ( ( ( finite_card_o @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: $o,B6: set_o] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_o @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_o @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_o @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_o ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2592_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: nat,B6: set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_nat @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_nat @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_nat @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2593_card__Suc__eq,axiom,
% 0.34/0.56 ! [A3: set_int,K: nat] :
% 0.34/0.56 ( ( ( finite_card_int @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 = ( ? [B4: int,B6: set_int] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_int @ B4 @ B6 ) )
% 0.34/0.56 & ~ ( member_int @ B4 @ B6 )
% 0.34/0.56 & ( ( finite_card_int @ B6 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B6 = bot_bot_set_int ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_eq
% 0.34/0.56 thf(fact_2594_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 0.34/0.56 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ B @ B8 ) )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ B @ B8 )
% 0.34/0.56 & ( ( finite711546835091564841at_nat @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2595_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_complex,K: nat] :
% 0.34/0.56 ( ( ( finite_card_complex @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: complex,B8: set_complex] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_complex @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_complex @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_complex @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_complex ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2596_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_list_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: list_nat,B8: set_list_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_list_nat @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_list_nat @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_list_nat @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_list_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2597_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: set_nat,B8: set_set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_set_nat @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_set_nat @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_set_nat @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2598_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_real,K: nat] :
% 0.34/0.56 ( ( ( finite_card_real @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: real,B8: set_real] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_real @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_real @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_real @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2599_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_o,K: nat] :
% 0.34/0.56 ( ( ( finite_card_o @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: $o,B8: set_o] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_o @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_o @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_o @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2600_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_nat,K: nat] :
% 0.34/0.56 ( ( ( finite_card_nat @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: nat,B8: set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_nat @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_nat @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_nat @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2601_card__eq__SucD,axiom,
% 0.34/0.56 ! [A3: set_int,K: nat] :
% 0.34/0.56 ( ( ( finite_card_int @ A3 )
% 0.34/0.56 = ( suc @ K ) )
% 0.34/0.56 => ? [B: int,B8: set_int] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_int @ B @ B8 ) )
% 0.34/0.56 & ~ ( member_int @ B @ B8 )
% 0.34/0.56 & ( ( finite_card_int @ B8 )
% 0.34/0.56 = K )
% 0.34/0.56 & ( ( K = zero_zero_nat )
% 0.34/0.56 => ( B8 = bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_eq_SucD
% 0.34/0.56 thf(fact_2602_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: product_prod_nat_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2603_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_complex] :
% 0.34/0.56 ( ( ( finite_card_complex @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: complex] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2604_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_list_nat] :
% 0.34/0.56 ( ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: list_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2605_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_set_nat] :
% 0.34/0.56 ( ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: set_nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2606_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_real] :
% 0.34/0.56 ( ( ( finite_card_real @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: real] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2607_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_o] :
% 0.34/0.56 ( ( ( finite_card_o @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: $o] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2608_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_nat] :
% 0.34/0.56 ( ( ( finite_card_nat @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: nat] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2609_card__1__singleton__iff,axiom,
% 0.34/0.56 ! [A3: set_int] :
% 0.34/0.56 ( ( ( finite_card_int @ A3 )
% 0.34/0.56 = ( suc @ zero_zero_nat ) )
% 0.34/0.56 = ( ? [X: int] :
% 0.34/0.56 ( A3
% 0.34/0.56 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_1_singleton_iff
% 0.34/0.56 thf(fact_2610_remove__induct,axiom,
% 0.34/0.56 ! [P: set_set_nat > $o,B2: set_set_nat] :
% 0.34/0.56 ( ( P @ bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2611_remove__induct,axiom,
% 0.34/0.56 ! [P: set_complex > $o,B2: set_complex] :
% 0.34/0.56 ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_complex )
% 0.34/0.56 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: complex] :
% 0.34/0.56 ( ( member_complex @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2612_remove__induct,axiom,
% 0.34/0.56 ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: product_prod_nat_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2613_remove__induct,axiom,
% 0.34/0.56 ! [P: set_Extended_enat > $o,B2: set_Extended_enat] :
% 0.34/0.56 ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ( ~ ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2614_remove__induct,axiom,
% 0.34/0.56 ! [P: set_real > $o,B2: set_real] :
% 0.34/0.56 ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ( ~ ( finite_finite_real @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_real )
% 0.34/0.56 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2615_remove__induct,axiom,
% 0.34/0.56 ! [P: set_o > $o,B2: set_o] :
% 0.34/0.56 ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ( ~ ( finite_finite_o @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2616_remove__induct,axiom,
% 0.34/0.56 ! [P: set_nat > $o,B2: set_nat] :
% 0.34/0.56 ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ( ~ ( finite_finite_nat @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2617_remove__induct,axiom,
% 0.34/0.56 ! [P: set_int > $o,B2: set_int] :
% 0.34/0.56 ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ( ~ ( finite_finite_int @ B2 )
% 0.34/0.56 => ( P @ B2 ) )
% 0.34/0.56 => ( ! [A5: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % remove_induct
% 0.34/0.56 thf(fact_2618_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_set_nat,P: set_set_nat > $o] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_set_nat )
% 0.34/0.56 => ( ! [A5: set_set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_set_nat )
% 0.34/0.56 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: set_nat] :
% 0.34/0.56 ( ( member_set_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2619_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_complex,P: set_complex > $o] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [A5: set_complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_complex )
% 0.34/0.56 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: complex] :
% 0.34/0.56 ( ( member_complex @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2620_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ B2 )
% 0.34/0.56 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: product_prod_nat_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2621_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ B2 )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [A5: set_Extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2622_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [A5: set_real] :
% 0.34/0.56 ( ( finite_finite_real @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_real )
% 0.34/0.56 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2623_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [A5: set_o] :
% 0.34/0.56 ( ( finite_finite_o @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_o )
% 0.34/0.56 => ( ( ord_less_eq_set_o @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2624_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [A5: set_nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_nat )
% 0.34/0.56 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2625_finite__remove__induct,axiom,
% 0.34/0.56 ! [B2: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ B2 )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [A5: set_int] :
% 0.34/0.56 ( ( finite_finite_int @ A5 )
% 0.34/0.56 => ( ( A5 != bot_bot_set_int )
% 0.34/0.56 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 0.34/0.56 => ( ! [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ A5 )
% 0.34/0.56 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
% 0.34/0.56 => ( P @ A5 ) ) ) ) )
% 0.34/0.56 => ( P @ B2 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_remove_induct
% 0.34/0.56 thf(fact_2626_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_real] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A3 ) )
% 0.34/0.56 = ( ? [A4: real,B6: set_real] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_real @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_real @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
% 0.34/0.56 & ( finite_finite_real @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2627_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_o] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_o @ A3 ) )
% 0.34/0.56 = ( ? [A4: $o,B6: set_o] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_o @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_o @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_o @ B6 ) )
% 0.34/0.56 & ( finite_finite_o @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2628_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_list_nat] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A3 ) )
% 0.34/0.56 = ( ? [A4: list_nat,B6: set_list_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_list_nat @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_list_nat @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B6 ) )
% 0.34/0.56 & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2629_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_set_nat] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A3 ) )
% 0.34/0.56 = ( ? [A4: set_nat,B6: set_set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_set_nat @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_set_nat @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B6 ) )
% 0.34/0.56 & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2630_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_nat] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A3 ) )
% 0.34/0.56 = ( ? [A4: nat,B6: set_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_nat @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_nat @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
% 0.34/0.56 & ( finite_finite_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2631_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_int] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A3 ) )
% 0.34/0.56 = ( ? [A4: int,B6: set_int] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_int @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_int @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
% 0.34/0.56 & ( finite_finite_int @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2632_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_complex] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A3 ) )
% 0.34/0.56 = ( ? [A4: complex,B6: set_complex] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_complex @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_complex @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B6 ) )
% 0.34/0.56 & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2633_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A3 ) )
% 0.34/0.56 = ( ? [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert8211810215607154385at_nat @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member8440522571783428010at_nat @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B6 ) )
% 0.34/0.56 & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2634_card__le__Suc__iff,axiom,
% 0.34/0.56 ! [N: nat,A3: set_Extended_enat] :
% 0.34/0.56 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ A3 ) )
% 0.34/0.56 = ( ? [A4: extended_enat,B6: set_Extended_enat] :
% 0.34/0.56 ( ( A3
% 0.34/0.56 = ( insert_Extended_enat @ A4 @ B6 ) )
% 0.34/0.56 & ~ ( member_Extended_enat @ A4 @ B6 )
% 0.34/0.56 & ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ B6 ) )
% 0.34/0.56 & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_le_Suc_iff
% 0.34/0.56 thf(fact_2635_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2636_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2637_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2638_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2639_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2640_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2641_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2642_card__Diff1__le,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_le
% 0.34/0.56 thf(fact_2643_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_complex,P: set_complex > $o] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_complex )
% 0.34/0.56 => ( ! [T4: set_complex] :
% 0.34/0.56 ( ( ord_less_set_complex @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: complex] :
% 0.34/0.56 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_complex @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2644_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ S )
% 0.34/0.56 => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.34/0.56 => ( ! [T4: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le7866589430770878221at_nat @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: product_prod_nat_nat] :
% 0.34/0.56 ( ( member8440522571783428010at_nat @ X4 @ ( minus_1356011639430497352at_nat @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert8211810215607154385at_nat @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2645_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_Extended_enat,P: set_Extended_enat > $o] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ S )
% 0.34/0.56 => ( ( P @ bot_bo7653980558646680370d_enat )
% 0.34/0.56 => ( ! [T4: set_Extended_enat] :
% 0.34/0.56 ( ( ord_le2529575680413868914d_enat @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: extended_enat] :
% 0.34/0.56 ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_Extended_enat @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2646_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_real,P: set_real > $o] :
% 0.34/0.56 ( ( finite_finite_real @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_real )
% 0.34/0.56 => ( ! [T4: set_real] :
% 0.34/0.56 ( ( ord_less_set_real @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: real] :
% 0.34/0.56 ( ( member_real @ X4 @ ( minus_minus_set_real @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_real @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2647_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_o,P: set_o > $o] :
% 0.34/0.56 ( ( finite_finite_o @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_o )
% 0.34/0.56 => ( ! [T4: set_o] :
% 0.34/0.56 ( ( ord_less_set_o @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: $o] :
% 0.34/0.56 ( ( member_o @ X4 @ ( minus_minus_set_o @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_o @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2648_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_int,P: set_int > $o] :
% 0.34/0.56 ( ( finite_finite_int @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_int )
% 0.34/0.56 => ( ! [T4: set_int] :
% 0.34/0.56 ( ( ord_less_set_int @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: int] :
% 0.34/0.56 ( ( member_int @ X4 @ ( minus_minus_set_int @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_int @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2649_finite__induct__select,axiom,
% 0.34/0.56 ! [S: set_nat,P: set_nat > $o] :
% 0.34/0.56 ( ( finite_finite_nat @ S )
% 0.34/0.56 => ( ( P @ bot_bot_set_nat )
% 0.34/0.56 => ( ! [T4: set_nat] :
% 0.34/0.56 ( ( ord_less_set_nat @ T4 @ S )
% 0.34/0.56 => ( ( P @ T4 )
% 0.34/0.56 => ? [X4: nat] :
% 0.34/0.56 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S @ T4 ) )
% 0.34/0.56 & ( P @ ( insert_nat @ X4 @ T4 ) ) ) ) )
% 0.34/0.56 => ( P @ S ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % finite_induct_select
% 0.34/0.56 thf(fact_2650_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.34/0.56 ( ( ord_le7866589430770878221at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.56 => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2651_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
% 0.34/0.56 ( ( ord_less_set_set_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_set_nat @ X2 @ B2 )
% 0.34/0.56 => ( ord_less_set_set_nat @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member_set_nat @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2652_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real,B2: set_real] :
% 0.34/0.56 ( ( ord_less_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_real @ X2 @ B2 )
% 0.34/0.56 => ( ord_less_set_real @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member_real @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2653_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o,B2: set_o] :
% 0.34/0.56 ( ( ord_less_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_o @ X2 @ B2 )
% 0.34/0.56 => ( ord_less_set_o @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member_o @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2654_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 0.34/0.56 ( ( ord_less_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_nat @ X2 @ B2 )
% 0.34/0.56 => ( ord_less_set_nat @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member_nat @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2655_psubset__insert__iff,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int,B2: set_int] :
% 0.34/0.56 ( ( ord_less_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 0.34/0.56 = ( ( ( member_int @ X2 @ B2 )
% 0.34/0.56 => ( ord_less_set_int @ A3 @ B2 ) )
% 0.34/0.56 & ( ~ ( member_int @ X2 @ B2 )
% 0.34/0.56 => ( ( ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
% 0.34/0.56 & ( ~ ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % psubset_insert_iff
% 0.34/0.56 thf(fact_2656_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.34/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.34/0.56 ( ! [A: $o,B: $o,X5: nat] :
% 0.34/0.56 ( X2
% 0.34/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 0.34/0.56 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.34/0.56 ( X2
% 0.34/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.34/0.56 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.34/0.56 ( X2
% 0.34/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ X5 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % VEBT_internal.naive_member.cases
% 0.34/0.56 thf(fact_2657_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.56 => ( ( member_list_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_list_nat @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2658_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_set_nat @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2659_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( member_complex @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_complex @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2660_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite711546835091564841at_nat @ A3 )
% 0.34/0.56 = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2661_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite121521170596916366d_enat @ A3 )
% 0.34/0.56 = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2662_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_real @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2663_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_o @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2664_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_int @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2665_card_Oremove,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( finite_card_nat @ A3 )
% 0.34/0.56 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.remove
% 0.34/0.56 thf(fact_2666_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.56 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2667_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2668_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2669_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.56 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2670_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2671_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2672_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2673_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2674_card_Oinsert__remove,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 0.34/0.56 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card.insert_remove
% 0.34/0.56 thf(fact_2675_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.56 => ( ( member_list_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) )
% 0.34/0.56 = ( finite_card_list_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2676_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 0.34/0.56 = ( finite_card_set_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2677_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( member_complex @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 0.34/0.56 = ( finite_card_complex @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2678_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.34/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.34/0.56 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.34/0.56 = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2679_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.34/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.34/0.56 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
% 0.34/0.56 = ( finite121521170596916366d_enat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2680_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_real,X2: real] :
% 0.34/0.56 ( ( finite_finite_real @ A3 )
% 0.34/0.56 => ( ( member_real @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 0.34/0.56 = ( finite_card_real @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2681_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_o,X2: $o] :
% 0.34/0.56 ( ( finite_finite_o @ A3 )
% 0.34/0.56 => ( ( member_o @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) )
% 0.34/0.56 = ( finite_card_o @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2682_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_int,X2: int] :
% 0.34/0.56 ( ( finite_finite_int @ A3 )
% 0.34/0.56 => ( ( member_int @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 0.34/0.56 = ( finite_card_int @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2683_card__Suc__Diff1,axiom,
% 0.34/0.56 ! [A3: set_nat,X2: nat] :
% 0.34/0.56 ( ( finite_finite_nat @ A3 )
% 0.34/0.56 => ( ( member_nat @ X2 @ A3 )
% 0.34/0.56 => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 0.34/0.56 = ( finite_card_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Suc_Diff1
% 0.34/0.56 thf(fact_2684_card__Diff1__less,axiom,
% 0.34/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.34/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.34/0.56 => ( ( member_list_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_less
% 0.34/0.56 thf(fact_2685_card__Diff1__less,axiom,
% 0.34/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.34/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.34/0.56 => ( ( member_set_nat @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_less
% 0.34/0.56 thf(fact_2686_card__Diff1__less,axiom,
% 0.34/0.56 ! [A3: set_complex,X2: complex] :
% 0.34/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.34/0.56 => ( ( member_complex @ X2 @ A3 )
% 0.34/0.56 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ).
% 0.34/0.56
% 0.34/0.56 % card_Diff1_less
% 0.34/0.56 thf(fact_2687_card__Diff1__less,axiom,
% 0.34/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.35/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.56 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2688_card__Diff1__less,axiom,
% 0.35/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.35/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.56 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2689_card__Diff1__less,axiom,
% 0.35/0.56 ! [A3: set_real,X2: real] :
% 0.35/0.56 ( ( finite_finite_real @ A3 )
% 0.35/0.56 => ( ( member_real @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2690_card__Diff1__less,axiom,
% 0.35/0.56 ! [A3: set_o,X2: $o] :
% 0.35/0.56 ( ( finite_finite_o @ A3 )
% 0.35/0.56 => ( ( member_o @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2691_card__Diff1__less,axiom,
% 0.35/0.56 ! [A3: set_int,X2: int] :
% 0.35/0.56 ( ( finite_finite_int @ A3 )
% 0.35/0.56 => ( ( member_int @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2692_card__Diff1__less,axiom,
% 0.35/0.56 ! [A3: set_nat,X2: nat] :
% 0.35/0.56 ( ( finite_finite_nat @ A3 )
% 0.35/0.56 => ( ( member_nat @ X2 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less
% 0.35/0.56 thf(fact_2693_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_list_nat,X2: list_nat,Y3: list_nat] :
% 0.35/0.56 ( ( finite8100373058378681591st_nat @ A3 )
% 0.35/0.56 => ( ( member_list_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( member_list_nat @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y3 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2694_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_set_nat,X2: set_nat,Y3: set_nat] :
% 0.35/0.56 ( ( finite1152437895449049373et_nat @ A3 )
% 0.35/0.56 => ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( member_set_nat @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2695_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_complex,X2: complex,Y3: complex] :
% 0.35/0.56 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.56 => ( ( member_complex @ X2 @ A3 )
% 0.35/0.56 => ( ( member_complex @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ ( insert_complex @ Y3 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2696_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.35/0.56 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.56 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2697_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_Extended_enat,X2: extended_enat,Y3: extended_enat] :
% 0.35/0.56 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.56 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.56 => ( ( member_Extended_enat @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) @ ( insert_Extended_enat @ Y3 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2698_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_real,X2: real,Y3: real] :
% 0.35/0.56 ( ( finite_finite_real @ A3 )
% 0.35/0.56 => ( ( member_real @ X2 @ A3 )
% 0.35/0.56 => ( ( member_real @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( insert_real @ Y3 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2699_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_o,X2: $o,Y3: $o] :
% 0.35/0.56 ( ( finite_finite_o @ A3 )
% 0.35/0.56 => ( ( member_o @ X2 @ A3 )
% 0.35/0.56 => ( ( member_o @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ ( insert_o @ Y3 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2700_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_int,X2: int,Y3: int] :
% 0.35/0.56 ( ( finite_finite_int @ A3 )
% 0.35/0.56 => ( ( member_int @ X2 @ A3 )
% 0.35/0.56 => ( ( member_int @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ ( insert_int @ Y3 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2701_card__Diff2__less,axiom,
% 0.35/0.56 ! [A3: set_nat,X2: nat,Y3: nat] :
% 0.35/0.56 ( ( finite_finite_nat @ A3 )
% 0.35/0.56 => ( ( member_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( member_nat @ Y3 @ A3 )
% 0.35/0.56 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff2_less
% 0.35/0.56 thf(fact_2702_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_list_nat,X2: list_nat] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) )
% 0.35/0.56 = ( ( finite8100373058378681591st_nat @ A3 )
% 0.35/0.56 & ( member_list_nat @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2703_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_set_nat,X2: set_nat] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) )
% 0.35/0.56 = ( ( finite1152437895449049373et_nat @ A3 )
% 0.35/0.56 & ( member_set_nat @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2704_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_complex,X2: complex] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) )
% 0.35/0.56 = ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.56 & ( member_complex @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2705_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) )
% 0.35/0.56 = ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.56 & ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2706_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_Extended_enat,X2: extended_enat] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) )
% 0.35/0.56 = ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.56 & ( member_Extended_enat @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2707_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_real,X2: real] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) )
% 0.35/0.56 = ( ( finite_finite_real @ A3 )
% 0.35/0.56 & ( member_real @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2708_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_o,X2: $o] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) )
% 0.35/0.56 = ( ( finite_finite_o @ A3 )
% 0.35/0.56 & ( member_o @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2709_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_int,X2: int] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) )
% 0.35/0.56 = ( ( finite_finite_int @ A3 )
% 0.35/0.56 & ( member_int @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2710_card__Diff1__less__iff,axiom,
% 0.35/0.56 ! [A3: set_nat,X2: nat] :
% 0.35/0.56 ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) )
% 0.35/0.56 = ( ( finite_finite_nat @ A3 )
% 0.35/0.56 & ( member_nat @ X2 @ A3 ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff1_less_iff
% 0.35/0.56 thf(fact_2711_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.56 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2712_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: complex,A3: set_complex] :
% 0.35/0.56 ( ( member_complex @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2713_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: list_nat,A3: set_list_nat] :
% 0.35/0.56 ( ( member_list_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2714_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: set_nat,A3: set_set_nat] :
% 0.35/0.56 ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2715_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: real,A3: set_real] :
% 0.35/0.56 ( ( member_real @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2716_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: $o,A3: set_o] :
% 0.35/0.56 ( ( member_o @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2717_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: int,A3: set_int] :
% 0.35/0.56 ( ( member_int @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2718_card__Diff__singleton,axiom,
% 0.35/0.56 ! [X2: nat,A3: set_nat] :
% 0.35/0.56 ( ( member_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton
% 0.35/0.56 thf(fact_2719_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.56 ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.56 = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2720_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: complex,A3: set_complex] :
% 0.35/0.56 ( ( ( member_complex @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_complex @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 0.35/0.56 = ( finite_card_complex @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2721_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: list_nat,A3: set_list_nat] :
% 0.35/0.56 ( ( ( member_list_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_list_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 0.35/0.56 = ( finite_card_list_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2722_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: set_nat,A3: set_set_nat] :
% 0.35/0.56 ( ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_set_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.56 = ( finite_card_set_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2723_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: real,A3: set_real] :
% 0.35/0.56 ( ( ( member_real @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_real @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.56 = ( finite_card_real @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2724_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: $o,A3: set_o] :
% 0.35/0.56 ( ( ( member_o @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_o @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.56 = ( finite_card_o @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2725_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: int,A3: set_int] :
% 0.35/0.56 ( ( ( member_int @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_int @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.56 = ( finite_card_int @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2726_card__Diff__singleton__if,axiom,
% 0.35/0.56 ! [X2: nat,A3: set_nat] :
% 0.35/0.56 ( ( ( member_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.56 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ ( member_nat @ X2 @ A3 )
% 0.35/0.56 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.56 = ( finite_card_nat @ A3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_Diff_singleton_if
% 0.35/0.56 thf(fact_2727_enumerate__Suc_H,axiom,
% 0.35/0.56 ! [S: set_nat,N: nat] :
% 0.35/0.56 ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
% 0.35/0.56 = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).
% 0.35/0.56
% 0.35/0.56 % enumerate_Suc'
% 0.35/0.56 thf(fact_2728_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2729_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_real,X2: real] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_real @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_real @ ( insert_real @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2730_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_o,X2: $o] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_o @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2731_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_complex,X2: complex] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2732_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_list_nat,X2: list_nat] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2733_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_set_nat,X2: set_nat] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2734_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_nat,X2: nat] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2735_card__insert__le__m1,axiom,
% 0.35/0.56 ! [N: nat,Y3: set_int,X2: int] :
% 0.35/0.56 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.56 => ( ( ord_less_eq_nat @ ( finite_card_int @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.56 => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X2 @ Y3 ) ) @ N ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_insert_le_m1
% 0.35/0.56 thf(fact_2736_vebt__pred_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 0.35/0.56 => ( ! [A: $o,Uw2: $o] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.56 => ( ! [A: $o,B: $o,Va2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 0.35/0.56 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_pred.cases
% 0.35/0.56 thf(fact_2737_vebt__succ_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [Uu2: $o,B: $o] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) )
% 0.35/0.56 => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
% 0.35/0.56 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_succ.cases
% 0.35/0.56 thf(fact_2738_VEBT__internal_Omembermima_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.35/0.56 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.35/0.56 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
% 0.35/0.56 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X5 ) )
% 0.35/0.56 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % VEBT_internal.membermima.cases
% 0.35/0.56 thf(fact_2739_vebt__member_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [A: $o,B: $o,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 0.35/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 0.35/0.56 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_member.cases
% 0.35/0.56 thf(fact_2740_vebt__delete_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [A: $o,B: $o] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) )
% 0.35/0.56 => ( ! [A: $o,B: $o] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.56 => ( ! [A: $o,B: $o,N3: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) )
% 0.35/0.56 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
% 0.35/0.56 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X5 ) )
% 0.35/0.56 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X5 ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_delete.cases
% 0.35/0.56 thf(fact_2741_vebt__insert_Ocases,axiom,
% 0.35/0.56 ! [X2: produc9072475918466114483BT_nat] :
% 0.35/0.56 ( ! [A: $o,B: $o,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 0.35/0.56 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X5 ) )
% 0.35/0.56 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X5 ) )
% 0.35/0.56 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ X5 ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 0.35/0.56 ( X2
% 0.35/0.56 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_insert.cases
% 0.35/0.56 thf(fact_2742_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2743_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2744_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2745_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2746_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2747_List_Ofinite__set,axiom,
% 0.35/0.56 ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % List.finite_set
% 0.35/0.56 thf(fact_2748_Bolzano,axiom,
% 0.35/0.56 ! [A2: real,B3: real,P: real > real > $o] :
% 0.35/0.56 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.56 => ( ! [A: real,B: real,C3: real] :
% 0.35/0.56 ( ( P @ A @ B )
% 0.35/0.56 => ( ( P @ B @ C3 )
% 0.35/0.56 => ( ( ord_less_eq_real @ A @ B )
% 0.35/0.56 => ( ( ord_less_eq_real @ B @ C3 )
% 0.35/0.56 => ( P @ A @ C3 ) ) ) ) )
% 0.35/0.56 => ( ! [X5: real] :
% 0.35/0.56 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.56 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.56 => ? [D3: real] :
% 0.35/0.56 ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.35/0.56 & ! [A: real,B: real] :
% 0.35/0.56 ( ( ( ord_less_eq_real @ A @ X5 )
% 0.35/0.56 & ( ord_less_eq_real @ X5 @ B )
% 0.35/0.56 & ( ord_less_real @ ( minus_minus_real @ B @ A ) @ D3 ) )
% 0.35/0.56 => ( P @ A @ B ) ) ) ) )
% 0.35/0.56 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % Bolzano
% 0.35/0.56 thf(fact_2749_card__length,axiom,
% 0.35/0.56 ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2750_card__length,axiom,
% 0.35/0.56 ! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2751_card__length,axiom,
% 0.35/0.56 ! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2752_card__length,axiom,
% 0.35/0.56 ! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2753_card__length,axiom,
% 0.35/0.56 ! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2754_card__length,axiom,
% 0.35/0.56 ! [Xs: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2755_card__length,axiom,
% 0.35/0.56 ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% 0.35/0.56
% 0.35/0.56 % card_length
% 0.35/0.56 thf(fact_2756_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: real,Xs: list_real] :
% 0.35/0.56 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2757_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: set_nat,Xs: list_set_nat] :
% 0.35/0.56 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2758_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: int,Xs: list_int] :
% 0.35/0.56 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2759_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.56 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2760_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: $o,Xs: list_o] :
% 0.35/0.56 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2761_length__pos__if__in__set,axiom,
% 0.35/0.56 ! [X2: nat,Xs: list_nat] :
% 0.35/0.56 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.35/0.56 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % length_pos_if_in_set
% 0.35/0.56 thf(fact_2762_the__elem__eq,axiom,
% 0.35/0.56 ! [X2: product_prod_nat_nat] :
% 0.35/0.56 ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.56 = X2 ) ).
% 0.35/0.56
% 0.35/0.56 % the_elem_eq
% 0.35/0.56 thf(fact_2763_the__elem__eq,axiom,
% 0.35/0.56 ! [X2: real] :
% 0.35/0.56 ( ( the_elem_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.56 = X2 ) ).
% 0.35/0.56
% 0.35/0.56 % the_elem_eq
% 0.35/0.56 thf(fact_2764_the__elem__eq,axiom,
% 0.35/0.56 ! [X2: $o] :
% 0.35/0.56 ( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.56 = X2 ) ).
% 0.35/0.56
% 0.35/0.56 % the_elem_eq
% 0.35/0.56 thf(fact_2765_the__elem__eq,axiom,
% 0.35/0.56 ! [X2: nat] :
% 0.35/0.56 ( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.56 = X2 ) ).
% 0.35/0.56
% 0.35/0.56 % the_elem_eq
% 0.35/0.56 thf(fact_2766_the__elem__eq,axiom,
% 0.35/0.56 ! [X2: int] :
% 0.35/0.56 ( ( the_elem_int @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.56 = X2 ) ).
% 0.35/0.56
% 0.35/0.56 % the_elem_eq
% 0.35/0.56 thf(fact_2767_case4_I1_J,axiom,
% 0.35/0.56 ! [X4: vEBT_VEBT] :
% 0.35/0.56 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
% 0.35/0.56 => ( ( vEBT_invar_vebt @ X4 @ na )
% 0.35/0.56 & ! [Xa: vEBT_VEBT] :
% 0.35/0.56 ( ( vEBT_invar_vebt @ Xa @ na )
% 0.35/0.56 => ( ( ( vEBT_VEBT_set_vebt @ X4 )
% 0.35/0.56 = ( vEBT_VEBT_set_vebt @ Xa ) )
% 0.35/0.56 => ( Xa = X4 ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % case4(1)
% 0.35/0.56 thf(fact_2768_vebt__maxt_Opelims,axiom,
% 0.35/0.56 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.35/0.56 ( ( ( vEBT_vebt_maxt @ X2 )
% 0.35/0.56 = Y3 )
% 0.35/0.56 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
% 0.35/0.56 => ( ! [A: $o,B: $o] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.56 => ( ( ( B
% 0.35/0.56 => ( Y3
% 0.35/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ B
% 0.35/0.56 => ( ( A
% 0.35/0.56 => ( Y3
% 0.35/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.56 & ( ~ A
% 0.35/0.56 => ( Y3 = none_nat ) ) ) ) )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
% 0.35/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.56 => ( ( Y3 = none_nat )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.56 => ( ( Y3
% 0.35/0.56 = ( some_nat @ Ma2 ) )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_maxt.pelims
% 0.35/0.56 thf(fact_2769_vebt__mint_Opelims,axiom,
% 0.35/0.56 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 0.35/0.56 ( ( ( vEBT_vebt_mint @ X2 )
% 0.35/0.56 = Y3 )
% 0.35/0.56 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
% 0.35/0.56 => ( ! [A: $o,B: $o] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.56 => ( ( ( A
% 0.35/0.56 => ( Y3
% 0.35/0.56 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.56 & ( ~ A
% 0.35/0.56 => ( ( B
% 0.35/0.56 => ( Y3
% 0.35/0.56 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.56 & ( ~ B
% 0.35/0.56 => ( Y3 = none_nat ) ) ) ) )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
% 0.35/0.56 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.56 => ( ( Y3 = none_nat )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.35/0.56 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.56 ( ( X2
% 0.35/0.56 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.56 => ( ( Y3
% 0.35/0.56 = ( some_nat @ Mi2 ) )
% 0.35/0.56 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % vebt_mint.pelims
% 0.35/0.56 thf(fact_2770_is__singletonI,axiom,
% 0.35/0.56 ! [X2: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singletonI
% 0.35/0.56 thf(fact_2771_is__singletonI,axiom,
% 0.35/0.56 ! [X2: real] : ( is_singleton_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singletonI
% 0.35/0.56 thf(fact_2772_is__singletonI,axiom,
% 0.35/0.56 ! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singletonI
% 0.35/0.56 thf(fact_2773_is__singletonI,axiom,
% 0.35/0.56 ! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singletonI
% 0.35/0.56 thf(fact_2774_is__singletonI,axiom,
% 0.35/0.56 ! [X2: int] : ( is_singleton_int @ ( insert_int @ X2 @ bot_bot_set_int ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singletonI
% 0.35/0.56 thf(fact_2775_case4_I5_J,axiom,
% 0.35/0.56 ( m
% 0.35/0.56 = ( suc @ na ) ) ).
% 0.35/0.56
% 0.35/0.56 % case4(5)
% 0.35/0.56 thf(fact_2776_is__singleton__the__elem,axiom,
% 0.35/0.56 ( is_sin2850979758926227957at_nat
% 0.35/0.56 = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 0.35/0.56 ( A6
% 0.35/0.56 = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singleton_the_elem
% 0.35/0.56 thf(fact_2777_is__singleton__the__elem,axiom,
% 0.35/0.56 ( is_singleton_real
% 0.35/0.56 = ( ^ [A6: set_real] :
% 0.35/0.56 ( A6
% 0.35/0.56 = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singleton_the_elem
% 0.35/0.56 thf(fact_2778_is__singleton__the__elem,axiom,
% 0.35/0.56 ( is_singleton_o
% 0.35/0.56 = ( ^ [A6: set_o] :
% 0.35/0.56 ( A6
% 0.35/0.56 = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singleton_the_elem
% 0.35/0.56 thf(fact_2779_is__singleton__the__elem,axiom,
% 0.35/0.56 ( is_singleton_nat
% 0.35/0.56 = ( ^ [A6: set_nat] :
% 0.35/0.56 ( A6
% 0.35/0.56 = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singleton_the_elem
% 0.35/0.56 thf(fact_2780_is__singleton__the__elem,axiom,
% 0.35/0.56 ( is_singleton_int
% 0.35/0.56 = ( ^ [A6: set_int] :
% 0.35/0.56 ( A6
% 0.35/0.56 = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.56
% 0.35/0.56 % is_singleton_the_elem
% 0.35/0.56 thf(fact_2781_is__singletonI_H,axiom,
% 0.35/0.56 ! [A3: set_set_nat] :
% 0.35/0.56 ( ( A3 != bot_bot_set_set_nat )
% 0.35/0.56 => ( ! [X5: set_nat,Y4: set_nat] :
% 0.35/0.57 ( ( member_set_nat @ X5 @ A3 )
% 0.35/0.57 => ( ( member_set_nat @ Y4 @ A3 )
% 0.35/0.57 => ( X5 = Y4 ) ) )
% 0.35/0.57 => ( is_singleton_set_nat @ A3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonI'
% 0.35/0.57 thf(fact_2782_is__singletonI_H,axiom,
% 0.35/0.57 ! [A3: set_real] :
% 0.35/0.57 ( ( A3 != bot_bot_set_real )
% 0.35/0.57 => ( ! [X5: real,Y4: real] :
% 0.35/0.57 ( ( member_real @ X5 @ A3 )
% 0.35/0.57 => ( ( member_real @ Y4 @ A3 )
% 0.35/0.57 => ( X5 = Y4 ) ) )
% 0.35/0.57 => ( is_singleton_real @ A3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonI'
% 0.35/0.57 thf(fact_2783_is__singletonI_H,axiom,
% 0.35/0.57 ! [A3: set_o] :
% 0.35/0.57 ( ( A3 != bot_bot_set_o )
% 0.35/0.57 => ( ! [X5: $o,Y4: $o] :
% 0.35/0.57 ( ( member_o @ X5 @ A3 )
% 0.35/0.57 => ( ( member_o @ Y4 @ A3 )
% 0.35/0.57 => ( X5 = Y4 ) ) )
% 0.35/0.57 => ( is_singleton_o @ A3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonI'
% 0.35/0.57 thf(fact_2784_is__singletonI_H,axiom,
% 0.35/0.57 ! [A3: set_nat] :
% 0.35/0.57 ( ( A3 != bot_bot_set_nat )
% 0.35/0.57 => ( ! [X5: nat,Y4: nat] :
% 0.35/0.57 ( ( member_nat @ X5 @ A3 )
% 0.35/0.57 => ( ( member_nat @ Y4 @ A3 )
% 0.35/0.57 => ( X5 = Y4 ) ) )
% 0.35/0.57 => ( is_singleton_nat @ A3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonI'
% 0.35/0.57 thf(fact_2785_is__singletonI_H,axiom,
% 0.35/0.57 ! [A3: set_int] :
% 0.35/0.57 ( ( A3 != bot_bot_set_int )
% 0.35/0.57 => ( ! [X5: int,Y4: int] :
% 0.35/0.57 ( ( member_int @ X5 @ A3 )
% 0.35/0.57 => ( ( member_int @ Y4 @ A3 )
% 0.35/0.57 => ( X5 = Y4 ) ) )
% 0.35/0.57 => ( is_singleton_int @ A3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonI'
% 0.35/0.57 thf(fact_2786_length__induct,axiom,
% 0.35/0.57 ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.35/0.57 ( ! [Xs3: list_VEBT_VEBT] :
% 0.35/0.57 ( ! [Ys: list_VEBT_VEBT] :
% 0.35/0.57 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.35/0.57 => ( P @ Ys ) )
% 0.35/0.57 => ( P @ Xs3 ) )
% 0.35/0.57 => ( P @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_induct
% 0.35/0.57 thf(fact_2787_length__induct,axiom,
% 0.35/0.57 ! [P: list_o > $o,Xs: list_o] :
% 0.35/0.57 ( ! [Xs3: list_o] :
% 0.35/0.57 ( ! [Ys: list_o] :
% 0.35/0.57 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 0.35/0.57 => ( P @ Ys ) )
% 0.35/0.57 => ( P @ Xs3 ) )
% 0.35/0.57 => ( P @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_induct
% 0.35/0.57 thf(fact_2788_length__induct,axiom,
% 0.35/0.57 ! [P: list_nat > $o,Xs: list_nat] :
% 0.35/0.57 ( ! [Xs3: list_nat] :
% 0.35/0.57 ( ! [Ys: list_nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 0.35/0.57 => ( P @ Ys ) )
% 0.35/0.57 => ( P @ Xs3 ) )
% 0.35/0.57 => ( P @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_induct
% 0.35/0.57 thf(fact_2789_finite__maxlen,axiom,
% 0.35/0.57 ! [M5: set_list_VEBT_VEBT] :
% 0.35/0.57 ( ( finite3004134309566078307T_VEBT @ M5 )
% 0.35/0.57 => ? [N3: nat] :
% 0.35/0.57 ! [X4: list_VEBT_VEBT] :
% 0.35/0.57 ( ( member2936631157270082147T_VEBT @ X4 @ M5 )
% 0.35/0.57 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X4 ) @ N3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_maxlen
% 0.35/0.57 thf(fact_2790_finite__maxlen,axiom,
% 0.35/0.57 ! [M5: set_list_o] :
% 0.35/0.57 ( ( finite_finite_list_o @ M5 )
% 0.35/0.57 => ? [N3: nat] :
% 0.35/0.57 ! [X4: list_o] :
% 0.35/0.57 ( ( member_list_o @ X4 @ M5 )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_maxlen
% 0.35/0.57 thf(fact_2791_finite__maxlen,axiom,
% 0.35/0.57 ! [M5: set_list_nat] :
% 0.35/0.57 ( ( finite8100373058378681591st_nat @ M5 )
% 0.35/0.57 => ? [N3: nat] :
% 0.35/0.57 ! [X4: list_nat] :
% 0.35/0.57 ( ( member_list_nat @ X4 @ M5 )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_maxlen
% 0.35/0.57 thf(fact_2792_is__singleton__def,axiom,
% 0.35/0.57 ( is_sin2850979758926227957at_nat
% 0.35/0.57 = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 0.35/0.57 ? [X: product_prod_nat_nat] :
% 0.35/0.57 ( A6
% 0.35/0.57 = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_def
% 0.35/0.57 thf(fact_2793_is__singleton__def,axiom,
% 0.35/0.57 ( is_singleton_real
% 0.35/0.57 = ( ^ [A6: set_real] :
% 0.35/0.57 ? [X: real] :
% 0.35/0.57 ( A6
% 0.35/0.57 = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_def
% 0.35/0.57 thf(fact_2794_is__singleton__def,axiom,
% 0.35/0.57 ( is_singleton_o
% 0.35/0.57 = ( ^ [A6: set_o] :
% 0.35/0.57 ? [X: $o] :
% 0.35/0.57 ( A6
% 0.35/0.57 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_def
% 0.35/0.57 thf(fact_2795_is__singleton__def,axiom,
% 0.35/0.57 ( is_singleton_nat
% 0.35/0.57 = ( ^ [A6: set_nat] :
% 0.35/0.57 ? [X: nat] :
% 0.35/0.57 ( A6
% 0.35/0.57 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_def
% 0.35/0.57 thf(fact_2796_is__singleton__def,axiom,
% 0.35/0.57 ( is_singleton_int
% 0.35/0.57 = ( ^ [A6: set_int] :
% 0.35/0.57 ? [X: int] :
% 0.35/0.57 ( A6
% 0.35/0.57 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_def
% 0.35/0.57 thf(fact_2797_is__singletonE,axiom,
% 0.35/0.57 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.57 ( ( is_sin2850979758926227957at_nat @ A3 )
% 0.35/0.57 => ~ ! [X5: product_prod_nat_nat] :
% 0.35/0.57 ( A3
% 0.35/0.57 != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonE
% 0.35/0.57 thf(fact_2798_is__singletonE,axiom,
% 0.35/0.57 ! [A3: set_real] :
% 0.35/0.57 ( ( is_singleton_real @ A3 )
% 0.35/0.57 => ~ ! [X5: real] :
% 0.35/0.57 ( A3
% 0.35/0.57 != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonE
% 0.35/0.57 thf(fact_2799_is__singletonE,axiom,
% 0.35/0.57 ! [A3: set_o] :
% 0.35/0.57 ( ( is_singleton_o @ A3 )
% 0.35/0.57 => ~ ! [X5: $o] :
% 0.35/0.57 ( A3
% 0.35/0.57 != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonE
% 0.35/0.57 thf(fact_2800_is__singletonE,axiom,
% 0.35/0.57 ! [A3: set_nat] :
% 0.35/0.57 ( ( is_singleton_nat @ A3 )
% 0.35/0.57 => ~ ! [X5: nat] :
% 0.35/0.57 ( A3
% 0.35/0.57 != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonE
% 0.35/0.57 thf(fact_2801_is__singletonE,axiom,
% 0.35/0.57 ! [A3: set_int] :
% 0.35/0.57 ( ( is_singleton_int @ A3 )
% 0.35/0.57 => ~ ! [X5: int] :
% 0.35/0.57 ( A3
% 0.35/0.57 != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singletonE
% 0.35/0.57 thf(fact_2802_is__singleton__altdef,axiom,
% 0.35/0.57 ( is_singleton_complex
% 0.35/0.57 = ( ^ [A6: set_complex] :
% 0.35/0.57 ( ( finite_card_complex @ A6 )
% 0.35/0.57 = one_one_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_altdef
% 0.35/0.57 thf(fact_2803_is__singleton__altdef,axiom,
% 0.35/0.57 ( is_sin2641923865335537900st_nat
% 0.35/0.57 = ( ^ [A6: set_list_nat] :
% 0.35/0.57 ( ( finite_card_list_nat @ A6 )
% 0.35/0.57 = one_one_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_altdef
% 0.35/0.57 thf(fact_2804_is__singleton__altdef,axiom,
% 0.35/0.57 ( is_singleton_set_nat
% 0.35/0.57 = ( ^ [A6: set_set_nat] :
% 0.35/0.57 ( ( finite_card_set_nat @ A6 )
% 0.35/0.57 = one_one_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_altdef
% 0.35/0.57 thf(fact_2805_is__singleton__altdef,axiom,
% 0.35/0.57 ( is_singleton_nat
% 0.35/0.57 = ( ^ [A6: set_nat] :
% 0.35/0.57 ( ( finite_card_nat @ A6 )
% 0.35/0.57 = one_one_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_altdef
% 0.35/0.57 thf(fact_2806_is__singleton__altdef,axiom,
% 0.35/0.57 ( is_singleton_int
% 0.35/0.57 = ( ^ [A6: set_int] :
% 0.35/0.57 ( ( finite_card_int @ A6 )
% 0.35/0.57 = one_one_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % is_singleton_altdef
% 0.35/0.57 thf(fact_2807_finite__list,axiom,
% 0.35/0.57 ! [A3: set_VEBT_VEBT] :
% 0.35/0.57 ( ( finite5795047828879050333T_VEBT @ A3 )
% 0.35/0.57 => ? [Xs3: list_VEBT_VEBT] :
% 0.35/0.57 ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2808_finite__list,axiom,
% 0.35/0.57 ! [A3: set_nat] :
% 0.35/0.57 ( ( finite_finite_nat @ A3 )
% 0.35/0.57 => ? [Xs3: list_nat] :
% 0.35/0.57 ( ( set_nat2 @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2809_finite__list,axiom,
% 0.35/0.57 ! [A3: set_int] :
% 0.35/0.57 ( ( finite_finite_int @ A3 )
% 0.35/0.57 => ? [Xs3: list_int] :
% 0.35/0.57 ( ( set_int2 @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2810_finite__list,axiom,
% 0.35/0.57 ! [A3: set_complex] :
% 0.35/0.57 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.57 => ? [Xs3: list_complex] :
% 0.35/0.57 ( ( set_complex2 @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2811_finite__list,axiom,
% 0.35/0.57 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.57 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.57 => ? [Xs3: list_P6011104703257516679at_nat] :
% 0.35/0.57 ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2812_finite__list,axiom,
% 0.35/0.57 ! [A3: set_Extended_enat] :
% 0.35/0.57 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.57 => ? [Xs3: list_Extended_enat] :
% 0.35/0.57 ( ( set_Extended_enat2 @ Xs3 )
% 0.35/0.57 = A3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % finite_list
% 0.35/0.57 thf(fact_2813_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_real,B2: set_real] :
% 0.35/0.57 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: real] :
% 0.35/0.57 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.35/0.57 => ( member_real @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2814_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_o,B2: set_o] :
% 0.35/0.57 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: $o] :
% 0.35/0.57 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( member_o @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2815_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_set_nat,B2: set_set_nat] :
% 0.35/0.57 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: set_nat] :
% 0.35/0.57 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.35/0.57 => ( member_set_nat @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2816_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.35/0.57 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: vEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2817_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_nat,B2: set_nat] :
% 0.35/0.57 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: nat] :
% 0.35/0.57 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( member_nat @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2818_subset__code_I1_J,axiom,
% 0.35/0.57 ! [Xs: list_int,B2: set_int] :
% 0.35/0.57 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 0.35/0.57 = ( ! [X: int] :
% 0.35/0.57 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( member_int @ X @ B2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % subset_code(1)
% 0.35/0.57 thf(fact_2819_case4_I8_J,axiom,
% 0.35/0.57 ( ( mi = ma )
% 0.35/0.57 => ! [X4: vEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
% 0.35/0.57 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % case4(8)
% 0.35/0.57 thf(fact_2820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT,Y3: $o] :
% 0.35/0.57 ( ( ( vEBT_VEBT_minNull @ X2 )
% 0.35/0.57 = Y3 )
% 0.35/0.57 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.35/0.57 => ( ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.57 => ( Y3
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.35/0.57 => ( ! [Uv2: $o] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.57 => ( ~ Y3
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.35/0.57 => ( ! [Uu2: $o] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.57 => ( ~ Y3
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.35/0.57 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.57 => ( Y3
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.35/0.57 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.35/0.57 => ( ~ Y3
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % VEBT_internal.minNull.pelims(1)
% 0.35/0.57 thf(fact_2821_case4_I13_J,axiom,
% 0.35/0.57 ( ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) )
% 0.35/0.57 = ( vEBT_VEBT_set_vebt @ sa ) ) ).
% 0.35/0.57
% 0.35/0.57 % case4(13)
% 0.35/0.57 thf(fact_2822_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT] :
% 0.35/0.57 ( ( vEBT_VEBT_minNull @ X2 )
% 0.35/0.57 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.35/0.57 => ( ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.57 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % VEBT_internal.minNull.pelims(2)
% 0.35/0.57 thf(fact_2823_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT] :
% 0.35/0.57 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 0.35/0.57 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 0.35/0.57 => ( ! [Uv2: $o] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.35/0.57 => ( ! [Uu2: $o] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.35/0.57 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.57 ( ( X2
% 0.35/0.57 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.35/0.57 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % VEBT_internal.minNull.pelims(3)
% 0.35/0.57 thf(fact_2824_int__power__div__base,axiom,
% 0.35/0.57 ! [M: nat,K: int] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.57 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.57 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.35/0.57 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % int_power_div_base
% 0.35/0.57 thf(fact_2825_set__removeAll,axiom,
% 0.35/0.57 ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.35/0.57 ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2826_set__removeAll,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.57 ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2827_set__removeAll,axiom,
% 0.35/0.57 ! [X2: real,Xs: list_real] :
% 0.35/0.57 ( ( set_real2 @ ( removeAll_real @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2828_set__removeAll,axiom,
% 0.35/0.57 ! [X2: $o,Xs: list_o] :
% 0.35/0.57 ( ( set_o2 @ ( removeAll_o @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2829_set__removeAll,axiom,
% 0.35/0.57 ! [X2: int,Xs: list_int] :
% 0.35/0.57 ( ( set_int2 @ ( removeAll_int @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2830_set__removeAll,axiom,
% 0.35/0.57 ! [X2: nat,Xs: list_nat] :
% 0.35/0.57 ( ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) )
% 0.35/0.57 = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % set_removeAll
% 0.35/0.57 thf(fact_2831_inthall,axiom,
% 0.35/0.57 ! [Xs: list_real,P: real > $o,N: nat] :
% 0.35/0.57 ( ! [X5: real] :
% 0.35/0.57 ( ( member_real @ X5 @ ( set_real2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2832_inthall,axiom,
% 0.35/0.57 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 0.35/0.57 ( ! [X5: set_nat] :
% 0.35/0.57 ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2833_inthall,axiom,
% 0.35/0.57 ! [Xs: list_int,P: int > $o,N: nat] :
% 0.35/0.57 ( ! [X5: int] :
% 0.35/0.57 ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2834_inthall,axiom,
% 0.35/0.57 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 0.35/0.57 ( ! [X5: vEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2835_inthall,axiom,
% 0.35/0.57 ! [Xs: list_o,P: $o > $o,N: nat] :
% 0.35/0.57 ( ! [X5: $o] :
% 0.35/0.57 ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2836_inthall,axiom,
% 0.35/0.57 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 0.35/0.57 ( ! [X5: nat] :
% 0.35/0.57 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % inthall
% 0.35/0.57 thf(fact_2837_nat__ivt__aux,axiom,
% 0.35/0.57 ! [N: nat,F: nat > int,K: int] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ N )
% 0.35/0.57 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.35/0.57 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.57 => ? [I2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.57 & ( ( F @ I2 )
% 0.35/0.57 = K ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_ivt_aux
% 0.35/0.57 thf(fact_2838_case4_I9_J,axiom,
% 0.35/0.57 ord_less_eq_nat @ mi @ ma ).
% 0.35/0.57
% 0.35/0.57 % case4(9)
% 0.35/0.57 thf(fact_2839_case4_I3_J,axiom,
% 0.35/0.57 vEBT_invar_vebt @ summary2 @ m ).
% 0.35/0.57
% 0.35/0.57 % case4(3)
% 0.35/0.57 thf(fact_2840_abs__idempotent,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
% 0.35/0.57 = ( abs_abs_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_idempotent
% 0.35/0.57 thf(fact_2841_abs__idempotent,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
% 0.35/0.57 = ( abs_abs_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_idempotent
% 0.35/0.57 thf(fact_2842_abs__idempotent,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
% 0.35/0.57 = ( abs_abs_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_idempotent
% 0.35/0.57 thf(fact_2843_abs__idempotent,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
% 0.35/0.57 = ( abs_abs_Code_integer @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_idempotent
% 0.35/0.57 thf(fact_2844_case4_I2_J,axiom,
% 0.35/0.57 ! [S2: vEBT_VEBT] :
% 0.35/0.57 ( ( vEBT_invar_vebt @ S2 @ m )
% 0.35/0.57 => ( ( ( vEBT_VEBT_set_vebt @ summary2 )
% 0.35/0.57 = ( vEBT_VEBT_set_vebt @ S2 ) )
% 0.35/0.57 => ( S2 = summary2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % case4(2)
% 0.35/0.57 thf(fact_2845_abs__0,axiom,
% 0.35/0.57 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0
% 0.35/0.57 thf(fact_2846_abs__0,axiom,
% 0.35/0.57 ( ( abs_abs_real @ zero_zero_real )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0
% 0.35/0.57 thf(fact_2847_abs__0,axiom,
% 0.35/0.57 ( ( abs_abs_rat @ zero_zero_rat )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0
% 0.35/0.57 thf(fact_2848_abs__0,axiom,
% 0.35/0.57 ( ( abs_abs_int @ zero_zero_int )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0
% 0.35/0.57 thf(fact_2849_abs__0__eq,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( zero_z3403309356797280102nteger
% 0.35/0.57 = ( abs_abs_Code_integer @ A2 ) )
% 0.35/0.57 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0_eq
% 0.35/0.57 thf(fact_2850_abs__0__eq,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( zero_zero_real
% 0.35/0.57 = ( abs_abs_real @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0_eq
% 0.35/0.57 thf(fact_2851_abs__0__eq,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( zero_zero_rat
% 0.35/0.57 = ( abs_abs_rat @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0_eq
% 0.35/0.57 thf(fact_2852_abs__0__eq,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( zero_zero_int
% 0.35/0.57 = ( abs_abs_int @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_0_eq
% 0.35/0.57 thf(fact_2853_abs__eq__0,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = zero_z3403309356797280102nteger )
% 0.35/0.57 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0
% 0.35/0.57 thf(fact_2854_abs__eq__0,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ( abs_abs_real @ A2 )
% 0.35/0.57 = zero_zero_real )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0
% 0.35/0.57 thf(fact_2855_abs__eq__0,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = zero_zero_rat )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0
% 0.35/0.57 thf(fact_2856_abs__eq__0,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ( abs_abs_int @ A2 )
% 0.35/0.57 = zero_zero_int )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0
% 0.35/0.57 thf(fact_2857_abs__zero,axiom,
% 0.35/0.57 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % abs_zero
% 0.35/0.57 thf(fact_2858_abs__zero,axiom,
% 0.35/0.57 ( ( abs_abs_real @ zero_zero_real )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % abs_zero
% 0.35/0.57 thf(fact_2859_abs__zero,axiom,
% 0.35/0.57 ( ( abs_abs_rat @ zero_zero_rat )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % abs_zero
% 0.35/0.57 thf(fact_2860_abs__zero,axiom,
% 0.35/0.57 ( ( abs_abs_int @ zero_zero_int )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % abs_zero
% 0.35/0.57 thf(fact_2861_abs__minus__cancel,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
% 0.35/0.57 = ( abs_abs_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_cancel
% 0.35/0.57 thf(fact_2862_abs__minus__cancel,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
% 0.35/0.57 = ( abs_abs_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_cancel
% 0.35/0.57 thf(fact_2863_abs__minus__cancel,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
% 0.35/0.57 = ( abs_abs_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_cancel
% 0.35/0.57 thf(fact_2864_abs__minus__cancel,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.35/0.57 = ( abs_abs_Code_integer @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_cancel
% 0.35/0.57 thf(fact_2865_abs__of__nat,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 0.35/0.57 = ( semiri681578069525770553at_rat @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nat
% 0.35/0.57 thf(fact_2866_abs__of__nat,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 0.35/0.57 = ( semiri4939895301339042750nteger @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nat
% 0.35/0.57 thf(fact_2867_abs__of__nat,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.57 = ( semiri1314217659103216013at_int @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nat
% 0.35/0.57 thf(fact_2868_abs__of__nat,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.57 = ( semiri5074537144036343181t_real @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nat
% 0.35/0.57 thf(fact_2869_abs__le__zero__iff,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
% 0.35/0.57 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_zero_iff
% 0.35/0.57 thf(fact_2870_abs__le__zero__iff,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ zero_zero_real )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_zero_iff
% 0.35/0.57 thf(fact_2871_abs__le__zero__iff,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_zero_iff
% 0.35/0.57 thf(fact_2872_abs__le__zero__iff,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_zero_iff
% 0.35/0.57 thf(fact_2873_abs__le__self__iff,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ A2 )
% 0.35/0.57 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_self_iff
% 0.35/0.57 thf(fact_2874_abs__le__self__iff,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ A2 )
% 0.35/0.57 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_self_iff
% 0.35/0.57 thf(fact_2875_abs__le__self__iff,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ A2 )
% 0.35/0.57 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_self_iff
% 0.35/0.57 thf(fact_2876_abs__le__self__iff,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
% 0.35/0.57 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_self_iff
% 0.35/0.57 thf(fact_2877_abs__of__nonneg,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.57 => ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonneg
% 0.35/0.57 thf(fact_2878_abs__of__nonneg,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.57 => ( ( abs_abs_real @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonneg
% 0.35/0.57 thf(fact_2879_abs__of__nonneg,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 => ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonneg
% 0.35/0.57 thf(fact_2880_abs__of__nonneg,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.57 => ( ( abs_abs_int @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonneg
% 0.35/0.57 thf(fact_2881_zero__less__abs__iff,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) )
% 0.35/0.57 = ( A2 != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_abs_iff
% 0.35/0.57 thf(fact_2882_zero__less__abs__iff,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A2 ) )
% 0.35/0.57 = ( A2 != zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_abs_iff
% 0.35/0.57 thf(fact_2883_zero__less__abs__iff,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) )
% 0.35/0.57 = ( A2 != zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_abs_iff
% 0.35/0.57 thf(fact_2884_zero__less__abs__iff,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
% 0.35/0.57 = ( A2 != zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_abs_iff
% 0.35/0.57 thf(fact_2885_zero__le__divide__abs__iff,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) )
% 0.35/0.57 = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.57 | ( B3 = zero_zero_real ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_le_divide_abs_iff
% 0.35/0.57 thf(fact_2886_zero__le__divide__abs__iff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) )
% 0.35/0.57 = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 | ( B3 = zero_zero_rat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_le_divide_abs_iff
% 0.35/0.57 thf(fact_2887_divide__le__0__abs__iff,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) @ zero_zero_real )
% 0.35/0.57 = ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.57 | ( B3 = zero_zero_real ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % divide_le_0_abs_iff
% 0.35/0.57 thf(fact_2888_divide__le__0__abs__iff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) @ zero_zero_rat )
% 0.35/0.57 = ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.57 | ( B3 = zero_zero_rat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % divide_le_0_abs_iff
% 0.35/0.57 thf(fact_2889_abs__of__nonpos,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.57 => ( ( abs_abs_real @ A2 )
% 0.35/0.57 = ( uminus_uminus_real @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonpos
% 0.35/0.57 thf(fact_2890_abs__of__nonpos,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.35/0.57 => ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonpos
% 0.35/0.57 thf(fact_2891_abs__of__nonpos,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.57 => ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonpos
% 0.35/0.57 thf(fact_2892_abs__of__nonpos,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.57 => ( ( abs_abs_int @ A2 )
% 0.35/0.57 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_nonpos
% 0.35/0.57 thf(fact_2893_zabs__less__one__iff,axiom,
% 0.35/0.57 ! [Z: int] :
% 0.35/0.57 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 0.35/0.57 = ( Z = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % zabs_less_one_iff
% 0.35/0.57 thf(fact_2894_aa,axiom,
% 0.35/0.57 ord_less_eq_set_nat @ ( insert_nat @ mi @ ( insert_nat @ ma @ bot_bot_set_nat ) ) @ ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % aa
% 0.35/0.57 thf(fact_2895_case4_I6_J,axiom,
% 0.35/0.57 ( deg
% 0.35/0.57 = ( plus_plus_nat @ na @ m ) ) ).
% 0.35/0.57
% 0.35/0.57 % case4(6)
% 0.35/0.57 thf(fact_2896_abs__le__D1,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D1
% 0.35/0.57 thf(fact_2897_abs__le__D1,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D1
% 0.35/0.57 thf(fact_2898_abs__le__D1,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D1
% 0.35/0.57 thf(fact_2899_abs__le__D1,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D1
% 0.35/0.57 thf(fact_2900_abs__ge__self,axiom,
% 0.35/0.57 ! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_self
% 0.35/0.57 thf(fact_2901_abs__ge__self,axiom,
% 0.35/0.57 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_self
% 0.35/0.57 thf(fact_2902_abs__ge__self,axiom,
% 0.35/0.57 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_self
% 0.35/0.57 thf(fact_2903_abs__ge__self,axiom,
% 0.35/0.57 ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_self
% 0.35/0.57 thf(fact_2904_abs__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = zero_z3403309356797280102nteger )
% 0.35/0.57 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0_iff
% 0.35/0.57 thf(fact_2905_abs__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ( abs_abs_real @ A2 )
% 0.35/0.57 = zero_zero_real )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0_iff
% 0.35/0.57 thf(fact_2906_abs__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = zero_zero_rat )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0_iff
% 0.35/0.57 thf(fact_2907_abs__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ( abs_abs_int @ A2 )
% 0.35/0.57 = zero_zero_int )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_0_iff
% 0.35/0.57 thf(fact_2908_abs__minus__commute,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
% 0.35/0.57 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_commute
% 0.35/0.57 thf(fact_2909_abs__minus__commute,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) )
% 0.35/0.57 = ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_commute
% 0.35/0.57 thf(fact_2910_abs__minus__commute,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_commute
% 0.35/0.57 thf(fact_2911_abs__minus__commute,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) )
% 0.35/0.57 = ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_commute
% 0.35/0.57 thf(fact_2912_infinite__int__iff__unbounded__le,axiom,
% 0.35/0.57 ! [S: set_int] :
% 0.35/0.57 ( ( ~ ( finite_finite_int @ S ) )
% 0.35/0.57 = ( ! [M2: int] :
% 0.35/0.57 ? [N2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N2 ) )
% 0.35/0.57 & ( member_int @ N2 @ S ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % infinite_int_iff_unbounded_le
% 0.35/0.57 thf(fact_2913_infinite__int__iff__unbounded,axiom,
% 0.35/0.57 ! [S: set_int] :
% 0.35/0.57 ( ( ~ ( finite_finite_int @ S ) )
% 0.35/0.57 = ( ! [M2: int] :
% 0.35/0.57 ? [N2: int] :
% 0.35/0.57 ( ( ord_less_int @ M2 @ ( abs_abs_int @ N2 ) )
% 0.35/0.57 & ( member_int @ N2 @ S ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % infinite_int_iff_unbounded
% 0.35/0.57 thf(fact_2914_abs__ge__zero,axiom,
% 0.35/0.57 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_zero
% 0.35/0.57 thf(fact_2915_abs__ge__zero,axiom,
% 0.35/0.57 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_zero
% 0.35/0.57 thf(fact_2916_abs__ge__zero,axiom,
% 0.35/0.57 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_zero
% 0.35/0.57 thf(fact_2917_abs__ge__zero,axiom,
% 0.35/0.57 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_zero
% 0.35/0.57 thf(fact_2918_abs__of__pos,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.57 => ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_pos
% 0.35/0.57 thf(fact_2919_abs__of__pos,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.57 => ( ( abs_abs_real @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_pos
% 0.35/0.57 thf(fact_2920_abs__of__pos,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 => ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_pos
% 0.35/0.57 thf(fact_2921_abs__of__pos,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.57 => ( ( abs_abs_int @ A2 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_pos
% 0.35/0.57 thf(fact_2922_abs__not__less__zero,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % abs_not_less_zero
% 0.35/0.57 thf(fact_2923_abs__not__less__zero,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % abs_not_less_zero
% 0.35/0.57 thf(fact_2924_abs__not__less__zero,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % abs_not_less_zero
% 0.35/0.57 thf(fact_2925_abs__not__less__zero,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % abs_not_less_zero
% 0.35/0.57 thf(fact_2926_abs__triangle__ineq2__sym,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2_sym
% 0.35/0.57 thf(fact_2927_abs__triangle__ineq2__sym,axiom,
% 0.35/0.57 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2_sym
% 0.35/0.57 thf(fact_2928_abs__triangle__ineq2__sym,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2_sym
% 0.35/0.57 thf(fact_2929_abs__triangle__ineq2__sym,axiom,
% 0.35/0.57 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2_sym
% 0.35/0.57 thf(fact_2930_abs__triangle__ineq3,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq3
% 0.35/0.57 thf(fact_2931_abs__triangle__ineq3,axiom,
% 0.35/0.57 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq3
% 0.35/0.57 thf(fact_2932_abs__triangle__ineq3,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq3
% 0.35/0.57 thf(fact_2933_abs__triangle__ineq3,axiom,
% 0.35/0.57 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq3
% 0.35/0.57 thf(fact_2934_abs__triangle__ineq2,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2
% 0.35/0.57 thf(fact_2935_abs__triangle__ineq2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2
% 0.35/0.57 thf(fact_2936_abs__triangle__ineq2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2
% 0.35/0.57 thf(fact_2937_abs__triangle__ineq2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_triangle_ineq2
% 0.35/0.57 thf(fact_2938_nonzero__abs__divide,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( B3 != zero_zero_rat )
% 0.35/0.57 => ( ( abs_abs_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( divide_divide_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nonzero_abs_divide
% 0.35/0.57 thf(fact_2939_nonzero__abs__divide,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( B3 != zero_zero_real )
% 0.35/0.57 => ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.35/0.57 = ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nonzero_abs_divide
% 0.35/0.57 thf(fact_2940_abs__ge__minus__self,axiom,
% 0.35/0.57 ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ ( abs_abs_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_minus_self
% 0.35/0.57 thf(fact_2941_abs__ge__minus__self,axiom,
% 0.35/0.57 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_minus_self
% 0.35/0.57 thf(fact_2942_abs__ge__minus__self,axiom,
% 0.35/0.57 ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ ( abs_abs_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_minus_self
% 0.35/0.57 thf(fact_2943_abs__ge__minus__self,axiom,
% 0.35/0.57 ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_ge_minus_self
% 0.35/0.57 thf(fact_2944_abs__le__iff,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_iff
% 0.35/0.57 thf(fact_2945_abs__le__iff,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 0.35/0.57 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_iff
% 0.35/0.57 thf(fact_2946_abs__le__iff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_iff
% 0.35/0.57 thf(fact_2947_abs__le__iff,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_iff
% 0.35/0.57 thf(fact_2948_abs__le__D2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D2
% 0.35/0.57 thf(fact_2949_abs__le__D2,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D2
% 0.35/0.57 thf(fact_2950_abs__le__D2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D2
% 0.35/0.57 thf(fact_2951_abs__le__D2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_le_D2
% 0.35/0.57 thf(fact_2952_abs__leI,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_leI
% 0.35/0.57 thf(fact_2953_abs__leI,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 0.35/0.57 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_leI
% 0.35/0.57 thf(fact_2954_abs__leI,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_leI
% 0.35/0.57 thf(fact_2955_abs__leI,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 0.35/0.57 => ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_leI
% 0.35/0.57 thf(fact_2956_abs__less__iff,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_less_iff
% 0.35/0.57 thf(fact_2957_abs__less__iff,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_less_iff
% 0.35/0.57 thf(fact_2958_abs__less__iff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.57 & ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_less_iff
% 0.35/0.57 thf(fact_2959_abs__less__iff,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 0.35/0.57 = ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
% 0.35/0.57 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_less_iff
% 0.35/0.57 thf(fact_2960_nth__equalityI,axiom,
% 0.35/0.57 ! [Xs: list_int,Ys2: list_int] :
% 0.35/0.57 ( ( ( size_size_list_int @ Xs )
% 0.35/0.57 = ( size_size_list_int @ Ys2 ) )
% 0.35/0.57 => ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( ( nth_int @ Xs @ I2 )
% 0.35/0.57 = ( nth_int @ Ys2 @ I2 ) ) )
% 0.35/0.57 => ( Xs = Ys2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_equalityI
% 0.35/0.57 thf(fact_2961_nth__equalityI,axiom,
% 0.35/0.57 ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.35/0.57 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.35/0.57 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.35/0.57 => ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.35/0.57 = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
% 0.35/0.57 => ( Xs = Ys2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_equalityI
% 0.35/0.57 thf(fact_2962_nth__equalityI,axiom,
% 0.35/0.57 ! [Xs: list_o,Ys2: list_o] :
% 0.35/0.57 ( ( ( size_size_list_o @ Xs )
% 0.35/0.57 = ( size_size_list_o @ Ys2 ) )
% 0.35/0.57 => ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( ( nth_o @ Xs @ I2 )
% 0.35/0.57 = ( nth_o @ Ys2 @ I2 ) ) )
% 0.35/0.57 => ( Xs = Ys2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_equalityI
% 0.35/0.57 thf(fact_2963_nth__equalityI,axiom,
% 0.35/0.57 ! [Xs: list_nat,Ys2: list_nat] :
% 0.35/0.57 ( ( ( size_size_list_nat @ Xs )
% 0.35/0.57 = ( size_size_list_nat @ Ys2 ) )
% 0.35/0.57 => ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( ( nth_nat @ Xs @ I2 )
% 0.35/0.57 = ( nth_nat @ Ys2 @ I2 ) ) )
% 0.35/0.57 => ( Xs = Ys2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_equalityI
% 0.35/0.57 thf(fact_2964_Skolem__list__nth,axiom,
% 0.35/0.57 ! [K: nat,P: nat > int > $o] :
% 0.35/0.57 ( ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ? [X8: int] : ( P @ I4 @ X8 ) ) )
% 0.35/0.57 = ( ? [Xs2: list_int] :
% 0.35/0.57 ( ( ( size_size_list_int @ Xs2 )
% 0.35/0.57 = K )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Skolem_list_nth
% 0.35/0.57 thf(fact_2965_Skolem__list__nth,axiom,
% 0.35/0.57 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.35/0.57 ( ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ? [X8: vEBT_VEBT] : ( P @ I4 @ X8 ) ) )
% 0.35/0.57 = ( ? [Xs2: list_VEBT_VEBT] :
% 0.35/0.57 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.35/0.57 = K )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Skolem_list_nth
% 0.35/0.57 thf(fact_2966_Skolem__list__nth,axiom,
% 0.35/0.57 ! [K: nat,P: nat > $o > $o] :
% 0.35/0.57 ( ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ? [X8: $o] : ( P @ I4 @ X8 ) ) )
% 0.35/0.57 = ( ? [Xs2: list_o] :
% 0.35/0.57 ( ( ( size_size_list_o @ Xs2 )
% 0.35/0.57 = K )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ( P @ I4 @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Skolem_list_nth
% 0.35/0.57 thf(fact_2967_Skolem__list__nth,axiom,
% 0.35/0.57 ! [K: nat,P: nat > nat > $o] :
% 0.35/0.57 ( ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
% 0.35/0.57 = ( ? [Xs2: list_nat] :
% 0.35/0.57 ( ( ( size_size_list_nat @ Xs2 )
% 0.35/0.57 = K )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ K )
% 0.35/0.57 => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Skolem_list_nth
% 0.35/0.57 thf(fact_2968_list__eq__iff__nth__eq,axiom,
% 0.35/0.57 ( ( ^ [Y6: list_int,Z3: list_int] : ( Y6 = Z3 ) )
% 0.35/0.57 = ( ^ [Xs2: list_int,Ys3: list_int] :
% 0.35/0.57 ( ( ( size_size_list_int @ Xs2 )
% 0.35/0.57 = ( size_size_list_int @ Ys3 ) )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 0.35/0.57 => ( ( nth_int @ Xs2 @ I4 )
% 0.35/0.57 = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_eq_iff_nth_eq
% 0.35/0.57 thf(fact_2969_list__eq__iff__nth__eq,axiom,
% 0.35/0.57 ( ( ^ [Y6: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : ( Y6 = Z3 ) )
% 0.35/0.57 = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.35/0.57 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.35/0.57 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.35/0.57 => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 0.35/0.57 = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_eq_iff_nth_eq
% 0.35/0.57 thf(fact_2970_list__eq__iff__nth__eq,axiom,
% 0.35/0.57 ( ( ^ [Y6: list_o,Z3: list_o] : ( Y6 = Z3 ) )
% 0.35/0.57 = ( ^ [Xs2: list_o,Ys3: list_o] :
% 0.35/0.57 ( ( ( size_size_list_o @ Xs2 )
% 0.35/0.57 = ( size_size_list_o @ Ys3 ) )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 0.35/0.57 => ( ( nth_o @ Xs2 @ I4 )
% 0.35/0.57 = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_eq_iff_nth_eq
% 0.35/0.57 thf(fact_2971_list__eq__iff__nth__eq,axiom,
% 0.35/0.57 ( ( ^ [Y6: list_nat,Z3: list_nat] : ( Y6 = Z3 ) )
% 0.35/0.57 = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 0.35/0.57 ( ( ( size_size_list_nat @ Xs2 )
% 0.35/0.57 = ( size_size_list_nat @ Ys3 ) )
% 0.35/0.57 & ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 0.35/0.57 => ( ( nth_nat @ Xs2 @ I4 )
% 0.35/0.57 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_eq_iff_nth_eq
% 0.35/0.57 thf(fact_2972_dense__eq0__I,axiom,
% 0.35/0.57 ! [X2: real] :
% 0.35/0.57 ( ! [E: real] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.57 => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E ) )
% 0.35/0.57 => ( X2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % dense_eq0_I
% 0.35/0.57 thf(fact_2973_dense__eq0__I,axiom,
% 0.35/0.57 ! [X2: rat] :
% 0.35/0.57 ( ! [E: rat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.35/0.57 => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E ) )
% 0.35/0.57 => ( X2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % dense_eq0_I
% 0.35/0.57 thf(fact_2974_eq__abs__iff_H,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( abs_abs_real @ B3 ) )
% 0.35/0.57 = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.57 & ( ( B3 = A2 )
% 0.35/0.57 | ( B3
% 0.35/0.57 = ( uminus_uminus_real @ A2 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % eq_abs_iff'
% 0.35/0.57 thf(fact_2975_eq__abs__iff_H,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( abs_abs_Code_integer @ B3 ) )
% 0.35/0.57 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.57 & ( ( B3 = A2 )
% 0.35/0.57 | ( B3
% 0.35/0.57 = ( uminus1351360451143612070nteger @ A2 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % eq_abs_iff'
% 0.35/0.57 thf(fact_2976_eq__abs__iff_H,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( abs_abs_rat @ B3 ) )
% 0.35/0.57 = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 & ( ( B3 = A2 )
% 0.35/0.57 | ( B3
% 0.35/0.57 = ( uminus_uminus_rat @ A2 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % eq_abs_iff'
% 0.35/0.57 thf(fact_2977_eq__abs__iff_H,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( abs_abs_int @ B3 ) )
% 0.35/0.57 = ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.57 & ( ( B3 = A2 )
% 0.35/0.57 | ( B3
% 0.35/0.57 = ( uminus_uminus_int @ A2 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % eq_abs_iff'
% 0.35/0.57 thf(fact_2978_abs__eq__iff_H,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ( abs_abs_real @ A2 )
% 0.35/0.57 = B3 )
% 0.35/0.57 = ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.57 & ( ( A2 = B3 )
% 0.35/0.57 | ( A2
% 0.35/0.57 = ( uminus_uminus_real @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_iff'
% 0.35/0.57 thf(fact_2979_abs__eq__iff_H,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = B3 )
% 0.35/0.57 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.57 & ( ( A2 = B3 )
% 0.35/0.57 | ( A2
% 0.35/0.57 = ( uminus1351360451143612070nteger @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_iff'
% 0.35/0.57 thf(fact_2980_abs__eq__iff_H,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = B3 )
% 0.35/0.57 = ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.57 & ( ( A2 = B3 )
% 0.35/0.57 | ( A2
% 0.35/0.57 = ( uminus_uminus_rat @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_iff'
% 0.35/0.57 thf(fact_2981_abs__eq__iff_H,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ( abs_abs_int @ A2 )
% 0.35/0.57 = B3 )
% 0.35/0.57 = ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.57 & ( ( A2 = B3 )
% 0.35/0.57 | ( A2
% 0.35/0.57 = ( uminus_uminus_int @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_eq_iff'
% 0.35/0.57 thf(fact_2982_abs__minus__le__zero,axiom,
% 0.35/0.57 ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A2 ) ) @ zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_le_zero
% 0.35/0.57 thf(fact_2983_abs__minus__le__zero,axiom,
% 0.35/0.57 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_le_zero
% 0.35/0.57 thf(fact_2984_abs__minus__le__zero,axiom,
% 0.35/0.57 ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A2 ) ) @ zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_le_zero
% 0.35/0.57 thf(fact_2985_abs__minus__le__zero,axiom,
% 0.35/0.57 ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % abs_minus_le_zero
% 0.35/0.57 thf(fact_2986_abs__div__pos,axiom,
% 0.35/0.57 ! [Y3: rat,X2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.35/0.57 => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y3 )
% 0.35/0.57 = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_div_pos
% 0.35/0.57 thf(fact_2987_abs__div__pos,axiom,
% 0.35/0.57 ! [Y3: real,X2: real] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.57 => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y3 )
% 0.35/0.57 = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_div_pos
% 0.35/0.57 thf(fact_2988_abs__if,axiom,
% 0.35/0.57 ( abs_abs_int
% 0.35/0.57 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if
% 0.35/0.57 thf(fact_2989_abs__if,axiom,
% 0.35/0.57 ( abs_abs_real
% 0.35/0.57 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if
% 0.35/0.57 thf(fact_2990_abs__if,axiom,
% 0.35/0.57 ( abs_abs_rat
% 0.35/0.57 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if
% 0.35/0.57 thf(fact_2991_abs__if,axiom,
% 0.35/0.57 ( abs_abs_Code_integer
% 0.35/0.57 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if
% 0.35/0.57 thf(fact_2992_abs__of__neg,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.57 => ( ( abs_abs_int @ A2 )
% 0.35/0.57 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_neg
% 0.35/0.57 thf(fact_2993_abs__of__neg,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.57 => ( ( abs_abs_real @ A2 )
% 0.35/0.57 = ( uminus_uminus_real @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_neg
% 0.35/0.57 thf(fact_2994_abs__of__neg,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.57 => ( ( abs_abs_rat @ A2 )
% 0.35/0.57 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_neg
% 0.35/0.57 thf(fact_2995_abs__of__neg,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.35/0.57 => ( ( abs_abs_Code_integer @ A2 )
% 0.35/0.57 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_of_neg
% 0.35/0.57 thf(fact_2996_abs__if__raw,axiom,
% 0.35/0.57 ( abs_abs_int
% 0.35/0.57 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if_raw
% 0.35/0.57 thf(fact_2997_abs__if__raw,axiom,
% 0.35/0.57 ( abs_abs_real
% 0.35/0.57 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if_raw
% 0.35/0.57 thf(fact_2998_abs__if__raw,axiom,
% 0.35/0.57 ( abs_abs_rat
% 0.35/0.57 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if_raw
% 0.35/0.57 thf(fact_2999_abs__if__raw,axiom,
% 0.35/0.57 ( abs_abs_Code_integer
% 0.35/0.57 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_if_raw
% 0.35/0.57 thf(fact_3000_length__removeAll__less__eq,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less_eq
% 0.35/0.57 thf(fact_3001_length__removeAll__less__eq,axiom,
% 0.35/0.57 ! [X2: $o,Xs: list_o] : ( ord_less_eq_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less_eq
% 0.35/0.57 thf(fact_3002_length__removeAll__less__eq,axiom,
% 0.35/0.57 ! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less_eq
% 0.35/0.57 thf(fact_3003_all__set__conv__all__nth,axiom,
% 0.35/0.57 ! [Xs: list_int,P: int > $o] :
% 0.35/0.57 ( ( ! [X: int] :
% 0.35/0.57 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( P @ X ) ) )
% 0.35/0.57 = ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_set_conv_all_nth
% 0.35/0.57 thf(fact_3004_all__set__conv__all__nth,axiom,
% 0.35/0.57 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.35/0.57 ( ( ! [X: vEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( P @ X ) ) )
% 0.35/0.57 = ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_set_conv_all_nth
% 0.35/0.57 thf(fact_3005_all__set__conv__all__nth,axiom,
% 0.35/0.57 ! [Xs: list_o,P: $o > $o] :
% 0.35/0.57 ( ( ! [X: $o] :
% 0.35/0.57 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( P @ X ) ) )
% 0.35/0.57 = ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_set_conv_all_nth
% 0.35/0.57 thf(fact_3006_all__set__conv__all__nth,axiom,
% 0.35/0.57 ! [Xs: list_nat,P: nat > $o] :
% 0.35/0.57 ( ( ! [X: nat] :
% 0.35/0.57 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X ) ) )
% 0.35/0.57 = ( ! [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_set_conv_all_nth
% 0.35/0.57 thf(fact_3007_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_real,P: real > $o,X2: real] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3008_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_set_nat,P: set_nat > $o,X2: set_nat] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3009_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_int,P: int > $o,X2: int] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3010_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3011_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_o,P: $o > $o,X2: $o] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3012_all__nth__imp__all__set,axiom,
% 0.35/0.57 ! [Xs: list_nat,P: nat > $o,X2: nat] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 0.35/0.57 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % all_nth_imp_all_set
% 0.35/0.57 thf(fact_3013_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: real,Xs: list_real] :
% 0.35/0.57 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 0.35/0.57 & ( ( nth_real @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3014_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: set_nat,Xs: list_set_nat] :
% 0.35/0.57 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.57 & ( ( nth_set_nat @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3015_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: int,Xs: list_int] :
% 0.35/0.57 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 & ( ( nth_int @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3016_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 & ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3017_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: $o,Xs: list_o] :
% 0.35/0.57 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 & ( ( nth_o @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3018_in__set__conv__nth,axiom,
% 0.35/0.57 ! [X2: nat,Xs: list_nat] :
% 0.35/0.57 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.35/0.57 = ( ? [I4: nat] :
% 0.35/0.57 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 & ( ( nth_nat @ Xs @ I4 )
% 0.35/0.57 = X2 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % in_set_conv_nth
% 0.35/0.57 thf(fact_3019_list__ball__nth,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_int,P: int > $o] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( ! [X5: int] :
% 0.35/0.57 ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_ball_nth
% 0.35/0.57 thf(fact_3020_list__ball__nth,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( ! [X5: vEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_ball_nth
% 0.35/0.57 thf(fact_3021_list__ball__nth,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_o,P: $o > $o] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( ! [X5: $o] :
% 0.35/0.57 ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_ball_nth
% 0.35/0.57 thf(fact_3022_list__ball__nth,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_nat,P: nat > $o] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( ! [X5: nat] :
% 0.35/0.57 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( P @ X5 ) )
% 0.35/0.57 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % list_ball_nth
% 0.35/0.57 thf(fact_3023_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_real] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.35/0.57 => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3024_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_set_nat] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.57 => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3025_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_int] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.57 => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3026_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.57 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3027_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_o] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.57 => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3028_nth__mem,axiom,
% 0.35/0.57 ! [N: nat,Xs: list_nat] :
% 0.35/0.57 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.57 => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nth_mem
% 0.35/0.57 thf(fact_3029_zabs__def,axiom,
% 0.35/0.57 ( abs_abs_int
% 0.35/0.57 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zabs_def
% 0.35/0.57 thf(fact_3030_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: real,Xs: list_real] :
% 0.35/0.57 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X2 @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3031_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: set_nat,Xs: list_set_nat] :
% 0.35/0.57 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3032_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: int,Xs: list_int] :
% 0.35/0.57 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3033_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.57 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3034_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: $o,Xs: list_o] :
% 0.35/0.57 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3035_length__removeAll__less,axiom,
% 0.35/0.57 ! [X2: nat,Xs: list_nat] :
% 0.35/0.57 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.35/0.57 => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % length_removeAll_less
% 0.35/0.57 thf(fact_3036_power__diff__power__eq,axiom,
% 0.35/0.57 ! [A2: int,N: nat,M: nat] :
% 0.35/0.57 ( ( A2 != zero_zero_int )
% 0.35/0.57 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.57 => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 0.35/0.57 = ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.35/0.57 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.57 => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 0.35/0.57 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_diff_power_eq
% 0.35/0.57 thf(fact_3037_power__diff__power__eq,axiom,
% 0.35/0.57 ! [A2: nat,N: nat,M: nat] :
% 0.35/0.57 ( ( A2 != zero_zero_nat )
% 0.35/0.57 => ( ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.57 => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 0.35/0.57 = ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
% 0.35/0.57 & ( ~ ( ord_less_eq_nat @ N @ M )
% 0.35/0.57 => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 0.35/0.57 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_diff_power_eq
% 0.35/0.57 thf(fact_3038_nat__intermed__int__val,axiom,
% 0.35/0.57 ! [M: nat,N: nat,F: nat > int,K: int] :
% 0.35/0.57 ( ! [I2: nat] :
% 0.35/0.57 ( ( ( ord_less_eq_nat @ M @ I2 )
% 0.35/0.57 & ( ord_less_nat @ I2 @ N ) )
% 0.35/0.57 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.57 => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.35/0.57 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.57 => ? [I2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ M @ I2 )
% 0.35/0.57 & ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.57 & ( ( F @ I2 )
% 0.35/0.57 = K ) ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_intermed_int_val
% 0.35/0.57 thf(fact_3039_of__nat__zero__less__power__iff,axiom,
% 0.35/0.57 ! [X2: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_zero_less_power_iff
% 0.35/0.57 thf(fact_3040_of__nat__zero__less__power__iff,axiom,
% 0.35/0.57 ! [X2: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_zero_less_power_iff
% 0.35/0.57 thf(fact_3041_of__nat__zero__less__power__iff,axiom,
% 0.35/0.57 ! [X2: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_zero_less_power_iff
% 0.35/0.57 thf(fact_3042_of__nat__zero__less__power__iff,axiom,
% 0.35/0.57 ! [X2: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_zero_less_power_iff
% 0.35/0.57 thf(fact_3043_power__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: real,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.57 => ( ( ord_less_real @ B3 @ one_one_real )
% 0.35/0.57 => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_decreasing_iff
% 0.35/0.57 thf(fact_3044_power__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: rat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.57 => ( ( ord_less_rat @ B3 @ one_one_rat )
% 0.35/0.57 => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_decreasing_iff
% 0.35/0.57 thf(fact_3045_power__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: nat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ B3 @ one_one_nat )
% 0.35/0.57 => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_decreasing_iff
% 0.35/0.57 thf(fact_3046_power__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: int,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.57 => ( ( ord_less_int @ B3 @ one_one_int )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_decreasing_iff
% 0.35/0.57 thf(fact_3047_zero__less__power__abs__iff,axiom,
% 0.35/0.57 ! [A2: code_integer,N: nat] :
% 0.35/0.57 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) )
% 0.35/0.57 = ( ( A2 != zero_z3403309356797280102nteger )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_power_abs_iff
% 0.35/0.57 thf(fact_3048_zero__less__power__abs__iff,axiom,
% 0.35/0.57 ! [A2: real,N: nat] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
% 0.35/0.57 = ( ( A2 != zero_zero_real )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_power_abs_iff
% 0.35/0.57 thf(fact_3049_zero__less__power__abs__iff,axiom,
% 0.35/0.57 ! [A2: rat,N: nat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) )
% 0.35/0.57 = ( ( A2 != zero_zero_rat )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_power_abs_iff
% 0.35/0.57 thf(fact_3050_zero__less__power__abs__iff,axiom,
% 0.35/0.57 ! [A2: int,N: nat] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) )
% 0.35/0.57 = ( ( A2 != zero_zero_int )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_power_abs_iff
% 0.35/0.57 thf(fact_3051_power__mono__iff,axiom,
% 0.35/0.57 ! [A2: real,B3: real,N: nat] :
% 0.35/0.57 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.57 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.57 => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_mono_iff
% 0.35/0.57 thf(fact_3052_power__mono__iff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat,N: nat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.57 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_mono_iff
% 0.35/0.57 thf(fact_3053_power__mono__iff,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.57 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.57 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_mono_iff
% 0.35/0.57 thf(fact_3054_power__mono__iff,axiom,
% 0.35/0.57 ! [A2: int,B3: int,N: nat] :
% 0.35/0.57 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.57 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_mono_iff
% 0.35/0.57 thf(fact_3055_power__increasing__iff,axiom,
% 0.35/0.57 ! [B3: real,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_increasing_iff
% 0.35/0.57 thf(fact_3056_power__increasing__iff,axiom,
% 0.35/0.57 ! [B3: rat,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_rat @ one_one_rat @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_increasing_iff
% 0.35/0.57 thf(fact_3057_power__increasing__iff,axiom,
% 0.35/0.57 ! [B3: nat,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ one_one_nat @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_increasing_iff
% 0.35/0.57 thf(fact_3058_power__increasing__iff,axiom,
% 0.35/0.57 ! [B3: int,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_int @ one_one_int @ B3 )
% 0.35/0.57 => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_increasing_iff
% 0.35/0.57 thf(fact_3059_power__strict__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: real,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.57 => ( ( ord_less_real @ B3 @ one_one_real )
% 0.35/0.57 => ( ( ord_less_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_decreasing_iff
% 0.35/0.57 thf(fact_3060_power__strict__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: rat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.57 => ( ( ord_less_rat @ B3 @ one_one_rat )
% 0.35/0.57 => ( ( ord_less_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_decreasing_iff
% 0.35/0.57 thf(fact_3061_power__strict__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: nat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ B3 @ one_one_nat )
% 0.35/0.57 => ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_decreasing_iff
% 0.35/0.57 thf(fact_3062_power__strict__decreasing__iff,axiom,
% 0.35/0.57 ! [B3: int,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.57 => ( ( ord_less_int @ B3 @ one_one_int )
% 0.35/0.57 => ( ( ord_less_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
% 0.35/0.57 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_decreasing_iff
% 0.35/0.57 thf(fact_3063_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3064_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3065_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3066_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_le_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3067_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.57 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.57 thf(fact_3068_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.57 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.57 thf(fact_3069_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.57 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.57 thf(fact_3070_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.35/0.57 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_le_of_nat_power_cancel_iff
% 0.35/0.57 thf(fact_3071_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3072_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3073_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3074_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.57 ! [X2: nat,B3: nat,W2: nat] :
% 0.35/0.57 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_power_less_of_nat_cancel_iff
% 0.35/0.57 thf(fact_3075_even__odd__cases,axiom,
% 0.35/0.57 ! [X2: nat] :
% 0.35/0.57 ( ! [N3: nat] :
% 0.35/0.57 ( X2
% 0.35/0.57 != ( plus_plus_nat @ N3 @ N3 ) )
% 0.35/0.57 => ~ ! [N3: nat] :
% 0.35/0.57 ( X2
% 0.35/0.57 != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % even_odd_cases
% 0.35/0.57 thf(fact_3076_local_Opower__def,axiom,
% 0.35/0.57 ( vEBT_VEBT_power
% 0.35/0.57 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % local.power_def
% 0.35/0.57 thf(fact_3077_power__shift,axiom,
% 0.35/0.57 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.57 ( ( ( power_power_nat @ X2 @ Y3 )
% 0.35/0.57 = Z )
% 0.35/0.57 = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.35/0.57 = ( some_nat @ Z ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_shift
% 0.35/0.57 thf(fact_3078_add__left__cancel,axiom,
% 0.35/0.57 ! [A2: real,B3: real,C: real] :
% 0.35/0.57 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.57 = ( plus_plus_real @ A2 @ C ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_left_cancel
% 0.35/0.57 thf(fact_3079_add__left__cancel,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.57 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.57 = ( plus_plus_rat @ A2 @ C ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_left_cancel
% 0.35/0.57 thf(fact_3080_add__left__cancel,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ A2 @ B3 )
% 0.35/0.57 = ( plus_plus_nat @ A2 @ C ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_left_cancel
% 0.35/0.57 thf(fact_3081_add__left__cancel,axiom,
% 0.35/0.57 ! [A2: int,B3: int,C: int] :
% 0.35/0.57 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.57 = ( plus_plus_int @ A2 @ C ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_left_cancel
% 0.35/0.57 thf(fact_3082_add__right__cancel,axiom,
% 0.35/0.57 ! [B3: real,A2: real,C: real] :
% 0.35/0.57 ( ( ( plus_plus_real @ B3 @ A2 )
% 0.35/0.57 = ( plus_plus_real @ C @ A2 ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_right_cancel
% 0.35/0.57 thf(fact_3083_add__right__cancel,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.57 ( ( ( plus_plus_rat @ B3 @ A2 )
% 0.35/0.57 = ( plus_plus_rat @ C @ A2 ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_right_cancel
% 0.35/0.57 thf(fact_3084_add__right__cancel,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ B3 @ A2 )
% 0.35/0.57 = ( plus_plus_nat @ C @ A2 ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_right_cancel
% 0.35/0.57 thf(fact_3085_add__right__cancel,axiom,
% 0.35/0.57 ! [B3: int,A2: int,C: int] :
% 0.35/0.57 ( ( ( plus_plus_int @ B3 @ A2 )
% 0.35/0.57 = ( plus_plus_int @ C @ A2 ) )
% 0.35/0.57 = ( B3 = C ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_right_cancel
% 0.35/0.57 thf(fact_3086_add__le__cancel__right,axiom,
% 0.35/0.57 ! [A2: real,C: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_right
% 0.35/0.57 thf(fact_3087_add__le__cancel__right,axiom,
% 0.35/0.57 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_right
% 0.35/0.57 thf(fact_3088_add__le__cancel__right,axiom,
% 0.35/0.57 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_right
% 0.35/0.57 thf(fact_3089_add__le__cancel__right,axiom,
% 0.35/0.57 ! [A2: int,C: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_right
% 0.35/0.57 thf(fact_3090_add__le__cancel__left,axiom,
% 0.35/0.57 ! [C: real,A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_left
% 0.35/0.57 thf(fact_3091_add__le__cancel__left,axiom,
% 0.35/0.57 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_left
% 0.35/0.57 thf(fact_3092_add__le__cancel__left,axiom,
% 0.35/0.57 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_left
% 0.35/0.57 thf(fact_3093_add__le__cancel__left,axiom,
% 0.35/0.57 ! [C: int,A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_cancel_left
% 0.35/0.57 thf(fact_3094_double__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ( plus_plus_real @ A2 @ A2 )
% 0.35/0.57 = zero_zero_real )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_eq_0_iff
% 0.35/0.57 thf(fact_3095_double__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ( plus_plus_rat @ A2 @ A2 )
% 0.35/0.57 = zero_zero_rat )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_eq_0_iff
% 0.35/0.57 thf(fact_3096_double__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ( plus_plus_int @ A2 @ A2 )
% 0.35/0.57 = zero_zero_int )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_eq_0_iff
% 0.35/0.57 thf(fact_3097_add__0,axiom,
% 0.35/0.57 ! [A2: literal] :
% 0.35/0.57 ( ( plus_plus_literal @ zero_zero_literal @ A2 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_0
% 0.35/0.57 thf(fact_3098_add__0,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_0
% 0.35/0.57 thf(fact_3099_add__0,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_0
% 0.35/0.57 thf(fact_3100_add__0,axiom,
% 0.35/0.57 ! [A2: nat] :
% 0.35/0.57 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_0
% 0.35/0.57 thf(fact_3101_add__0,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_0
% 0.35/0.57 thf(fact_3102_zero__eq__add__iff__both__eq__0,axiom,
% 0.35/0.57 ! [X2: nat,Y3: nat] :
% 0.35/0.57 ( ( zero_zero_nat
% 0.35/0.57 = ( plus_plus_nat @ X2 @ Y3 ) )
% 0.35/0.57 = ( ( X2 = zero_zero_nat )
% 0.35/0.57 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_eq_add_iff_both_eq_0
% 0.35/0.57 thf(fact_3103_add__eq__0__iff__both__eq__0,axiom,
% 0.35/0.57 ! [X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.35/0.57 = zero_zero_nat )
% 0.35/0.57 = ( ( X2 = zero_zero_nat )
% 0.35/0.57 & ( Y3 = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_eq_0_iff_both_eq_0
% 0.35/0.57 thf(fact_3104_add__cancel__right__right,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.57 = ( B3 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_right
% 0.35/0.57 thf(fact_3105_add__cancel__right__right,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( B3 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_right
% 0.35/0.57 thf(fact_3106_add__cancel__right__right,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_nat @ A2 @ B3 ) )
% 0.35/0.57 = ( B3 = zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_right
% 0.35/0.57 thf(fact_3107_add__cancel__right__right,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.57 = ( B3 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_right
% 0.35/0.57 thf(fact_3108_add__cancel__right__left,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_real @ B3 @ A2 ) )
% 0.35/0.57 = ( B3 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_left
% 0.35/0.57 thf(fact_3109_add__cancel__right__left,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_rat @ B3 @ A2 ) )
% 0.35/0.57 = ( B3 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_left
% 0.35/0.57 thf(fact_3110_add__cancel__right__left,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_nat @ B3 @ A2 ) )
% 0.35/0.57 = ( B3 = zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_left
% 0.35/0.57 thf(fact_3111_add__cancel__right__left,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( A2
% 0.35/0.57 = ( plus_plus_int @ B3 @ A2 ) )
% 0.35/0.57 = ( B3 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_right_left
% 0.35/0.57 thf(fact_3112_add__cancel__left__right,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_right
% 0.35/0.57 thf(fact_3113_add__cancel__left__right,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_right
% 0.35/0.57 thf(fact_3114_add__cancel__left__right,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ A2 @ B3 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_right
% 0.35/0.57 thf(fact_3115_add__cancel__left__right,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_right
% 0.35/0.57 thf(fact_3116_add__cancel__left__left,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( ( plus_plus_real @ B3 @ A2 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_left
% 0.35/0.57 thf(fact_3117_add__cancel__left__left,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( ( plus_plus_rat @ B3 @ A2 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_left
% 0.35/0.57 thf(fact_3118_add__cancel__left__left,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ B3 @ A2 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_left
% 0.35/0.57 thf(fact_3119_add__cancel__left__left,axiom,
% 0.35/0.57 ! [B3: int,A2: int] :
% 0.35/0.57 ( ( ( plus_plus_int @ B3 @ A2 )
% 0.35/0.57 = A2 )
% 0.35/0.57 = ( B3 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_cancel_left_left
% 0.35/0.57 thf(fact_3120_double__zero__sym,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( zero_zero_real
% 0.35/0.57 = ( plus_plus_real @ A2 @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_zero_sym
% 0.35/0.57 thf(fact_3121_double__zero__sym,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( zero_zero_rat
% 0.35/0.57 = ( plus_plus_rat @ A2 @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_zero_sym
% 0.35/0.57 thf(fact_3122_double__zero__sym,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( zero_zero_int
% 0.35/0.57 = ( plus_plus_int @ A2 @ A2 ) )
% 0.35/0.57 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_zero_sym
% 0.35/0.57 thf(fact_3123_add_Oright__neutral,axiom,
% 0.35/0.57 ! [A2: literal] :
% 0.35/0.57 ( ( plus_plus_literal @ A2 @ zero_zero_literal )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_neutral
% 0.35/0.57 thf(fact_3124_add_Oright__neutral,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_neutral
% 0.35/0.57 thf(fact_3125_add_Oright__neutral,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_neutral
% 0.35/0.57 thf(fact_3126_add_Oright__neutral,axiom,
% 0.35/0.57 ! [A2: nat] :
% 0.35/0.57 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_neutral
% 0.35/0.57 thf(fact_3127_add_Oright__neutral,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_neutral
% 0.35/0.57 thf(fact_3128_add__less__cancel__left,axiom,
% 0.35/0.57 ! [C: real,A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_left
% 0.35/0.57 thf(fact_3129_add__less__cancel__left,axiom,
% 0.35/0.57 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_left
% 0.35/0.57 thf(fact_3130_add__less__cancel__left,axiom,
% 0.35/0.57 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_left
% 0.35/0.57 thf(fact_3131_add__less__cancel__left,axiom,
% 0.35/0.57 ! [C: int,A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 0.35/0.57 = ( ord_less_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_left
% 0.35/0.57 thf(fact_3132_add__less__cancel__right,axiom,
% 0.35/0.57 ! [A2: real,C: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_right
% 0.35/0.57 thf(fact_3133_add__less__cancel__right,axiom,
% 0.35/0.57 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_right
% 0.35/0.57 thf(fact_3134_add__less__cancel__right,axiom,
% 0.35/0.57 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_right
% 0.35/0.57 thf(fact_3135_add__less__cancel__right,axiom,
% 0.35/0.57 ! [A2: int,C: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.57 = ( ord_less_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_cancel_right
% 0.35/0.57 thf(fact_3136_add__diff__cancel,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel
% 0.35/0.57 thf(fact_3137_add__diff__cancel,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel
% 0.35/0.57 thf(fact_3138_add__diff__cancel,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel
% 0.35/0.57 thf(fact_3139_diff__add__cancel,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % diff_add_cancel
% 0.35/0.57 thf(fact_3140_diff__add__cancel,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % diff_add_cancel
% 0.35/0.57 thf(fact_3141_diff__add__cancel,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % diff_add_cancel
% 0.35/0.57 thf(fact_3142_add__diff__cancel__left,axiom,
% 0.35/0.57 ! [C: real,A2: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 0.35/0.57 = ( minus_minus_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left
% 0.35/0.57 thf(fact_3143_add__diff__cancel__left,axiom,
% 0.35/0.57 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 0.35/0.57 = ( minus_minus_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left
% 0.35/0.57 thf(fact_3144_add__diff__cancel__left,axiom,
% 0.35/0.57 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.35/0.57 = ( minus_minus_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left
% 0.35/0.57 thf(fact_3145_add__diff__cancel__left,axiom,
% 0.35/0.57 ! [C: int,A2: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 0.35/0.57 = ( minus_minus_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left
% 0.35/0.57 thf(fact_3146_add__diff__cancel__left_H,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ A2 )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left'
% 0.35/0.57 thf(fact_3147_add__diff__cancel__left_H,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ A2 )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left'
% 0.35/0.57 thf(fact_3148_add__diff__cancel__left_H,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ A2 )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left'
% 0.35/0.57 thf(fact_3149_add__diff__cancel__left_H,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ A2 )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_left'
% 0.35/0.57 thf(fact_3150_add__diff__cancel__right,axiom,
% 0.35/0.57 ! [A2: real,C: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.57 = ( minus_minus_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right
% 0.35/0.57 thf(fact_3151_add__diff__cancel__right,axiom,
% 0.35/0.57 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.57 = ( minus_minus_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right
% 0.35/0.57 thf(fact_3152_add__diff__cancel__right,axiom,
% 0.35/0.57 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.35/0.57 = ( minus_minus_nat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right
% 0.35/0.57 thf(fact_3153_add__diff__cancel__right,axiom,
% 0.35/0.57 ! [A2: int,C: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.57 = ( minus_minus_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right
% 0.35/0.57 thf(fact_3154_add__diff__cancel__right_H,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right'
% 0.35/0.57 thf(fact_3155_add__diff__cancel__right_H,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right'
% 0.35/0.57 thf(fact_3156_add__diff__cancel__right_H,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right'
% 0.35/0.57 thf(fact_3157_add__diff__cancel__right_H,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % add_diff_cancel_right'
% 0.35/0.57 thf(fact_3158_add__minus__cancel,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_minus_cancel
% 0.35/0.57 thf(fact_3159_add__minus__cancel,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_minus_cancel
% 0.35/0.57 thf(fact_3160_add__minus__cancel,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_minus_cancel
% 0.35/0.57 thf(fact_3161_add__minus__cancel,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_minus_cancel
% 0.35/0.57 thf(fact_3162_add__minus__cancel,axiom,
% 0.35/0.57 ! [A2: complex,B3: complex] :
% 0.35/0.57 ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % add_minus_cancel
% 0.35/0.57 thf(fact_3163_minus__add__cancel,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_cancel
% 0.35/0.57 thf(fact_3164_minus__add__cancel,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_cancel
% 0.35/0.57 thf(fact_3165_minus__add__cancel,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_cancel
% 0.35/0.57 thf(fact_3166_minus__add__cancel,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_cancel
% 0.35/0.57 thf(fact_3167_minus__add__cancel,axiom,
% 0.35/0.57 ! [A2: complex,B3: complex] :
% 0.35/0.57 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B3 ) )
% 0.35/0.57 = B3 ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_cancel
% 0.35/0.57 thf(fact_3168_minus__add__distrib,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.57 = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_distrib
% 0.35/0.57 thf(fact_3169_minus__add__distrib,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.57 = ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_distrib
% 0.35/0.57 thf(fact_3170_minus__add__distrib,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_distrib
% 0.35/0.57 thf(fact_3171_minus__add__distrib,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 0.35/0.57 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_distrib
% 0.35/0.57 thf(fact_3172_minus__add__distrib,axiom,
% 0.35/0.57 ! [A2: complex,B3: complex] :
% 0.35/0.57 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
% 0.35/0.57 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % minus_add_distrib
% 0.35/0.57 thf(fact_3173_abs__add__abs,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) )
% 0.35/0.57 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_add_abs
% 0.35/0.57 thf(fact_3174_abs__add__abs,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) )
% 0.35/0.57 = ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_add_abs
% 0.35/0.57 thf(fact_3175_abs__add__abs,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) )
% 0.35/0.57 = ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_add_abs
% 0.35/0.57 thf(fact_3176_abs__add__abs,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) )
% 0.35/0.57 = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % abs_add_abs
% 0.35/0.57 thf(fact_3177_add__Suc__right,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 0.35/0.57 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_Suc_right
% 0.35/0.57 thf(fact_3178_Nat_Oadd__0__right,axiom,
% 0.35/0.57 ! [M: nat] :
% 0.35/0.57 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.35/0.57 = M ) ).
% 0.35/0.57
% 0.35/0.57 % Nat.add_0_right
% 0.35/0.57 thf(fact_3179_add__is__0,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( ( plus_plus_nat @ M @ N )
% 0.35/0.57 = zero_zero_nat )
% 0.35/0.57 = ( ( M = zero_zero_nat )
% 0.35/0.57 & ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_is_0
% 0.35/0.57 thf(fact_3180_nat__add__left__cancel__less,axiom,
% 0.35/0.57 ! [K: nat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.35/0.57 = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_add_left_cancel_less
% 0.35/0.57 thf(fact_3181_nat__add__left__cancel__le,axiom,
% 0.35/0.57 ! [K: nat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.35/0.57 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_add_left_cancel_le
% 0.35/0.57 thf(fact_3182_diff__diff__left,axiom,
% 0.35/0.57 ! [I: nat,J: nat,K: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.35/0.57 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_diff_left
% 0.35/0.57 thf(fact_3183_add__le__same__cancel1,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel1
% 0.35/0.57 thf(fact_3184_add__le__same__cancel1,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel1
% 0.35/0.57 thf(fact_3185_add__le__same__cancel1,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel1
% 0.35/0.57 thf(fact_3186_add__le__same__cancel1,axiom,
% 0.35/0.57 ! [B3: int,A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel1
% 0.35/0.57 thf(fact_3187_add__le__same__cancel2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel2
% 0.35/0.57 thf(fact_3188_add__le__same__cancel2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel2
% 0.35/0.57 thf(fact_3189_add__le__same__cancel2,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel2
% 0.35/0.57 thf(fact_3190_add__le__same__cancel2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_le_same_cancel2
% 0.35/0.57 thf(fact_3191_le__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel1
% 0.35/0.57 thf(fact_3192_le__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel1
% 0.35/0.57 thf(fact_3193_le__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel1
% 0.35/0.57 thf(fact_3194_le__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel1
% 0.35/0.57 thf(fact_3195_le__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel2
% 0.35/0.57 thf(fact_3196_le__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel2
% 0.35/0.57 thf(fact_3197_le__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel2
% 0.35/0.57 thf(fact_3198_le__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_same_cancel2
% 0.35/0.57 thf(fact_3199_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
% 0.35/0.57 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_le_zero_iff_single_add_le_zero
% 0.35/0.57 thf(fact_3200_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 0.35/0.57 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_le_zero_iff_single_add_le_zero
% 0.35/0.57 thf(fact_3201_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 0.35/0.57 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_le_zero_iff_single_add_le_zero
% 0.35/0.57 thf(fact_3202_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_le_double_add_iff_zero_le_single_add
% 0.35/0.57 thf(fact_3203_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_le_double_add_iff_zero_le_single_add
% 0.35/0.57 thf(fact_3204_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_le_double_add_iff_zero_le_single_add
% 0.35/0.57 thf(fact_3205_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_double_add_iff_zero_less_single_add
% 0.35/0.57 thf(fact_3206_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_double_add_iff_zero_less_single_add
% 0.35/0.57 thf(fact_3207_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 0.35/0.57 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % zero_less_double_add_iff_zero_less_single_add
% 0.35/0.57 thf(fact_3208_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
% 0.35/0.57 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_less_zero_iff_single_add_less_zero
% 0.35/0.57 thf(fact_3209_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 0.35/0.57 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_less_zero_iff_single_add_less_zero
% 0.35/0.57 thf(fact_3210_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 0.35/0.57 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % double_add_less_zero_iff_single_add_less_zero
% 0.35/0.57 thf(fact_3211_less__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel2
% 0.35/0.57 thf(fact_3212_less__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel2
% 0.35/0.57 thf(fact_3213_less__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel2
% 0.35/0.57 thf(fact_3214_less__add__same__cancel2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
% 0.35/0.57 = ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel2
% 0.35/0.57 thf(fact_3215_less__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel1
% 0.35/0.57 thf(fact_3216_less__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel1
% 0.35/0.57 thf(fact_3217_less__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel1
% 0.35/0.57 thf(fact_3218_less__add__same__cancel1,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.57 = ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % less_add_same_cancel1
% 0.35/0.57 thf(fact_3219_add__less__same__cancel2,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel2
% 0.35/0.57 thf(fact_3220_add__less__same__cancel2,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel2
% 0.35/0.57 thf(fact_3221_add__less__same__cancel2,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel2
% 0.35/0.57 thf(fact_3222_add__less__same__cancel2,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel2
% 0.35/0.57 thf(fact_3223_add__less__same__cancel1,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( ord_less_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel1
% 0.35/0.57 thf(fact_3224_add__less__same__cancel1,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( ord_less_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel1
% 0.35/0.57 thf(fact_3225_add__less__same__cancel1,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat] :
% 0.35/0.57 ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel1
% 0.35/0.57 thf(fact_3226_add__less__same__cancel1,axiom,
% 0.35/0.57 ! [B3: int,A2: int] :
% 0.35/0.57 ( ( ord_less_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 0.35/0.57 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_less_same_cancel1
% 0.35/0.57 thf(fact_3227_le__add__diff__inverse2,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse2
% 0.35/0.57 thf(fact_3228_le__add__diff__inverse2,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse2
% 0.35/0.57 thf(fact_3229_le__add__diff__inverse2,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse2
% 0.35/0.57 thf(fact_3230_le__add__diff__inverse2,axiom,
% 0.35/0.57 ! [B3: int,A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse2
% 0.35/0.57 thf(fact_3231_le__add__diff__inverse,axiom,
% 0.35/0.57 ! [B3: real,A2: real] :
% 0.35/0.57 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse
% 0.35/0.57 thf(fact_3232_le__add__diff__inverse,axiom,
% 0.35/0.57 ! [B3: rat,A2: rat] :
% 0.35/0.57 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse
% 0.35/0.57 thf(fact_3233_le__add__diff__inverse,axiom,
% 0.35/0.57 ! [B3: nat,A2: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse
% 0.35/0.57 thf(fact_3234_le__add__diff__inverse,axiom,
% 0.35/0.57 ! [B3: int,A2: int] :
% 0.35/0.57 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.57 => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
% 0.35/0.57 = A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % le_add_diff_inverse
% 0.35/0.57 thf(fact_3235_diff__add__zero,axiom,
% 0.35/0.57 ! [A2: nat,B3: nat] :
% 0.35/0.57 ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 0.35/0.57 = zero_zero_nat ) ).
% 0.35/0.57
% 0.35/0.57 % diff_add_zero
% 0.35/0.57 thf(fact_3236_add_Oright__inverse,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_inverse
% 0.35/0.57 thf(fact_3237_add_Oright__inverse,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_inverse
% 0.35/0.57 thf(fact_3238_add_Oright__inverse,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_inverse
% 0.35/0.57 thf(fact_3239_add_Oright__inverse,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_inverse
% 0.35/0.57 thf(fact_3240_add_Oright__inverse,axiom,
% 0.35/0.57 ! [A2: complex] :
% 0.35/0.57 ( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
% 0.35/0.57 = zero_zero_complex ) ).
% 0.35/0.57
% 0.35/0.57 % add.right_inverse
% 0.35/0.57 thf(fact_3241_ab__left__minus,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % ab_left_minus
% 0.35/0.57 thf(fact_3242_ab__left__minus,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % ab_left_minus
% 0.35/0.57 thf(fact_3243_ab__left__minus,axiom,
% 0.35/0.57 ! [A2: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % ab_left_minus
% 0.35/0.57 thf(fact_3244_ab__left__minus,axiom,
% 0.35/0.57 ! [A2: code_integer] :
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % ab_left_minus
% 0.35/0.57 thf(fact_3245_ab__left__minus,axiom,
% 0.35/0.57 ! [A2: complex] :
% 0.35/0.57 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
% 0.35/0.57 = zero_zero_complex ) ).
% 0.35/0.57
% 0.35/0.57 % ab_left_minus
% 0.35/0.57 thf(fact_3246_power__inject__exp,axiom,
% 0.35/0.57 ! [A2: real,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.57 => ( ( ( power_power_real @ A2 @ M )
% 0.35/0.57 = ( power_power_real @ A2 @ N ) )
% 0.35/0.57 = ( M = N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_inject_exp
% 0.35/0.57 thf(fact_3247_power__inject__exp,axiom,
% 0.35/0.57 ! [A2: rat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.57 => ( ( ( power_power_rat @ A2 @ M )
% 0.35/0.57 = ( power_power_rat @ A2 @ N ) )
% 0.35/0.57 = ( M = N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_inject_exp
% 0.35/0.57 thf(fact_3248_power__inject__exp,axiom,
% 0.35/0.57 ! [A2: nat,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.57 => ( ( ( power_power_nat @ A2 @ M )
% 0.35/0.57 = ( power_power_nat @ A2 @ N ) )
% 0.35/0.57 = ( M = N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_inject_exp
% 0.35/0.57 thf(fact_3249_power__inject__exp,axiom,
% 0.35/0.57 ! [A2: int,M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.57 => ( ( ( power_power_int @ A2 @ M )
% 0.35/0.57 = ( power_power_int @ A2 @ N ) )
% 0.35/0.57 = ( M = N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_inject_exp
% 0.35/0.57 thf(fact_3250_power__0__Suc,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % power_0_Suc
% 0.35/0.57 thf(fact_3251_power__0__Suc,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % power_0_Suc
% 0.35/0.57 thf(fact_3252_power__0__Suc,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.57 = zero_zero_nat ) ).
% 0.35/0.57
% 0.35/0.57 % power_0_Suc
% 0.35/0.57 thf(fact_3253_power__0__Suc,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % power_0_Suc
% 0.35/0.57 thf(fact_3254_power__0__Suc,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 0.35/0.57 = zero_zero_complex ) ).
% 0.35/0.57
% 0.35/0.57 % power_0_Suc
% 0.35/0.57 thf(fact_3255_diff__minus__eq__add,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 0.35/0.57 = ( plus_plus_int @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_minus_eq_add
% 0.35/0.57 thf(fact_3256_diff__minus__eq__add,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 0.35/0.57 = ( plus_plus_real @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_minus_eq_add
% 0.35/0.57 thf(fact_3257_diff__minus__eq__add,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 0.35/0.57 = ( plus_plus_rat @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_minus_eq_add
% 0.35/0.57 thf(fact_3258_diff__minus__eq__add,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 0.35/0.57 = ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_minus_eq_add
% 0.35/0.57 thf(fact_3259_diff__minus__eq__add,axiom,
% 0.35/0.57 ! [A2: complex,B3: complex] :
% 0.35/0.57 ( ( minus_minus_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
% 0.35/0.57 = ( plus_plus_complex @ A2 @ B3 ) ) ).
% 0.35/0.57
% 0.35/0.57 % diff_minus_eq_add
% 0.35/0.57 thf(fact_3260_uminus__add__conv__diff,axiom,
% 0.35/0.57 ! [A2: int,B3: int] :
% 0.35/0.57 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 0.35/0.57 = ( minus_minus_int @ B3 @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % uminus_add_conv_diff
% 0.35/0.57 thf(fact_3261_uminus__add__conv__diff,axiom,
% 0.35/0.57 ! [A2: real,B3: real] :
% 0.35/0.57 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 0.35/0.57 = ( minus_minus_real @ B3 @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % uminus_add_conv_diff
% 0.35/0.57 thf(fact_3262_uminus__add__conv__diff,axiom,
% 0.35/0.57 ! [A2: rat,B3: rat] :
% 0.35/0.57 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 0.35/0.57 = ( minus_minus_rat @ B3 @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % uminus_add_conv_diff
% 0.35/0.57 thf(fact_3263_uminus__add__conv__diff,axiom,
% 0.35/0.57 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 0.35/0.57 = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % uminus_add_conv_diff
% 0.35/0.57 thf(fact_3264_uminus__add__conv__diff,axiom,
% 0.35/0.57 ! [A2: complex,B3: complex] :
% 0.35/0.57 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
% 0.35/0.57 = ( minus_minus_complex @ B3 @ A2 ) ) ).
% 0.35/0.57
% 0.35/0.57 % uminus_add_conv_diff
% 0.35/0.57 thf(fact_3265_power__Suc0__right,axiom,
% 0.35/0.57 ! [A2: int] :
% 0.35/0.57 ( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % power_Suc0_right
% 0.35/0.57 thf(fact_3266_power__Suc0__right,axiom,
% 0.35/0.57 ! [A2: nat] :
% 0.35/0.57 ( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % power_Suc0_right
% 0.35/0.57 thf(fact_3267_power__Suc0__right,axiom,
% 0.35/0.57 ! [A2: real] :
% 0.35/0.57 ( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % power_Suc0_right
% 0.35/0.57 thf(fact_3268_power__Suc0__right,axiom,
% 0.35/0.57 ! [A2: complex] :
% 0.35/0.57 ( ( power_power_complex @ A2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.57 = A2 ) ).
% 0.35/0.57
% 0.35/0.57 % power_Suc0_right
% 0.35/0.57 thf(fact_3269_of__nat__add,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.57 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_add
% 0.35/0.57 thf(fact_3270_of__nat__add,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.57 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_add
% 0.35/0.57 thf(fact_3271_of__nat__add,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.57 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_add
% 0.35/0.57 thf(fact_3272_of__nat__add,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.57 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % of_nat_add
% 0.35/0.57 thf(fact_3273_add__gr__0,axiom,
% 0.35/0.57 ! [M: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.57 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % add_gr_0
% 0.35/0.57 thf(fact_3274_power__Suc__0,axiom,
% 0.35/0.57 ! [N: nat] :
% 0.35/0.57 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.57 = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_Suc_0
% 0.35/0.57 thf(fact_3275_nat__power__eq__Suc__0__iff,axiom,
% 0.35/0.57 ! [X2: nat,M: nat] :
% 0.35/0.57 ( ( ( power_power_nat @ X2 @ M )
% 0.35/0.57 = ( suc @ zero_zero_nat ) )
% 0.35/0.57 = ( ( M = zero_zero_nat )
% 0.35/0.57 | ( X2
% 0.35/0.57 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_power_eq_Suc_0_iff
% 0.35/0.57 thf(fact_3276_nat__zero__less__power__iff,axiom,
% 0.35/0.57 ! [X2: nat,N: nat] :
% 0.35/0.57 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
% 0.35/0.57 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 0.35/0.57 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % nat_zero_less_power_iff
% 0.35/0.57 thf(fact_3277_Nat_Odiff__diff__right,axiom,
% 0.35/0.57 ! [K: nat,J: nat,I: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.57 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.57 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Nat.diff_diff_right
% 0.35/0.57 thf(fact_3278_Nat_Oadd__diff__assoc2,axiom,
% 0.35/0.57 ! [K: nat,J: nat,I: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.57 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.57 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Nat.add_diff_assoc2
% 0.35/0.57 thf(fact_3279_Nat_Oadd__diff__assoc,axiom,
% 0.35/0.57 ! [K: nat,J: nat,I: nat] :
% 0.35/0.57 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.57 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.57 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % Nat.add_diff_assoc
% 0.35/0.57 thf(fact_3280_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.57 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(8)
% 0.35/0.57 thf(fact_3281_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.57 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(8)
% 0.35/0.57 thf(fact_3282_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.57 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(8)
% 0.35/0.57 thf(fact_3283_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(8)
% 0.35/0.57 thf(fact_3284_add__neg__numeral__special_I8_J,axiom,
% 0.35/0.57 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.35/0.57 = zero_zero_complex ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(8)
% 0.35/0.57 thf(fact_3285_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.57 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.57 = zero_zero_int ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(7)
% 0.35/0.57 thf(fact_3286_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.57 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.57 = zero_zero_real ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(7)
% 0.35/0.57 thf(fact_3287_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.57 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.57 = zero_zero_rat ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(7)
% 0.35/0.57 thf(fact_3288_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.57 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.57 = zero_z3403309356797280102nteger ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(7)
% 0.35/0.57 thf(fact_3289_add__neg__numeral__special_I7_J,axiom,
% 0.35/0.57 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.35/0.57 = zero_zero_complex ) ).
% 0.35/0.57
% 0.35/0.57 % add_neg_numeral_special(7)
% 0.35/0.57 thf(fact_3290_power__strict__increasing__iff,axiom,
% 0.35/0.57 ! [B3: real,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.57 => ( ( ord_less_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_increasing_iff
% 0.35/0.57 thf(fact_3291_power__strict__increasing__iff,axiom,
% 0.35/0.57 ! [B3: rat,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_rat @ one_one_rat @ B3 )
% 0.35/0.57 => ( ( ord_less_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_increasing_iff
% 0.35/0.57 thf(fact_3292_power__strict__increasing__iff,axiom,
% 0.35/0.57 ! [B3: nat,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_nat @ one_one_nat @ B3 )
% 0.35/0.57 => ( ( ord_less_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_increasing_iff
% 0.35/0.57 thf(fact_3293_power__strict__increasing__iff,axiom,
% 0.35/0.57 ! [B3: int,X2: nat,Y3: nat] :
% 0.35/0.57 ( ( ord_less_int @ one_one_int @ B3 )
% 0.35/0.57 => ( ( ord_less_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
% 0.35/0.57 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_strict_increasing_iff
% 0.35/0.57 thf(fact_3294_power__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: rat,N: nat] :
% 0.35/0.57 ( ( ( power_power_rat @ A2 @ N )
% 0.35/0.57 = zero_zero_rat )
% 0.35/0.57 = ( ( A2 = zero_zero_rat )
% 0.35/0.57 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.57
% 0.35/0.57 % power_eq_0_iff
% 0.35/0.57 thf(fact_3295_power__eq__0__iff,axiom,
% 0.35/0.57 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ( power_power_int @ A2 @ N )
% 0.35/0.58 = zero_zero_int )
% 0.35/0.58 = ( ( A2 = zero_zero_int )
% 0.35/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_eq_0_iff
% 0.35/0.58 thf(fact_3296_power__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ( power_power_nat @ A2 @ N )
% 0.35/0.58 = zero_zero_nat )
% 0.35/0.58 = ( ( A2 = zero_zero_nat )
% 0.35/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_eq_0_iff
% 0.35/0.58 thf(fact_3297_power__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ( power_power_real @ A2 @ N )
% 0.35/0.58 = zero_zero_real )
% 0.35/0.58 = ( ( A2 = zero_zero_real )
% 0.35/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_eq_0_iff
% 0.35/0.58 thf(fact_3298_power__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: complex,N: nat] :
% 0.35/0.58 ( ( ( power_power_complex @ A2 @ N )
% 0.35/0.58 = zero_zero_complex )
% 0.35/0.58 = ( ( A2 = zero_zero_complex )
% 0.35/0.58 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_eq_0_iff
% 0.35/0.58 thf(fact_3299_of__nat__Suc,axiom,
% 0.35/0.58 ! [M: nat] :
% 0.35/0.58 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.35/0.58 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_Suc
% 0.35/0.58 thf(fact_3300_of__nat__Suc,axiom,
% 0.35/0.58 ! [M: nat] :
% 0.35/0.58 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.35/0.58 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_Suc
% 0.35/0.58 thf(fact_3301_of__nat__Suc,axiom,
% 0.35/0.58 ! [M: nat] :
% 0.35/0.58 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.35/0.58 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_Suc
% 0.35/0.58 thf(fact_3302_of__nat__Suc,axiom,
% 0.35/0.58 ! [M: nat] :
% 0.35/0.58 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.35/0.58 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_Suc
% 0.35/0.58 thf(fact_3303_of__nat__Suc,axiom,
% 0.35/0.58 ! [M: nat] :
% 0.35/0.58 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.35/0.58 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_Suc
% 0.35/0.58 thf(fact_3304_diff__Suc__diff__eq2,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.35/0.58 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_Suc_diff_eq2
% 0.35/0.58 thf(fact_3305_diff__Suc__diff__eq1,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.35/0.58 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_Suc_diff_eq1
% 0.35/0.58 thf(fact_3306_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.58 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.58 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.35/0.58 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.58 thf(fact_3307_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.58 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.35/0.58 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.58 thf(fact_3308_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.58 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.58 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.35/0.58 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.58 thf(fact_3309_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.35/0.58 ! [B3: nat,W2: nat,X2: nat] :
% 0.35/0.58 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.35/0.58 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % of_nat_less_of_nat_power_cancel_iff
% 0.35/0.58 thf(fact_3310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.35/0.58 thf(fact_3311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.35/0.58 thf(fact_3312_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.35/0.58 thf(fact_3313_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.35/0.58 thf(fact_3314_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ( plus_plus_real @ I @ K )
% 0.35/0.58 = ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(4)
% 0.35/0.58 thf(fact_3315_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ( plus_plus_rat @ I @ K )
% 0.35/0.58 = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(4)
% 0.35/0.58 thf(fact_3316_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ( plus_plus_nat @ I @ K )
% 0.35/0.58 = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(4)
% 0.35/0.58 thf(fact_3317_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ( plus_plus_int @ I @ K )
% 0.35/0.58 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(4)
% 0.35/0.58 thf(fact_3318_group__cancel_Oadd1,axiom,
% 0.35/0.58 ! [A3: real,K: real,A2: real,B3: real] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_real @ K @ A2 ) )
% 0.35/0.58 => ( ( plus_plus_real @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add1
% 0.35/0.58 thf(fact_3319_group__cancel_Oadd1,axiom,
% 0.35/0.58 ! [A3: rat,K: rat,A2: rat,B3: rat] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_rat @ K @ A2 ) )
% 0.35/0.58 => ( ( plus_plus_rat @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add1
% 0.35/0.58 thf(fact_3320_group__cancel_Oadd1,axiom,
% 0.35/0.58 ! [A3: nat,K: nat,A2: nat,B3: nat] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_nat @ K @ A2 ) )
% 0.35/0.58 => ( ( plus_plus_nat @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add1
% 0.35/0.58 thf(fact_3321_group__cancel_Oadd1,axiom,
% 0.35/0.58 ! [A3: int,K: int,A2: int,B3: int] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_int @ K @ A2 ) )
% 0.35/0.58 => ( ( plus_plus_int @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add1
% 0.35/0.58 thf(fact_3322_group__cancel_Oadd2,axiom,
% 0.35/0.58 ! [B2: real,K: real,B3: real,A2: real] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_real @ K @ B3 ) )
% 0.35/0.58 => ( ( plus_plus_real @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add2
% 0.35/0.58 thf(fact_3323_group__cancel_Oadd2,axiom,
% 0.35/0.58 ! [B2: rat,K: rat,B3: rat,A2: rat] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_rat @ K @ B3 ) )
% 0.35/0.58 => ( ( plus_plus_rat @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add2
% 0.35/0.58 thf(fact_3324_group__cancel_Oadd2,axiom,
% 0.35/0.58 ! [B2: nat,K: nat,B3: nat,A2: nat] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_nat @ K @ B3 ) )
% 0.35/0.58 => ( ( plus_plus_nat @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add2
% 0.35/0.58 thf(fact_3325_group__cancel_Oadd2,axiom,
% 0.35/0.58 ! [B2: int,K: int,B3: int,A2: int] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_int @ K @ B3 ) )
% 0.35/0.58 => ( ( plus_plus_int @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.add2
% 0.35/0.58 thf(fact_3326_add_Oassoc,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.assoc
% 0.35/0.58 thf(fact_3327_add_Oassoc,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.assoc
% 0.35/0.58 thf(fact_3328_add_Oassoc,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.assoc
% 0.35/0.58 thf(fact_3329_add_Oassoc,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.assoc
% 0.35/0.58 thf(fact_3330_add_Oleft__cancel,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_real @ A2 @ C ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_cancel
% 0.35/0.58 thf(fact_3331_add_Oleft__cancel,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_rat @ A2 @ C ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_cancel
% 0.35/0.58 thf(fact_3332_add_Oleft__cancel,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_int @ A2 @ C ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_cancel
% 0.35/0.58 thf(fact_3333_add_Oright__cancel,axiom,
% 0.35/0.58 ! [B3: real,A2: real,C: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_real @ C @ A2 ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.right_cancel
% 0.35/0.58 thf(fact_3334_add_Oright__cancel,axiom,
% 0.35/0.58 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_rat @ C @ A2 ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.right_cancel
% 0.35/0.58 thf(fact_3335_add_Oright__cancel,axiom,
% 0.35/0.58 ! [B3: int,A2: int,C: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_int @ C @ A2 ) )
% 0.35/0.58 = ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.right_cancel
% 0.35/0.58 thf(fact_3336_add_Ocommute,axiom,
% 0.35/0.58 ( plus_plus_real
% 0.35/0.58 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.commute
% 0.35/0.58 thf(fact_3337_add_Ocommute,axiom,
% 0.35/0.58 ( plus_plus_rat
% 0.35/0.58 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.commute
% 0.35/0.58 thf(fact_3338_add_Ocommute,axiom,
% 0.35/0.58 ( plus_plus_nat
% 0.35/0.58 = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.commute
% 0.35/0.58 thf(fact_3339_add_Ocommute,axiom,
% 0.35/0.58 ( plus_plus_int
% 0.35/0.58 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.commute
% 0.35/0.58 thf(fact_3340_add_Oleft__commute,axiom,
% 0.35/0.58 ! [B3: real,A2: real,C: real] :
% 0.35/0.58 ( ( plus_plus_real @ B3 @ ( plus_plus_real @ A2 @ C ) )
% 0.35/0.58 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_commute
% 0.35/0.58 thf(fact_3341_add_Oleft__commute,axiom,
% 0.35/0.58 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) )
% 0.35/0.58 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_commute
% 0.35/0.58 thf(fact_3342_add_Oleft__commute,axiom,
% 0.35/0.58 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) )
% 0.35/0.58 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_commute
% 0.35/0.58 thf(fact_3343_add_Oleft__commute,axiom,
% 0.35/0.58 ! [B3: int,A2: int,C: int] :
% 0.35/0.58 ( ( plus_plus_int @ B3 @ ( plus_plus_int @ A2 @ C ) )
% 0.35/0.58 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.left_commute
% 0.35/0.58 thf(fact_3344_add__left__imp__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_real @ A2 @ C ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_imp_eq
% 0.35/0.58 thf(fact_3345_add__left__imp__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_rat @ A2 @ C ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_imp_eq
% 0.35/0.58 thf(fact_3346_add__left__imp__eq,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ( plus_plus_nat @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_nat @ A2 @ C ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_imp_eq
% 0.35/0.58 thf(fact_3347_add__left__imp__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = ( plus_plus_int @ A2 @ C ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_imp_eq
% 0.35/0.58 thf(fact_3348_add__right__imp__eq,axiom,
% 0.35/0.58 ! [B3: real,A2: real,C: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_real @ C @ A2 ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_imp_eq
% 0.35/0.58 thf(fact_3349_add__right__imp__eq,axiom,
% 0.35/0.58 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_rat @ C @ A2 ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_imp_eq
% 0.35/0.58 thf(fact_3350_add__right__imp__eq,axiom,
% 0.35/0.58 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.58 ( ( ( plus_plus_nat @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_nat @ C @ A2 ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_imp_eq
% 0.35/0.58 thf(fact_3351_add__right__imp__eq,axiom,
% 0.35/0.58 ! [B3: int,A2: int,C: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ B3 @ A2 )
% 0.35/0.58 = ( plus_plus_int @ C @ A2 ) )
% 0.35/0.58 => ( B3 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_imp_eq
% 0.35/0.58 thf(fact_3352_add__le__imp__le__right,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_right
% 0.35/0.58 thf(fact_3353_add__le__imp__le__right,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_right
% 0.35/0.58 thf(fact_3354_add__le__imp__le__right,axiom,
% 0.35/0.58 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_right
% 0.35/0.58 thf(fact_3355_add__le__imp__le__right,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_right
% 0.35/0.58 thf(fact_3356_add__le__imp__le__left,axiom,
% 0.35/0.58 ! [C: real,A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_left
% 0.35/0.58 thf(fact_3357_add__le__imp__le__left,axiom,
% 0.35/0.58 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_left
% 0.35/0.58 thf(fact_3358_add__le__imp__le__left,axiom,
% 0.35/0.58 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_left
% 0.35/0.58 thf(fact_3359_add__le__imp__le__left,axiom,
% 0.35/0.58 ! [C: int,A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_left
% 0.35/0.58 thf(fact_3360_le__iff__add,axiom,
% 0.35/0.58 ( ord_less_eq_nat
% 0.35/0.58 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.58 ? [C5: nat] :
% 0.35/0.58 ( B4
% 0.35/0.58 = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_iff_add
% 0.35/0.58 thf(fact_3361_add__right__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_mono
% 0.35/0.58 thf(fact_3362_add__right__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_mono
% 0.35/0.58 thf(fact_3363_add__right__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_mono
% 0.35/0.58 thf(fact_3364_add__right__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_right_mono
% 0.35/0.58 thf(fact_3365_less__eqE,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ~ ! [C3: nat] :
% 0.35/0.58 ( B3
% 0.35/0.58 != ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_eqE
% 0.35/0.58 thf(fact_3366_add__left__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_mono
% 0.35/0.58 thf(fact_3367_add__left__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_mono
% 0.35/0.58 thf(fact_3368_add__left__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_mono
% 0.35/0.58 thf(fact_3369_add__left__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_left_mono
% 0.35/0.58 thf(fact_3370_add__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono
% 0.35/0.58 thf(fact_3371_add__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono
% 0.35/0.58 thf(fact_3372_add__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono
% 0.35/0.58 thf(fact_3373_add__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono
% 0.35/0.58 thf(fact_3374_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_eq_real @ I @ J )
% 0.35/0.58 & ( ord_less_eq_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(1)
% 0.35/0.58 thf(fact_3375_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_eq_rat @ I @ J )
% 0.35/0.58 & ( ord_less_eq_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(1)
% 0.35/0.58 thf(fact_3376_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 & ( ord_less_eq_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(1)
% 0.35/0.58 thf(fact_3377_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_eq_int @ I @ J )
% 0.35/0.58 & ( ord_less_eq_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(1)
% 0.35/0.58 thf(fact_3378_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_eq_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(2)
% 0.35/0.58 thf(fact_3379_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_eq_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(2)
% 0.35/0.58 thf(fact_3380_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_eq_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(2)
% 0.35/0.58 thf(fact_3381_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_eq_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(2)
% 0.35/0.58 thf(fact_3382_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_eq_real @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(3)
% 0.35/0.58 thf(fact_3383_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_eq_rat @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(3)
% 0.35/0.58 thf(fact_3384_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(3)
% 0.35/0.58 thf(fact_3385_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_eq_int @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_semiring(3)
% 0.35/0.58 thf(fact_3386_verit__sum__simplify,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % verit_sum_simplify
% 0.35/0.58 thf(fact_3387_verit__sum__simplify,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % verit_sum_simplify
% 0.35/0.58 thf(fact_3388_verit__sum__simplify,axiom,
% 0.35/0.58 ! [A2: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % verit_sum_simplify
% 0.35/0.58 thf(fact_3389_verit__sum__simplify,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % verit_sum_simplify
% 0.35/0.58 thf(fact_3390_add_Ogroup__left__neutral,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.group_left_neutral
% 0.35/0.58 thf(fact_3391_add_Ogroup__left__neutral,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.group_left_neutral
% 0.35/0.58 thf(fact_3392_add_Ogroup__left__neutral,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.group_left_neutral
% 0.35/0.58 thf(fact_3393_add_Ocomm__neutral,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.comm_neutral
% 0.35/0.58 thf(fact_3394_add_Ocomm__neutral,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.comm_neutral
% 0.35/0.58 thf(fact_3395_add_Ocomm__neutral,axiom,
% 0.35/0.58 ! [A2: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.comm_neutral
% 0.35/0.58 thf(fact_3396_add_Ocomm__neutral,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % add.comm_neutral
% 0.35/0.58 thf(fact_3397_comm__monoid__add__class_Oadd__0,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % comm_monoid_add_class.add_0
% 0.35/0.58 thf(fact_3398_comm__monoid__add__class_Oadd__0,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % comm_monoid_add_class.add_0
% 0.35/0.58 thf(fact_3399_comm__monoid__add__class_Oadd__0,axiom,
% 0.35/0.58 ! [A2: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % comm_monoid_add_class.add_0
% 0.35/0.58 thf(fact_3400_comm__monoid__add__class_Oadd__0,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 0.35/0.58 = A2 ) ).
% 0.35/0.58
% 0.35/0.58 % comm_monoid_add_class.add_0
% 0.35/0.58 thf(fact_3401_add__mono__thms__linordered__field_I5_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_real @ I @ J )
% 0.35/0.58 & ( ord_less_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(5)
% 0.35/0.58 thf(fact_3402_add__mono__thms__linordered__field_I5_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_rat @ I @ J )
% 0.35/0.58 & ( ord_less_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(5)
% 0.35/0.58 thf(fact_3403_add__mono__thms__linordered__field_I5_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_nat @ I @ J )
% 0.35/0.58 & ( ord_less_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(5)
% 0.35/0.58 thf(fact_3404_add__mono__thms__linordered__field_I5_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_int @ I @ J )
% 0.35/0.58 & ( ord_less_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(5)
% 0.35/0.58 thf(fact_3405_add__mono__thms__linordered__field_I2_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(2)
% 0.35/0.58 thf(fact_3406_add__mono__thms__linordered__field_I2_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(2)
% 0.35/0.58 thf(fact_3407_add__mono__thms__linordered__field_I2_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(2)
% 0.35/0.58 thf(fact_3408_add__mono__thms__linordered__field_I2_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( I = J )
% 0.35/0.58 & ( ord_less_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(2)
% 0.35/0.58 thf(fact_3409_add__mono__thms__linordered__field_I1_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_real @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(1)
% 0.35/0.58 thf(fact_3410_add__mono__thms__linordered__field_I1_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_rat @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(1)
% 0.35/0.58 thf(fact_3411_add__mono__thms__linordered__field_I1_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_nat @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(1)
% 0.35/0.58 thf(fact_3412_add__mono__thms__linordered__field_I1_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_int @ I @ J )
% 0.35/0.58 & ( K = L ) )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(1)
% 0.35/0.58 thf(fact_3413_add__strict__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_real @ C @ D )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_mono
% 0.35/0.58 thf(fact_3414_add__strict__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_rat @ C @ D )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_mono
% 0.35/0.58 thf(fact_3415_add__strict__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_nat @ C @ D )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_mono
% 0.35/0.58 thf(fact_3416_add__strict__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_int @ C @ D )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_mono
% 0.35/0.58 thf(fact_3417_add__strict__left__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_left_mono
% 0.35/0.58 thf(fact_3418_add__strict__left__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_left_mono
% 0.35/0.58 thf(fact_3419_add__strict__left__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_left_mono
% 0.35/0.58 thf(fact_3420_add__strict__left__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_left_mono
% 0.35/0.58 thf(fact_3421_add__strict__right__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_right_mono
% 0.35/0.58 thf(fact_3422_add__strict__right__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_right_mono
% 0.35/0.58 thf(fact_3423_add__strict__right__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_right_mono
% 0.35/0.58 thf(fact_3424_add__strict__right__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_right_mono
% 0.35/0.58 thf(fact_3425_add__less__imp__less__left,axiom,
% 0.35/0.58 ! [C: real,A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_real @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_left
% 0.35/0.58 thf(fact_3426_add__less__imp__less__left,axiom,
% 0.35/0.58 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_left
% 0.35/0.58 thf(fact_3427_add__less__imp__less__left,axiom,
% 0.35/0.58 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_nat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_left
% 0.35/0.58 thf(fact_3428_add__less__imp__less__left,axiom,
% 0.35/0.58 ! [C: int,A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 0.35/0.58 => ( ord_less_int @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_left
% 0.35/0.58 thf(fact_3429_add__less__imp__less__right,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_real @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_right
% 0.35/0.58 thf(fact_3430_add__less__imp__less__right,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_right
% 0.35/0.58 thf(fact_3431_add__less__imp__less__right,axiom,
% 0.35/0.58 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_nat @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_right
% 0.35/0.58 thf(fact_3432_add__less__imp__less__right,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.58 => ( ord_less_int @ A2 @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_imp_less_right
% 0.35/0.58 thf(fact_3433_group__cancel_Osub1,axiom,
% 0.35/0.58 ! [A3: real,K: real,A2: real,B3: real] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_real @ K @ A2 ) )
% 0.35/0.58 => ( ( minus_minus_real @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub1
% 0.35/0.58 thf(fact_3434_group__cancel_Osub1,axiom,
% 0.35/0.58 ! [A3: rat,K: rat,A2: rat,B3: rat] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_rat @ K @ A2 ) )
% 0.35/0.58 => ( ( minus_minus_rat @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub1
% 0.35/0.58 thf(fact_3435_group__cancel_Osub1,axiom,
% 0.35/0.58 ! [A3: int,K: int,A2: int,B3: int] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_int @ K @ A2 ) )
% 0.35/0.58 => ( ( minus_minus_int @ A3 @ B3 )
% 0.35/0.58 = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub1
% 0.35/0.58 thf(fact_3436_diff__eq__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ( minus_minus_real @ A2 @ B3 )
% 0.35/0.58 = C )
% 0.35/0.58 = ( A2
% 0.35/0.58 = ( plus_plus_real @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_eq_eq
% 0.35/0.58 thf(fact_3437_diff__eq__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ( minus_minus_rat @ A2 @ B3 )
% 0.35/0.58 = C )
% 0.35/0.58 = ( A2
% 0.35/0.58 = ( plus_plus_rat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_eq_eq
% 0.35/0.58 thf(fact_3438_diff__eq__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ( minus_minus_int @ A2 @ B3 )
% 0.35/0.58 = C )
% 0.35/0.58 = ( A2
% 0.35/0.58 = ( plus_plus_int @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_eq_eq
% 0.35/0.58 thf(fact_3439_eq__diff__eq,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( minus_minus_real @ C @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_diff_eq
% 0.35/0.58 thf(fact_3440_eq__diff__eq,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( minus_minus_rat @ C @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_diff_eq
% 0.35/0.58 thf(fact_3441_eq__diff__eq,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( minus_minus_int @ C @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = C ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_diff_eq
% 0.35/0.58 thf(fact_3442_add__diff__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( plus_plus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_diff_eq
% 0.35/0.58 thf(fact_3443_add__diff__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_diff_eq
% 0.35/0.58 thf(fact_3444_add__diff__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_diff_eq
% 0.35/0.58 thf(fact_3445_diff__diff__eq2,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( minus_minus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq2
% 0.35/0.58 thf(fact_3446_diff__diff__eq2,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq2
% 0.35/0.58 thf(fact_3447_diff__diff__eq2,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq2
% 0.35/0.58 thf(fact_3448_diff__add__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq
% 0.35/0.58 thf(fact_3449_diff__add__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq
% 0.35/0.58 thf(fact_3450_diff__add__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq
% 0.35/0.58 thf(fact_3451_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq_diff_diff_swap
% 0.35/0.58 thf(fact_3452_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq_diff_diff_swap
% 0.35/0.58 thf(fact_3453_diff__add__eq__diff__diff__swap,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) )
% 0.35/0.58 = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_eq_diff_diff_swap
% 0.35/0.58 thf(fact_3454_add__implies__diff,axiom,
% 0.35/0.58 ! [C: real,B3: real,A2: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ C @ B3 )
% 0.35/0.58 = A2 )
% 0.35/0.58 => ( C
% 0.35/0.58 = ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_implies_diff
% 0.35/0.58 thf(fact_3455_add__implies__diff,axiom,
% 0.35/0.58 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ C @ B3 )
% 0.35/0.58 = A2 )
% 0.35/0.58 => ( C
% 0.35/0.58 = ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_implies_diff
% 0.35/0.58 thf(fact_3456_add__implies__diff,axiom,
% 0.35/0.58 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.58 ( ( ( plus_plus_nat @ C @ B3 )
% 0.35/0.58 = A2 )
% 0.35/0.58 => ( C
% 0.35/0.58 = ( minus_minus_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_implies_diff
% 0.35/0.58 thf(fact_3457_add__implies__diff,axiom,
% 0.35/0.58 ! [C: int,B3: int,A2: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ C @ B3 )
% 0.35/0.58 = A2 )
% 0.35/0.58 => ( C
% 0.35/0.58 = ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_implies_diff
% 0.35/0.58 thf(fact_3458_diff__diff__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq
% 0.35/0.58 thf(fact_3459_diff__diff__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq
% 0.35/0.58 thf(fact_3460_diff__diff__eq,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq
% 0.35/0.58 thf(fact_3461_diff__diff__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_diff_eq
% 0.35/0.58 thf(fact_3462_group__cancel_Oneg1,axiom,
% 0.35/0.58 ! [A3: int,K: int,A2: int] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_int @ K @ A2 ) )
% 0.35/0.58 => ( ( uminus_uminus_int @ A3 )
% 0.35/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.neg1
% 0.35/0.58 thf(fact_3463_group__cancel_Oneg1,axiom,
% 0.35/0.58 ! [A3: real,K: real,A2: real] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_real @ K @ A2 ) )
% 0.35/0.58 => ( ( uminus_uminus_real @ A3 )
% 0.35/0.58 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.neg1
% 0.35/0.58 thf(fact_3464_group__cancel_Oneg1,axiom,
% 0.35/0.58 ! [A3: rat,K: rat,A2: rat] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_rat @ K @ A2 ) )
% 0.35/0.58 => ( ( uminus_uminus_rat @ A3 )
% 0.35/0.58 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.neg1
% 0.35/0.58 thf(fact_3465_group__cancel_Oneg1,axiom,
% 0.35/0.58 ! [A3: code_integer,K: code_integer,A2: code_integer] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_p5714425477246183910nteger @ K @ A2 ) )
% 0.35/0.58 => ( ( uminus1351360451143612070nteger @ A3 )
% 0.35/0.58 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.neg1
% 0.35/0.58 thf(fact_3466_group__cancel_Oneg1,axiom,
% 0.35/0.58 ! [A3: complex,K: complex,A2: complex] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_complex @ K @ A2 ) )
% 0.35/0.58 => ( ( uminus1482373934393186551omplex @ A3 )
% 0.35/0.58 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.neg1
% 0.35/0.58 thf(fact_3467_add_Oinverse__distrib__swap,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
% 0.35/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_distrib_swap
% 0.35/0.58 thf(fact_3468_add_Oinverse__distrib__swap,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
% 0.35/0.58 = ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_distrib_swap
% 0.35/0.58 thf(fact_3469_add_Oinverse__distrib__swap,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
% 0.35/0.58 = ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_distrib_swap
% 0.35/0.58 thf(fact_3470_add_Oinverse__distrib__swap,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.58 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 0.35/0.58 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_distrib_swap
% 0.35/0.58 thf(fact_3471_add_Oinverse__distrib__swap,axiom,
% 0.35/0.58 ! [A2: complex,B3: complex] :
% 0.35/0.58 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
% 0.35/0.58 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_distrib_swap
% 0.35/0.58 thf(fact_3472_nat__arith_Osuc1,axiom,
% 0.35/0.58 ! [A3: nat,K: nat,A2: nat] :
% 0.35/0.58 ( ( A3
% 0.35/0.58 = ( plus_plus_nat @ K @ A2 ) )
% 0.35/0.58 => ( ( suc @ A3 )
% 0.35/0.58 = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_arith.suc1
% 0.35/0.58 thf(fact_3473_add__Suc,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.35/0.58 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_Suc
% 0.35/0.58 thf(fact_3474_add__Suc__shift,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 0.35/0.58 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_Suc_shift
% 0.35/0.58 thf(fact_3475_add__eq__self__zero,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( ( plus_plus_nat @ M @ N )
% 0.35/0.58 = M )
% 0.35/0.58 => ( N = zero_zero_nat ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_self_zero
% 0.35/0.58 thf(fact_3476_plus__nat_Oadd__0,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 0.35/0.58 = N ) ).
% 0.35/0.58
% 0.35/0.58 % plus_nat.add_0
% 0.35/0.58 thf(fact_3477_add__lessD1,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.35/0.58 => ( ord_less_nat @ I @ K ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_lessD1
% 0.35/0.58 thf(fact_3478_add__less__mono,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ J )
% 0.35/0.58 => ( ( ord_less_nat @ K @ L )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_mono
% 0.35/0.58 thf(fact_3479_not__add__less1,axiom,
% 0.35/0.58 ! [I: nat,J: nat] :
% 0.35/0.58 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.35/0.58
% 0.35/0.58 % not_add_less1
% 0.35/0.58 thf(fact_3480_not__add__less2,axiom,
% 0.35/0.58 ! [J: nat,I: nat] :
% 0.35/0.58 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.35/0.58
% 0.35/0.58 % not_add_less2
% 0.35/0.58 thf(fact_3481_add__less__mono1,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ J )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_mono1
% 0.35/0.58 thf(fact_3482_trans__less__add1,axiom,
% 0.35/0.58 ! [I: nat,J: nat,M: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ J )
% 0.35/0.58 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % trans_less_add1
% 0.35/0.58 thf(fact_3483_trans__less__add2,axiom,
% 0.35/0.58 ! [I: nat,J: nat,M: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ J )
% 0.35/0.58 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % trans_less_add2
% 0.35/0.58 thf(fact_3484_less__add__eq__less,axiom,
% 0.35/0.58 ! [K: nat,L: nat,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ K @ L )
% 0.35/0.58 => ( ( ( plus_plus_nat @ M @ L )
% 0.35/0.58 = ( plus_plus_nat @ K @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_eq_less
% 0.35/0.58 thf(fact_3485_nat__le__iff__add,axiom,
% 0.35/0.58 ( ord_less_eq_nat
% 0.35/0.58 = ( ^ [M2: nat,N2: nat] :
% 0.35/0.58 ? [K3: nat] :
% 0.35/0.58 ( N2
% 0.35/0.58 = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_le_iff_add
% 0.35/0.58 thf(fact_3486_trans__le__add2,axiom,
% 0.35/0.58 ! [I: nat,J: nat,M: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % trans_le_add2
% 0.35/0.58 thf(fact_3487_trans__le__add1,axiom,
% 0.35/0.58 ! [I: nat,J: nat,M: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % trans_le_add1
% 0.35/0.58 thf(fact_3488_add__le__mono1,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_mono1
% 0.35/0.58 thf(fact_3489_add__le__mono,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 => ( ( ord_less_eq_nat @ K @ L )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_mono
% 0.35/0.58 thf(fact_3490_le__Suc__ex,axiom,
% 0.35/0.58 ! [K: nat,L: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ L )
% 0.35/0.58 => ? [N3: nat] :
% 0.35/0.58 ( L
% 0.35/0.58 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_Suc_ex
% 0.35/0.58 thf(fact_3491_add__leD2,axiom,
% 0.35/0.58 ! [M: nat,K: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_nat @ K @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_leD2
% 0.35/0.58 thf(fact_3492_add__leD1,axiom,
% 0.35/0.58 ! [M: nat,K: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_leD1
% 0.35/0.58 thf(fact_3493_le__add2,axiom,
% 0.35/0.58 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_add2
% 0.35/0.58 thf(fact_3494_le__add1,axiom,
% 0.35/0.58 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_add1
% 0.35/0.58 thf(fact_3495_add__leE,axiom,
% 0.35/0.58 ! [M: nat,K: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 0.35/0.58 => ~ ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.58 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_leE
% 0.35/0.58 thf(fact_3496_Nat_Odiff__cancel,axiom,
% 0.35/0.58 ! [K: nat,M: nat,N: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 0.35/0.58 = ( minus_minus_nat @ M @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % Nat.diff_cancel
% 0.35/0.58 thf(fact_3497_diff__cancel2,axiom,
% 0.35/0.58 ! [M: nat,K: nat,N: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.58 = ( minus_minus_nat @ M @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_cancel2
% 0.35/0.58 thf(fact_3498_diff__add__inverse,axiom,
% 0.35/0.58 ! [N: nat,M: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 0.35/0.58 = M ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_inverse
% 0.35/0.58 thf(fact_3499_diff__add__inverse2,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 0.35/0.58 = M ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_inverse2
% 0.35/0.58 thf(fact_3500_nat__power__less__imp__less,axiom,
% 0.35/0.58 ! [I: nat,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.35/0.58 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_power_less_imp_less
% 0.35/0.58 thf(fact_3501_abs__real__def,axiom,
% 0.35/0.58 ( abs_abs_real
% 0.35/0.58 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_real_def
% 0.35/0.58 thf(fact_3502_add__decreasing,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_eq_real @ C @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing
% 0.35/0.58 thf(fact_3503_add__decreasing,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_eq_rat @ C @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing
% 0.35/0.58 thf(fact_3504_add__decreasing,axiom,
% 0.35/0.58 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_eq_nat @ C @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing
% 0.35/0.58 thf(fact_3505_add__decreasing,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_eq_int @ C @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing
% 0.35/0.58 thf(fact_3506_add__increasing,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ B3 @ C )
% 0.35/0.58 => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing
% 0.35/0.58 thf(fact_3507_add__increasing,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.35/0.58 => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing
% 0.35/0.58 thf(fact_3508_add__increasing,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.35/0.58 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing
% 0.35/0.58 thf(fact_3509_add__increasing,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ B3 @ C )
% 0.35/0.58 => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing
% 0.35/0.58 thf(fact_3510_add__decreasing2,axiom,
% 0.35/0.58 ! [C: real,A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing2
% 0.35/0.58 thf(fact_3511_add__decreasing2,axiom,
% 0.35/0.58 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing2
% 0.35/0.58 thf(fact_3512_add__decreasing2,axiom,
% 0.35/0.58 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing2
% 0.35/0.58 thf(fact_3513_add__decreasing2,axiom,
% 0.35/0.58 ! [C: int,A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_decreasing2
% 0.35/0.58 thf(fact_3514_add__increasing2,axiom,
% 0.35/0.58 ! [C: real,B3: real,A2: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.58 => ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.58 => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing2
% 0.35/0.58 thf(fact_3515_add__increasing2,axiom,
% 0.35/0.58 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.58 => ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.58 => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing2
% 0.35/0.58 thf(fact_3516_add__increasing2,axiom,
% 0.35/0.58 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.58 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.58 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing2
% 0.35/0.58 thf(fact_3517_add__increasing2,axiom,
% 0.35/0.58 ! [C: int,B3: int,A2: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.58 => ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.58 => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_increasing2
% 0.35/0.58 thf(fact_3518_add__nonneg__nonneg,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_nonneg
% 0.35/0.58 thf(fact_3519_add__nonneg__nonneg,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_nonneg
% 0.35/0.58 thf(fact_3520_add__nonneg__nonneg,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_nonneg
% 0.35/0.58 thf(fact_3521_add__nonneg__nonneg,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_nonneg
% 0.35/0.58 thf(fact_3522_add__nonpos__nonpos,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.35/0.58 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_nonpos
% 0.35/0.58 thf(fact_3523_add__nonpos__nonpos,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 0.35/0.58 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_nonpos
% 0.35/0.58 thf(fact_3524_add__nonpos__nonpos,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_nonpos
% 0.35/0.58 thf(fact_3525_add__nonpos__nonpos,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_nonpos
% 0.35/0.58 thf(fact_3526_add__nonneg__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.58 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_real )
% 0.35/0.58 = ( ( X2 = zero_zero_real )
% 0.35/0.58 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_eq_0_iff
% 0.35/0.58 thf(fact_3527_add__nonneg__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: rat,Y3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.58 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_rat )
% 0.35/0.58 = ( ( X2 = zero_zero_rat )
% 0.35/0.58 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_eq_0_iff
% 0.35/0.58 thf(fact_3528_add__nonneg__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: nat,Y3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.35/0.58 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_nat )
% 0.35/0.58 = ( ( X2 = zero_zero_nat )
% 0.35/0.58 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_eq_0_iff
% 0.35/0.58 thf(fact_3529_add__nonneg__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: int,Y3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.58 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_int )
% 0.35/0.58 = ( ( X2 = zero_zero_int )
% 0.35/0.58 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_eq_0_iff
% 0.35/0.58 thf(fact_3530_add__nonpos__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.35/0.58 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_real )
% 0.35/0.58 = ( ( X2 = zero_zero_real )
% 0.35/0.58 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_eq_0_iff
% 0.35/0.58 thf(fact_3531_add__nonpos__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: rat,Y3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.35/0.58 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_rat )
% 0.35/0.58 = ( ( X2 = zero_zero_rat )
% 0.35/0.58 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_eq_0_iff
% 0.35/0.58 thf(fact_3532_add__nonpos__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: nat,Y3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.35/0.58 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_nat )
% 0.35/0.58 = ( ( X2 = zero_zero_nat )
% 0.35/0.58 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_eq_0_iff
% 0.35/0.58 thf(fact_3533_add__nonpos__eq__0__iff,axiom,
% 0.35/0.58 ! [X2: int,Y3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.35/0.58 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 0.35/0.58 = zero_zero_int )
% 0.35/0.58 = ( ( X2 = zero_zero_int )
% 0.35/0.58 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_eq_0_iff
% 0.35/0.58 thf(fact_3534_add__less__le__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_le_mono
% 0.35/0.58 thf(fact_3535_add__less__le__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_le_mono
% 0.35/0.58 thf(fact_3536_add__less__le__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_le_mono
% 0.35/0.58 thf(fact_3537_add__less__le__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_le_mono
% 0.35/0.58 thf(fact_3538_add__le__less__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_real @ C @ D )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_less_mono
% 0.35/0.58 thf(fact_3539_add__le__less__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_rat @ C @ D )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_less_mono
% 0.35/0.58 thf(fact_3540_add__le__less__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_nat @ C @ D )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_less_mono
% 0.35/0.58 thf(fact_3541_add__le__less__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_int @ C @ D )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_less_mono
% 0.35/0.58 thf(fact_3542_add__mono__thms__linordered__field_I3_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_real @ I @ J )
% 0.35/0.58 & ( ord_less_eq_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(3)
% 0.35/0.58 thf(fact_3543_add__mono__thms__linordered__field_I3_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_rat @ I @ J )
% 0.35/0.58 & ( ord_less_eq_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(3)
% 0.35/0.58 thf(fact_3544_add__mono__thms__linordered__field_I3_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_nat @ I @ J )
% 0.35/0.58 & ( ord_less_eq_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(3)
% 0.35/0.58 thf(fact_3545_add__mono__thms__linordered__field_I3_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_int @ I @ J )
% 0.35/0.58 & ( ord_less_eq_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(3)
% 0.35/0.58 thf(fact_3546_add__mono__thms__linordered__field_I4_J,axiom,
% 0.35/0.58 ! [I: real,J: real,K: real,L: real] :
% 0.35/0.58 ( ( ( ord_less_eq_real @ I @ J )
% 0.35/0.58 & ( ord_less_real @ K @ L ) )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(4)
% 0.35/0.58 thf(fact_3547_add__mono__thms__linordered__field_I4_J,axiom,
% 0.35/0.58 ! [I: rat,J: rat,K: rat,L: rat] :
% 0.35/0.58 ( ( ( ord_less_eq_rat @ I @ J )
% 0.35/0.58 & ( ord_less_rat @ K @ L ) )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(4)
% 0.35/0.58 thf(fact_3548_add__mono__thms__linordered__field_I4_J,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.58 ( ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 & ( ord_less_nat @ K @ L ) )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(4)
% 0.35/0.58 thf(fact_3549_add__mono__thms__linordered__field_I4_J,axiom,
% 0.35/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.35/0.58 ( ( ( ord_less_eq_int @ I @ J )
% 0.35/0.58 & ( ord_less_int @ K @ L ) )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono_thms_linordered_field(4)
% 0.35/0.58 thf(fact_3550_pos__add__strict,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ B3 @ C )
% 0.35/0.58 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % pos_add_strict
% 0.35/0.58 thf(fact_3551_pos__add__strict,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_rat @ B3 @ C )
% 0.35/0.58 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % pos_add_strict
% 0.35/0.58 thf(fact_3552_pos__add__strict,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_nat @ B3 @ C )
% 0.35/0.58 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % pos_add_strict
% 0.35/0.58 thf(fact_3553_pos__add__strict,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_int @ B3 @ C )
% 0.35/0.58 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % pos_add_strict
% 0.35/0.58 thf(fact_3554_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ~ ! [C3: nat] :
% 0.35/0.58 ( ( B3
% 0.35/0.58 = ( plus_plus_nat @ A2 @ C3 ) )
% 0.35/0.58 => ( C3 = zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % canonically_ordered_monoid_add_class.lessE
% 0.35/0.58 thf(fact_3555_add__pos__pos,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_pos
% 0.35/0.58 thf(fact_3556_add__pos__pos,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_pos
% 0.35/0.58 thf(fact_3557_add__pos__pos,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_pos
% 0.35/0.58 thf(fact_3558_add__pos__pos,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_pos
% 0.35/0.58 thf(fact_3559_add__neg__neg,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_neg
% 0.35/0.58 thf(fact_3560_add__neg__neg,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_neg
% 0.35/0.58 thf(fact_3561_add__neg__neg,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_neg
% 0.35/0.58 thf(fact_3562_add__neg__neg,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_neg
% 0.35/0.58 thf(fact_3563_add__less__zeroD,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.58 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_zeroD
% 0.35/0.58 thf(fact_3564_add__less__zeroD,axiom,
% 0.35/0.58 ! [X2: rat,Y3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 0.35/0.58 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_zeroD
% 0.35/0.58 thf(fact_3565_add__less__zeroD,axiom,
% 0.35/0.58 ! [X2: int,Y3: int] :
% 0.35/0.58 ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_int @ X2 @ zero_zero_int )
% 0.35/0.58 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_less_zeroD
% 0.35/0.58 thf(fact_3566_power__gt__expt,axiom,
% 0.35/0.58 ! [N: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.58 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_gt_expt
% 0.35/0.58 thf(fact_3567_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ( minus_minus_nat @ B3 @ A2 )
% 0.35/0.58 = C )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.35/0.58 thf(fact_3568_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.35/0.58 thf(fact_3569_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 0.35/0.58 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.35/0.58 thf(fact_3570_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
% 0.35/0.58 = ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.35/0.58 thf(fact_3571_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
% 0.35/0.58 = ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.35/0.58 thf(fact_3572_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
% 0.35/0.58 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.35/0.58 thf(fact_3573_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 0.35/0.58 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.35/0.58 thf(fact_3574_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 0.35/0.58 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.35/0.58 thf(fact_3575_le__add__diff,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_add_diff
% 0.35/0.58 thf(fact_3576_diff__add,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add
% 0.35/0.58 thf(fact_3577_le__diff__eq,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_diff_eq
% 0.35/0.58 thf(fact_3578_le__diff__eq,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_diff_eq
% 0.35/0.58 thf(fact_3579_le__diff__eq,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_diff_eq
% 0.35/0.58 thf(fact_3580_diff__le__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_le_eq
% 0.35/0.58 thf(fact_3581_diff__le__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_le_eq
% 0.35/0.58 thf(fact_3582_diff__le__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_le_eq
% 0.35/0.58 thf(fact_3583_add__le__add__imp__diff__le,axiom,
% 0.35/0.58 ! [I: real,K: real,N: real,J: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.35/0.58 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 0.35/0.58 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_add_imp_diff_le
% 0.35/0.58 thf(fact_3584_add__le__add__imp__diff__le,axiom,
% 0.35/0.58 ! [I: rat,K: rat,N: rat,J: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.35/0.58 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 0.35/0.58 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_add_imp_diff_le
% 0.35/0.58 thf(fact_3585_add__le__add__imp__diff__le,axiom,
% 0.35/0.58 ! [I: nat,K: nat,N: nat,J: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.58 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.58 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_add_imp_diff_le
% 0.35/0.58 thf(fact_3586_add__le__add__imp__diff__le,axiom,
% 0.35/0.58 ! [I: int,K: int,N: int,J: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.35/0.58 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.58 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.35/0.58 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_add_imp_diff_le
% 0.35/0.58 thf(fact_3587_add__le__imp__le__diff,axiom,
% 0.35/0.58 ! [I: real,K: real,N: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_diff
% 0.35/0.58 thf(fact_3588_add__le__imp__le__diff,axiom,
% 0.35/0.58 ! [I: rat,K: rat,N: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_diff
% 0.35/0.58 thf(fact_3589_add__le__imp__le__diff,axiom,
% 0.35/0.58 ! [I: nat,K: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_diff
% 0.35/0.58 thf(fact_3590_add__le__imp__le__diff,axiom,
% 0.35/0.58 ! [I: int,K: int,N: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.35/0.58 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_le_imp_le_diff
% 0.35/0.58 thf(fact_3591_less__add__one,axiom,
% 0.35/0.58 ! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_one
% 0.35/0.58 thf(fact_3592_less__add__one,axiom,
% 0.35/0.58 ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_one
% 0.35/0.58 thf(fact_3593_less__add__one,axiom,
% 0.35/0.58 ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_one
% 0.35/0.58 thf(fact_3594_less__add__one,axiom,
% 0.35/0.58 ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_one
% 0.35/0.58 thf(fact_3595_add__mono1,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono1
% 0.35/0.58 thf(fact_3596_add__mono1,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B3 @ one_one_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono1
% 0.35/0.58 thf(fact_3597_add__mono1,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono1
% 0.35/0.58 thf(fact_3598_add__mono1,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_mono1
% 0.35/0.58 thf(fact_3599_nat__one__le__power,axiom,
% 0.35/0.58 ! [I: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.35/0.58 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_one_le_power
% 0.35/0.58 thf(fact_3600_less__diff__eq,axiom,
% 0.35/0.58 ! [A2: real,C: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_diff_eq
% 0.35/0.58 thf(fact_3601_less__diff__eq,axiom,
% 0.35/0.58 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_diff_eq
% 0.35/0.58 thf(fact_3602_less__diff__eq,axiom,
% 0.35/0.58 ! [A2: int,C: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
% 0.35/0.58 = ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_diff_eq
% 0.35/0.58 thf(fact_3603_diff__less__eq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_less_eq
% 0.35/0.58 thf(fact_3604_diff__less__eq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_less_eq
% 0.35/0.58 thf(fact_3605_diff__less__eq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 0.35/0.58 = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_less_eq
% 0.35/0.58 thf(fact_3606_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ~ ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
% 0.35/0.58 = A2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % linordered_semidom_class.add_diff_inverse
% 0.35/0.58 thf(fact_3607_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ~ ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
% 0.35/0.58 = A2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % linordered_semidom_class.add_diff_inverse
% 0.35/0.58 thf(fact_3608_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ~ ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.35/0.58 = A2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % linordered_semidom_class.add_diff_inverse
% 0.35/0.58 thf(fact_3609_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ~ ( ord_less_int @ A2 @ B3 )
% 0.35/0.58 => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
% 0.35/0.58 = A2 ) ) ).
% 0.35/0.58
% 0.35/0.58 % linordered_semidom_class.add_diff_inverse
% 0.35/0.58 thf(fact_3610_add__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = zero_zero_int )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( uminus_uminus_int @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_0_iff
% 0.35/0.58 thf(fact_3611_add__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = zero_zero_real )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( uminus_uminus_real @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_0_iff
% 0.35/0.58 thf(fact_3612_add__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = zero_zero_rat )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_0_iff
% 0.35/0.58 thf(fact_3613_add__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.58 ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 0.35/0.58 = zero_z3403309356797280102nteger )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_0_iff
% 0.35/0.58 thf(fact_3614_add__eq__0__iff,axiom,
% 0.35/0.58 ! [A2: complex,B3: complex] :
% 0.35/0.58 ( ( ( plus_plus_complex @ A2 @ B3 )
% 0.35/0.58 = zero_zero_complex )
% 0.35/0.58 = ( B3
% 0.35/0.58 = ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_0_iff
% 0.35/0.58 thf(fact_3615_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 0.35/0.58 = zero_zero_int ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_left_minus
% 0.35/0.58 thf(fact_3616_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 0.35/0.58 = zero_zero_real ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_left_minus
% 0.35/0.58 thf(fact_3617_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 0.35/0.58 = zero_zero_rat ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_left_minus
% 0.35/0.58 thf(fact_3618_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.58 ! [A2: code_integer] :
% 0.35/0.58 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 0.35/0.58 = zero_z3403309356797280102nteger ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_left_minus
% 0.35/0.58 thf(fact_3619_ab__group__add__class_Oab__left__minus,axiom,
% 0.35/0.58 ! [A2: complex] :
% 0.35/0.58 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
% 0.35/0.58 = zero_zero_complex ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_left_minus
% 0.35/0.58 thf(fact_3620_add_Oinverse__unique,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = zero_zero_int )
% 0.35/0.58 => ( ( uminus_uminus_int @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_unique
% 0.35/0.58 thf(fact_3621_add_Oinverse__unique,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = zero_zero_real )
% 0.35/0.58 => ( ( uminus_uminus_real @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_unique
% 0.35/0.58 thf(fact_3622_add_Oinverse__unique,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = zero_zero_rat )
% 0.35/0.58 => ( ( uminus_uminus_rat @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_unique
% 0.35/0.58 thf(fact_3623_add_Oinverse__unique,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.58 ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 0.35/0.58 = zero_z3403309356797280102nteger )
% 0.35/0.58 => ( ( uminus1351360451143612070nteger @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_unique
% 0.35/0.58 thf(fact_3624_add_Oinverse__unique,axiom,
% 0.35/0.58 ! [A2: complex,B3: complex] :
% 0.35/0.58 ( ( ( plus_plus_complex @ A2 @ B3 )
% 0.35/0.58 = zero_zero_complex )
% 0.35/0.58 => ( ( uminus1482373934393186551omplex @ A2 )
% 0.35/0.58 = B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % add.inverse_unique
% 0.35/0.58 thf(fact_3625_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( uminus_uminus_int @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = zero_zero_int ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_neg_iff_add_eq_0
% 0.35/0.58 thf(fact_3626_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( uminus_uminus_real @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = zero_zero_real ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_neg_iff_add_eq_0
% 0.35/0.58 thf(fact_3627_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( uminus_uminus_rat @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = zero_zero_rat ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_neg_iff_add_eq_0
% 0.35/0.58 thf(fact_3628_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( uminus1351360451143612070nteger @ B3 ) )
% 0.35/0.58 = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 0.35/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_neg_iff_add_eq_0
% 0.35/0.58 thf(fact_3629_eq__neg__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: complex,B3: complex] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( uminus1482373934393186551omplex @ B3 ) )
% 0.35/0.58 = ( ( plus_plus_complex @ A2 @ B3 )
% 0.35/0.58 = zero_zero_complex ) ) ).
% 0.35/0.58
% 0.35/0.58 % eq_neg_iff_add_eq_0
% 0.35/0.58 thf(fact_3630_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ( uminus_uminus_int @ A2 )
% 0.35/0.58 = B3 )
% 0.35/0.58 = ( ( plus_plus_int @ A2 @ B3 )
% 0.35/0.58 = zero_zero_int ) ) ).
% 0.35/0.58
% 0.35/0.58 % neg_eq_iff_add_eq_0
% 0.35/0.58 thf(fact_3631_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ( uminus_uminus_real @ A2 )
% 0.35/0.58 = B3 )
% 0.35/0.58 = ( ( plus_plus_real @ A2 @ B3 )
% 0.35/0.58 = zero_zero_real ) ) ).
% 0.35/0.58
% 0.35/0.58 % neg_eq_iff_add_eq_0
% 0.35/0.58 thf(fact_3632_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ( uminus_uminus_rat @ A2 )
% 0.35/0.58 = B3 )
% 0.35/0.58 = ( ( plus_plus_rat @ A2 @ B3 )
% 0.35/0.58 = zero_zero_rat ) ) ).
% 0.35/0.58
% 0.35/0.58 % neg_eq_iff_add_eq_0
% 0.35/0.58 thf(fact_3633_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.58 ( ( ( uminus1351360451143612070nteger @ A2 )
% 0.35/0.58 = B3 )
% 0.35/0.58 = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 0.35/0.58 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.58
% 0.35/0.58 % neg_eq_iff_add_eq_0
% 0.35/0.58 thf(fact_3634_neg__eq__iff__add__eq__0,axiom,
% 0.35/0.58 ! [A2: complex,B3: complex] :
% 0.35/0.58 ( ( ( uminus1482373934393186551omplex @ A2 )
% 0.35/0.58 = B3 )
% 0.35/0.58 = ( ( plus_plus_complex @ A2 @ B3 )
% 0.35/0.58 = zero_zero_complex ) ) ).
% 0.35/0.58
% 0.35/0.58 % neg_eq_iff_add_eq_0
% 0.35/0.58 thf(fact_3635_group__cancel_Osub2,axiom,
% 0.35/0.58 ! [B2: int,K: int,B3: int,A2: int] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_int @ K @ B3 ) )
% 0.35/0.58 => ( ( minus_minus_int @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub2
% 0.35/0.58 thf(fact_3636_group__cancel_Osub2,axiom,
% 0.35/0.58 ! [B2: real,K: real,B3: real,A2: real] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_real @ K @ B3 ) )
% 0.35/0.58 => ( ( minus_minus_real @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub2
% 0.35/0.58 thf(fact_3637_group__cancel_Osub2,axiom,
% 0.35/0.58 ! [B2: rat,K: rat,B3: rat,A2: rat] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_rat @ K @ B3 ) )
% 0.35/0.58 => ( ( minus_minus_rat @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub2
% 0.35/0.58 thf(fact_3638_group__cancel_Osub2,axiom,
% 0.35/0.58 ! [B2: code_integer,K: code_integer,B3: code_integer,A2: code_integer] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_p5714425477246183910nteger @ K @ B3 ) )
% 0.35/0.58 => ( ( minus_8373710615458151222nteger @ A2 @ B2 )
% 0.35/0.58 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub2
% 0.35/0.58 thf(fact_3639_group__cancel_Osub2,axiom,
% 0.35/0.58 ! [B2: complex,K: complex,B3: complex,A2: complex] :
% 0.35/0.58 ( ( B2
% 0.35/0.58 = ( plus_plus_complex @ K @ B3 ) )
% 0.35/0.58 => ( ( minus_minus_complex @ A2 @ B2 )
% 0.35/0.58 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % group_cancel.sub2
% 0.35/0.58 thf(fact_3640_diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_int
% 0.35/0.58 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_conv_add_uminus
% 0.35/0.58 thf(fact_3641_diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_real
% 0.35/0.58 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_conv_add_uminus
% 0.35/0.58 thf(fact_3642_diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_rat
% 0.35/0.58 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_conv_add_uminus
% 0.35/0.58 thf(fact_3643_diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_8373710615458151222nteger
% 0.35/0.58 = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_conv_add_uminus
% 0.35/0.58 thf(fact_3644_diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_complex
% 0.35/0.58 = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % diff_conv_add_uminus
% 0.35/0.58 thf(fact_3645_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_int
% 0.35/0.58 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.58 thf(fact_3646_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_real
% 0.35/0.58 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.58 thf(fact_3647_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_rat
% 0.35/0.58 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.58 thf(fact_3648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_8373710615458151222nteger
% 0.35/0.58 = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.58 thf(fact_3649_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.35/0.58 ( minus_minus_complex
% 0.35/0.58 = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.35/0.58 thf(fact_3650_abs__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq
% 0.35/0.58 thf(fact_3651_abs__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq
% 0.35/0.58 thf(fact_3652_abs__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq
% 0.35/0.58 thf(fact_3653_abs__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq
% 0.35/0.58 thf(fact_3654_add__is__1,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( ( plus_plus_nat @ M @ N )
% 0.35/0.58 = ( suc @ zero_zero_nat ) )
% 0.35/0.58 = ( ( ( M
% 0.35/0.58 = ( suc @ zero_zero_nat ) )
% 0.35/0.58 & ( N = zero_zero_nat ) )
% 0.35/0.58 | ( ( M = zero_zero_nat )
% 0.35/0.58 & ( N
% 0.35/0.58 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_is_1
% 0.35/0.58 thf(fact_3655_one__is__add,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( ( suc @ zero_zero_nat )
% 0.35/0.58 = ( plus_plus_nat @ M @ N ) )
% 0.35/0.58 = ( ( ( M
% 0.35/0.58 = ( suc @ zero_zero_nat ) )
% 0.35/0.58 & ( N = zero_zero_nat ) )
% 0.35/0.58 | ( ( M = zero_zero_nat )
% 0.35/0.58 & ( N
% 0.35/0.58 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % one_is_add
% 0.35/0.58 thf(fact_3656_less__imp__Suc__add,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ M @ N )
% 0.35/0.58 => ? [K2: nat] :
% 0.35/0.58 ( N
% 0.35/0.58 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_imp_Suc_add
% 0.35/0.58 thf(fact_3657_less__iff__Suc__add,axiom,
% 0.35/0.58 ( ord_less_nat
% 0.35/0.58 = ( ^ [M2: nat,N2: nat] :
% 0.35/0.58 ? [K3: nat] :
% 0.35/0.58 ( N2
% 0.35/0.58 = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_iff_Suc_add
% 0.35/0.58 thf(fact_3658_less__add__Suc2,axiom,
% 0.35/0.58 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_Suc2
% 0.35/0.58 thf(fact_3659_less__add__Suc1,axiom,
% 0.35/0.58 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_add_Suc1
% 0.35/0.58 thf(fact_3660_less__natE,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ M @ N )
% 0.35/0.58 => ~ ! [Q4: nat] :
% 0.35/0.58 ( N
% 0.35/0.58 != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_natE
% 0.35/0.58 thf(fact_3661_real__arch__pow,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.58 => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % real_arch_pow
% 0.35/0.58 thf(fact_3662_less__imp__add__positive,axiom,
% 0.35/0.58 ! [I: nat,J: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ J )
% 0.35/0.58 => ? [K2: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.35/0.58 & ( ( plus_plus_nat @ I @ K2 )
% 0.35/0.58 = J ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_imp_add_positive
% 0.35/0.58 thf(fact_3663_mono__nat__linear__lb,axiom,
% 0.35/0.58 ! [F: nat > nat,M: nat,K: nat] :
% 0.35/0.58 ( ! [M4: nat,N3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ M4 @ N3 )
% 0.35/0.58 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 0.35/0.58 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % mono_nat_linear_lb
% 0.35/0.58 thf(fact_3664_diff__add__0,axiom,
% 0.35/0.58 ! [N: nat,M: nat] :
% 0.35/0.58 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 0.35/0.58 = zero_zero_nat ) ).
% 0.35/0.58
% 0.35/0.58 % diff_add_0
% 0.35/0.58 thf(fact_3665_add__diff__inverse__nat,axiom,
% 0.35/0.58 ! [M: nat,N: nat] :
% 0.35/0.58 ( ~ ( ord_less_nat @ M @ N )
% 0.35/0.58 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.58 = M ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_diff_inverse_nat
% 0.35/0.58 thf(fact_3666_less__diff__conv,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.58 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_diff_conv
% 0.35/0.58 thf(fact_3667_Suc__eq__plus1,axiom,
% 0.35/0.58 ( suc
% 0.35/0.58 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % Suc_eq_plus1
% 0.35/0.58 thf(fact_3668_plus__1__eq__Suc,axiom,
% 0.35/0.58 ( ( plus_plus_nat @ one_one_nat )
% 0.35/0.58 = suc ) ).
% 0.35/0.58
% 0.35/0.58 % plus_1_eq_Suc
% 0.35/0.58 thf(fact_3669_Suc__eq__plus1__left,axiom,
% 0.35/0.58 ( suc
% 0.35/0.58 = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.35/0.58
% 0.35/0.58 % Suc_eq_plus1_left
% 0.35/0.58 thf(fact_3670_Nat_Ole__imp__diff__is__add,axiom,
% 0.35/0.58 ! [I: nat,J: nat,K: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.58 => ( ( ( minus_minus_nat @ J @ I )
% 0.35/0.58 = K )
% 0.35/0.58 = ( J
% 0.35/0.58 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % Nat.le_imp_diff_is_add
% 0.35/0.58 thf(fact_3671_Nat_Odiff__add__assoc2,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.35/0.58 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % Nat.diff_add_assoc2
% 0.35/0.58 thf(fact_3672_Nat_Odiff__add__assoc,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.35/0.58 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % Nat.diff_add_assoc
% 0.35/0.58 thf(fact_3673_Nat_Ole__diff__conv2,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.35/0.58 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % Nat.le_diff_conv2
% 0.35/0.58 thf(fact_3674_le__diff__conv,axiom,
% 0.35/0.58 ! [J: nat,K: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.58 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % le_diff_conv
% 0.35/0.58 thf(fact_3675_add__strict__increasing2,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ B3 @ C )
% 0.35/0.58 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing2
% 0.35/0.58 thf(fact_3676_add__strict__increasing2,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_rat @ B3 @ C )
% 0.35/0.58 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing2
% 0.35/0.58 thf(fact_3677_add__strict__increasing2,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_nat @ B3 @ C )
% 0.35/0.58 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing2
% 0.35/0.58 thf(fact_3678_add__strict__increasing2,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_int @ B3 @ C )
% 0.35/0.58 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing2
% 0.35/0.58 thf(fact_3679_add__strict__increasing,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ B3 @ C )
% 0.35/0.58 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing
% 0.35/0.58 thf(fact_3680_add__strict__increasing,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ B3 @ C )
% 0.35/0.58 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing
% 0.35/0.58 thf(fact_3681_add__strict__increasing,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ B3 @ C )
% 0.35/0.58 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing
% 0.35/0.58 thf(fact_3682_add__strict__increasing,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int] :
% 0.35/0.58 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ B3 @ C )
% 0.35/0.58 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_strict_increasing
% 0.35/0.58 thf(fact_3683_add__pos__nonneg,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_nonneg
% 0.35/0.58 thf(fact_3684_add__pos__nonneg,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_nonneg
% 0.35/0.58 thf(fact_3685_add__pos__nonneg,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_nonneg
% 0.35/0.58 thf(fact_3686_add__pos__nonneg,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_pos_nonneg
% 0.35/0.58 thf(fact_3687_add__nonpos__neg,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_neg
% 0.35/0.58 thf(fact_3688_add__nonpos__neg,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_neg
% 0.35/0.58 thf(fact_3689_add__nonpos__neg,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_neg
% 0.35/0.58 thf(fact_3690_add__nonpos__neg,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonpos_neg
% 0.35/0.58 thf(fact_3691_add__nonneg__pos,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_pos
% 0.35/0.58 thf(fact_3692_add__nonneg__pos,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_pos
% 0.35/0.58 thf(fact_3693_add__nonneg__pos,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_pos
% 0.35/0.58 thf(fact_3694_add__nonneg__pos,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_nonneg_pos
% 0.35/0.58 thf(fact_3695_add__neg__nonpos,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.58 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.35/0.58 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_nonpos
% 0.35/0.58 thf(fact_3696_add__neg__nonpos,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.58 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 0.35/0.58 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_nonpos
% 0.35/0.58 thf(fact_3697_add__neg__nonpos,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 0.35/0.58 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_nonpos
% 0.35/0.58 thf(fact_3698_add__neg__nonpos,axiom,
% 0.35/0.58 ! [A2: int,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.58 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.58 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_neg_nonpos
% 0.35/0.58 thf(fact_3699_field__le__epsilon,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ! [E: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.58 => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y3 @ E ) ) )
% 0.35/0.58 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % field_le_epsilon
% 0.35/0.58 thf(fact_3700_field__le__epsilon,axiom,
% 0.35/0.58 ! [X2: rat,Y3: rat] :
% 0.35/0.58 ( ! [E: rat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.35/0.58 => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y3 @ E ) ) )
% 0.35/0.58 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % field_le_epsilon
% 0.35/0.58 thf(fact_3701_discrete,axiom,
% 0.35/0.58 ( ord_less_nat
% 0.35/0.58 = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % discrete
% 0.35/0.58 thf(fact_3702_discrete,axiom,
% 0.35/0.58 ( ord_less_int
% 0.35/0.58 = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % discrete
% 0.35/0.58 thf(fact_3703_zero__less__two,axiom,
% 0.35/0.58 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_two
% 0.35/0.58 thf(fact_3704_zero__less__two,axiom,
% 0.35/0.58 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_two
% 0.35/0.58 thf(fact_3705_zero__less__two,axiom,
% 0.35/0.58 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_two
% 0.35/0.58 thf(fact_3706_zero__less__two,axiom,
% 0.35/0.58 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_two
% 0.35/0.58 thf(fact_3707_div__add__self2,axiom,
% 0.35/0.58 ! [B3: int,A2: int] :
% 0.35/0.58 ( ( B3 != zero_zero_int )
% 0.35/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 0.35/0.58 = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % div_add_self2
% 0.35/0.58 thf(fact_3708_div__add__self2,axiom,
% 0.35/0.58 ! [B3: nat,A2: nat] :
% 0.35/0.58 ( ( B3 != zero_zero_nat )
% 0.35/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % div_add_self2
% 0.35/0.58 thf(fact_3709_div__add__self1,axiom,
% 0.35/0.58 ! [B3: int,A2: int] :
% 0.35/0.58 ( ( B3 != zero_zero_int )
% 0.35/0.58 => ( ( divide_divide_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 0.35/0.58 = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % div_add_self1
% 0.35/0.58 thf(fact_3710_div__add__self1,axiom,
% 0.35/0.58 ! [B3: nat,A2: nat] :
% 0.35/0.58 ( ( B3 != zero_zero_nat )
% 0.35/0.58 => ( ( divide_divide_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 0.35/0.58 = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % div_add_self1
% 0.35/0.58 thf(fact_3711_gt__half__sum,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % gt_half_sum
% 0.35/0.58 thf(fact_3712_gt__half__sum,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B3 ) ) ).
% 0.35/0.58
% 0.35/0.58 % gt_half_sum
% 0.35/0.58 thf(fact_3713_less__half__sum,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_rat @ A2 @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_half_sum
% 0.35/0.58 thf(fact_3714_less__half__sum,axiom,
% 0.35/0.58 ! [A2: real,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.58 => ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_half_sum
% 0.35/0.58 thf(fact_3715_abs__diff__le__iff,axiom,
% 0.35/0.58 ! [X2: code_integer,A2: code_integer,R2: code_integer] :
% 0.35/0.58 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_le_iff
% 0.35/0.58 thf(fact_3716_abs__diff__le__iff,axiom,
% 0.35/0.58 ! [X2: real,A2: real,R2: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_le_iff
% 0.35/0.58 thf(fact_3717_abs__diff__le__iff,axiom,
% 0.35/0.58 ! [X2: rat,A2: rat,R2: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_le_iff
% 0.35/0.58 thf(fact_3718_abs__diff__le__iff,axiom,
% 0.35/0.58 ! [X2: int,A2: int,R2: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_le_iff
% 0.35/0.58 thf(fact_3719_abs__diff__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_triangle_ineq
% 0.35/0.58 thf(fact_3720_abs__diff__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_triangle_ineq
% 0.35/0.58 thf(fact_3721_abs__diff__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_triangle_ineq
% 0.35/0.58 thf(fact_3722_abs__diff__triangle__ineq,axiom,
% 0.35/0.58 ! [A2: int,B3: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_triangle_ineq
% 0.35/0.58 thf(fact_3723_abs__triangle__ineq4,axiom,
% 0.35/0.58 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq4
% 0.35/0.58 thf(fact_3724_abs__triangle__ineq4,axiom,
% 0.35/0.58 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq4
% 0.35/0.58 thf(fact_3725_abs__triangle__ineq4,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq4
% 0.35/0.58 thf(fact_3726_abs__triangle__ineq4,axiom,
% 0.35/0.58 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_triangle_ineq4
% 0.35/0.58 thf(fact_3727_abs__diff__less__iff,axiom,
% 0.35/0.58 ! [X2: code_integer,A2: code_integer,R2: code_integer] :
% 0.35/0.58 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_less_iff
% 0.35/0.58 thf(fact_3728_abs__diff__less__iff,axiom,
% 0.35/0.58 ! [X2: real,A2: real,R2: real] :
% 0.35/0.58 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_less_iff
% 0.35/0.58 thf(fact_3729_abs__diff__less__iff,axiom,
% 0.35/0.58 ! [X2: rat,A2: rat,R2: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_less_iff
% 0.35/0.58 thf(fact_3730_abs__diff__less__iff,axiom,
% 0.35/0.58 ! [X2: int,A2: int,R2: int] :
% 0.35/0.58 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
% 0.35/0.58 = ( ( ord_less_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
% 0.35/0.58 & ( ord_less_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_diff_less_iff
% 0.35/0.58 thf(fact_3731_real__arch__pow__inv,axiom,
% 0.35/0.58 ! [Y3: real,X2: real] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.58 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.58 => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % real_arch_pow_inv
% 0.35/0.58 thf(fact_3732_nat__diff__split__asm,axiom,
% 0.35/0.58 ! [P: nat > $o,A2: nat,B3: nat] :
% 0.35/0.58 ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.35/0.58 = ( ~ ( ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 & ~ ( P @ zero_zero_nat ) )
% 0.35/0.58 | ? [D5: nat] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( plus_plus_nat @ B3 @ D5 ) )
% 0.35/0.58 & ~ ( P @ D5 ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_diff_split_asm
% 0.35/0.58 thf(fact_3733_nat__diff__split,axiom,
% 0.35/0.58 ! [P: nat > $o,A2: nat,B3: nat] :
% 0.35/0.58 ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.35/0.58 = ( ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.58 => ( P @ zero_zero_nat ) )
% 0.35/0.58 & ! [D5: nat] :
% 0.35/0.58 ( ( A2
% 0.35/0.58 = ( plus_plus_nat @ B3 @ D5 ) )
% 0.35/0.58 => ( P @ D5 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat_diff_split
% 0.35/0.58 thf(fact_3734_less__diff__conv2,axiom,
% 0.35/0.58 ! [K: nat,J: nat,I: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ K @ J )
% 0.35/0.58 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.35/0.58 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % less_diff_conv2
% 0.35/0.58 thf(fact_3735_abs__add__one__gt__zero,axiom,
% 0.35/0.58 ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_add_one_gt_zero
% 0.35/0.58 thf(fact_3736_abs__add__one__gt__zero,axiom,
% 0.35/0.58 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_add_one_gt_zero
% 0.35/0.58 thf(fact_3737_abs__add__one__gt__zero,axiom,
% 0.35/0.58 ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_add_one_gt_zero
% 0.35/0.58 thf(fact_3738_abs__add__one__gt__zero,axiom,
% 0.35/0.58 ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % abs_add_one_gt_zero
% 0.35/0.58 thf(fact_3739_add__eq__if,axiom,
% 0.35/0.58 ( plus_plus_nat
% 0.35/0.58 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % add_eq_if
% 0.35/0.58 thf(fact_3740_frac__add,axiom,
% 0.35/0.58 ! [X2: real,Y3: real] :
% 0.35/0.58 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.35/0.58 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.35/0.58 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 0.35/0.58 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.35/0.58 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.35/0.58 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % frac_add
% 0.35/0.58 thf(fact_3741_frac__add,axiom,
% 0.35/0.58 ! [X2: rat,Y3: rat] :
% 0.35/0.58 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.35/0.58 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
% 0.35/0.58 = ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 0.35/0.58 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.35/0.58 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
% 0.35/0.58 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % frac_add
% 0.35/0.58 thf(fact_3742_option_Osize__gen_I2_J,axiom,
% 0.35/0.58 ! [X2: nat > nat,X22: nat] :
% 0.35/0.58 ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
% 0.35/0.58 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % option.size_gen(2)
% 0.35/0.58 thf(fact_3743_option_Osize__gen_I2_J,axiom,
% 0.35/0.58 ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.35/0.58 ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.35/0.58 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % option.size_gen(2)
% 0.35/0.58 thf(fact_3744_option_Osize__gen_I2_J,axiom,
% 0.35/0.58 ! [X2: num > nat,X22: num] :
% 0.35/0.58 ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
% 0.35/0.58 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % option.size_gen(2)
% 0.35/0.58 thf(fact_3745_power__not__zero,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( A2 != zero_zero_rat )
% 0.35/0.58 => ( ( power_power_rat @ A2 @ N )
% 0.35/0.58 != zero_zero_rat ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_not_zero
% 0.35/0.58 thf(fact_3746_power__not__zero,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( A2 != zero_zero_int )
% 0.35/0.58 => ( ( power_power_int @ A2 @ N )
% 0.35/0.58 != zero_zero_int ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_not_zero
% 0.35/0.58 thf(fact_3747_power__not__zero,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( A2 != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_nat @ A2 @ N )
% 0.35/0.58 != zero_zero_nat ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_not_zero
% 0.35/0.58 thf(fact_3748_power__not__zero,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( A2 != zero_zero_real )
% 0.35/0.58 => ( ( power_power_real @ A2 @ N )
% 0.35/0.58 != zero_zero_real ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_not_zero
% 0.35/0.58 thf(fact_3749_power__not__zero,axiom,
% 0.35/0.58 ! [A2: complex,N: nat] :
% 0.35/0.58 ( ( A2 != zero_zero_complex )
% 0.35/0.58 => ( ( power_power_complex @ A2 @ N )
% 0.35/0.58 != zero_zero_complex ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_not_zero
% 0.35/0.58 thf(fact_3750_nat0__intermed__int__val,axiom,
% 0.35/0.58 ! [N: nat,F: nat > int,K: int] :
% 0.35/0.58 ( ! [I2: nat] :
% 0.35/0.58 ( ( ord_less_nat @ I2 @ N )
% 0.35/0.58 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 0.35/0.58 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.35/0.58 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 0.35/0.58 => ? [I2: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ I2 @ N )
% 0.35/0.58 & ( ( F @ I2 )
% 0.35/0.58 = K ) ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % nat0_intermed_int_val
% 0.35/0.58 thf(fact_3751_power__mono,axiom,
% 0.35/0.58 ! [A2: real,B3: real,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_mono
% 0.35/0.58 thf(fact_3752_power__mono,axiom,
% 0.35/0.58 ! [A2: rat,B3: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_mono
% 0.35/0.58 thf(fact_3753_power__mono,axiom,
% 0.35/0.58 ! [A2: nat,B3: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_mono
% 0.35/0.58 thf(fact_3754_power__mono,axiom,
% 0.35/0.58 ! [A2: int,B3: int,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_mono
% 0.35/0.58 thf(fact_3755_zero__le__power,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power
% 0.35/0.58 thf(fact_3756_zero__le__power,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power
% 0.35/0.58 thf(fact_3757_zero__le__power,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power
% 0.35/0.58 thf(fact_3758_zero__le__power,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power
% 0.35/0.58 thf(fact_3759_zero__less__power,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_power
% 0.35/0.58 thf(fact_3760_zero__less__power,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_power
% 0.35/0.58 thf(fact_3761_zero__less__power,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_power
% 0.35/0.58 thf(fact_3762_zero__less__power,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_less_power
% 0.35/0.58 thf(fact_3763_one__le__power,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_real @ one_one_real @ A2 )
% 0.35/0.58 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % one_le_power
% 0.35/0.58 thf(fact_3764_one__le__power,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.35/0.58 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % one_le_power
% 0.35/0.58 thf(fact_3765_one__le__power,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.35/0.58 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % one_le_power
% 0.35/0.58 thf(fact_3766_one__le__power,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.35/0.58 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % one_le_power
% 0.35/0.58 thf(fact_3767_power__0,axiom,
% 0.35/0.58 ! [A2: rat] :
% 0.35/0.58 ( ( power_power_rat @ A2 @ zero_zero_nat )
% 0.35/0.58 = one_one_rat ) ).
% 0.35/0.58
% 0.35/0.58 % power_0
% 0.35/0.58 thf(fact_3768_power__0,axiom,
% 0.35/0.58 ! [A2: int] :
% 0.35/0.58 ( ( power_power_int @ A2 @ zero_zero_nat )
% 0.35/0.58 = one_one_int ) ).
% 0.35/0.58
% 0.35/0.58 % power_0
% 0.35/0.58 thf(fact_3769_power__0,axiom,
% 0.35/0.58 ! [A2: nat] :
% 0.35/0.58 ( ( power_power_nat @ A2 @ zero_zero_nat )
% 0.35/0.58 = one_one_nat ) ).
% 0.35/0.58
% 0.35/0.58 % power_0
% 0.35/0.58 thf(fact_3770_power__0,axiom,
% 0.35/0.58 ! [A2: real] :
% 0.35/0.58 ( ( power_power_real @ A2 @ zero_zero_nat )
% 0.35/0.58 = one_one_real ) ).
% 0.35/0.58
% 0.35/0.58 % power_0
% 0.35/0.58 thf(fact_3771_power__0,axiom,
% 0.35/0.58 ! [A2: complex] :
% 0.35/0.58 ( ( power_power_complex @ A2 @ zero_zero_nat )
% 0.35/0.58 = one_one_complex ) ).
% 0.35/0.58
% 0.35/0.58 % power_0
% 0.35/0.58 thf(fact_3772_power__less__imp__less__base,axiom,
% 0.35/0.58 ! [A2: real,N: nat,B3: real] :
% 0.35/0.58 ( ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_base
% 0.35/0.58 thf(fact_3773_power__less__imp__less__base,axiom,
% 0.35/0.58 ! [A2: rat,N: nat,B3: rat] :
% 0.35/0.58 ( ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_base
% 0.35/0.58 thf(fact_3774_power__less__imp__less__base,axiom,
% 0.35/0.58 ! [A2: nat,N: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_base
% 0.35/0.58 thf(fact_3775_power__less__imp__less__base,axiom,
% 0.35/0.58 ! [A2: int,N: nat,B3: int] :
% 0.35/0.58 ( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_base
% 0.35/0.58 thf(fact_3776_power__le__one,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 0.35/0.58 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_one
% 0.35/0.58 thf(fact_3777_power__le__one,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.35/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_one
% 0.35/0.58 thf(fact_3778_power__le__one,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.35/0.58 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_one
% 0.35/0.58 thf(fact_3779_power__le__one,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.35/0.58 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ one_one_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_one
% 0.35/0.58 thf(fact_3780_power__inject__base,axiom,
% 0.35/0.58 ! [A2: real,N: nat,B3: real] :
% 0.35/0.58 ( ( ( power_power_real @ A2 @ ( suc @ N ) )
% 0.35/0.58 = ( power_power_real @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( A2 = B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_inject_base
% 0.35/0.58 thf(fact_3781_power__inject__base,axiom,
% 0.35/0.58 ! [A2: rat,N: nat,B3: rat] :
% 0.35/0.58 ( ( ( power_power_rat @ A2 @ ( suc @ N ) )
% 0.35/0.58 = ( power_power_rat @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( A2 = B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_inject_base
% 0.35/0.58 thf(fact_3782_power__inject__base,axiom,
% 0.35/0.58 ! [A2: nat,N: nat,B3: nat] :
% 0.35/0.58 ( ( ( power_power_nat @ A2 @ ( suc @ N ) )
% 0.35/0.58 = ( power_power_nat @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( A2 = B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_inject_base
% 0.35/0.58 thf(fact_3783_power__inject__base,axiom,
% 0.35/0.58 ! [A2: int,N: nat,B3: int] :
% 0.35/0.58 ( ( ( power_power_int @ A2 @ ( suc @ N ) )
% 0.35/0.58 = ( power_power_int @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( A2 = B3 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_inject_base
% 0.35/0.58 thf(fact_3784_power__le__imp__le__base,axiom,
% 0.35/0.58 ! [A2: real,N: nat,B3: real] :
% 0.35/0.58 ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ ( power_power_real @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.58 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_imp_le_base
% 0.35/0.58 thf(fact_3785_power__le__imp__le__base,axiom,
% 0.35/0.58 ! [A2: rat,N: nat,B3: rat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ ( power_power_rat @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.58 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_imp_le_base
% 0.35/0.58 thf(fact_3786_power__le__imp__le__base,axiom,
% 0.35/0.58 ! [A2: nat,N: nat,B3: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.58 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_imp_le_base
% 0.35/0.58 thf(fact_3787_power__le__imp__le__base,axiom,
% 0.35/0.58 ! [A2: int,N: nat,B3: int] :
% 0.35/0.58 ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ ( power_power_int @ B3 @ ( suc @ N ) ) )
% 0.35/0.58 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.58 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_le_imp_le_base
% 0.35/0.58 thf(fact_3788_power__gt1,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.58 => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ ( suc @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_gt1
% 0.35/0.58 thf(fact_3789_power__gt1,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.58 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_gt1
% 0.35/0.58 thf(fact_3790_power__gt1,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.58 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_gt1
% 0.35/0.58 thf(fact_3791_power__gt1,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.58 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_gt1
% 0.35/0.58 thf(fact_3792_power__0__left,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ( N = zero_zero_nat )
% 0.35/0.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.35/0.58 = one_one_rat ) )
% 0.35/0.58 & ( ( N != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.35/0.58 = zero_zero_rat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_0_left
% 0.35/0.58 thf(fact_3793_power__0__left,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ( N = zero_zero_nat )
% 0.35/0.58 => ( ( power_power_int @ zero_zero_int @ N )
% 0.35/0.58 = one_one_int ) )
% 0.35/0.58 & ( ( N != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_int @ zero_zero_int @ N )
% 0.35/0.58 = zero_zero_int ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_0_left
% 0.35/0.58 thf(fact_3794_power__0__left,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ( N = zero_zero_nat )
% 0.35/0.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.35/0.58 = one_one_nat ) )
% 0.35/0.58 & ( ( N != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.35/0.58 = zero_zero_nat ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_0_left
% 0.35/0.58 thf(fact_3795_power__0__left,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ( N = zero_zero_nat )
% 0.35/0.58 => ( ( power_power_real @ zero_zero_real @ N )
% 0.35/0.58 = one_one_real ) )
% 0.35/0.58 & ( ( N != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_real @ zero_zero_real @ N )
% 0.35/0.58 = zero_zero_real ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_0_left
% 0.35/0.58 thf(fact_3796_power__0__left,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ( N = zero_zero_nat )
% 0.35/0.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.35/0.58 = one_one_complex ) )
% 0.35/0.58 & ( ( N != zero_zero_nat )
% 0.35/0.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.35/0.58 = zero_zero_complex ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_0_left
% 0.35/0.58 thf(fact_3797_power__less__imp__less__exp,axiom,
% 0.35/0.58 ! [A2: real,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_exp
% 0.35/0.58 thf(fact_3798_power__less__imp__less__exp,axiom,
% 0.35/0.58 ! [A2: rat,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_exp
% 0.35/0.58 thf(fact_3799_power__less__imp__less__exp,axiom,
% 0.35/0.58 ! [A2: nat,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_exp
% 0.35/0.58 thf(fact_3800_power__less__imp__less__exp,axiom,
% 0.35/0.58 ! [A2: int,M: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.58 => ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 0.35/0.58 => ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_less_imp_less_exp
% 0.35/0.58 thf(fact_3801_power__strict__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: real] :
% 0.35/0.58 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.58 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_strict_increasing
% 0.35/0.58 thf(fact_3802_power__strict__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: rat] :
% 0.35/0.58 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.58 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_strict_increasing
% 0.35/0.58 thf(fact_3803_power__strict__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: nat] :
% 0.35/0.58 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.58 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_strict_increasing
% 0.35/0.58 thf(fact_3804_power__strict__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: int] :
% 0.35/0.58 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.58 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_strict_increasing
% 0.35/0.58 thf(fact_3805_zero__le__power__abs,axiom,
% 0.35/0.58 ! [A2: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power_abs
% 0.35/0.58 thf(fact_3806_zero__le__power__abs,axiom,
% 0.35/0.58 ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power_abs
% 0.35/0.58 thf(fact_3807_zero__le__power__abs,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power_abs
% 0.35/0.58 thf(fact_3808_zero__le__power__abs,axiom,
% 0.35/0.58 ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_le_power_abs
% 0.35/0.58 thf(fact_3809_power__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: real] :
% 0.35/0.58 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_eq_real @ one_one_real @ A2 )
% 0.35/0.58 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_increasing
% 0.35/0.58 thf(fact_3810_power__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: rat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.35/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_increasing
% 0.35/0.58 thf(fact_3811_power__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.35/0.58 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_increasing
% 0.35/0.58 thf(fact_3812_power__increasing,axiom,
% 0.35/0.58 ! [N: nat,N6: nat,A2: int] :
% 0.35/0.58 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.58 => ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.35/0.58 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_increasing
% 0.35/0.58 thf(fact_3813_zero__power,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58 => ( ( power_power_rat @ zero_zero_rat @ N )
% 0.35/0.58 = zero_zero_rat ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_power
% 0.35/0.58 thf(fact_3814_zero__power,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58 => ( ( power_power_int @ zero_zero_int @ N )
% 0.35/0.58 = zero_zero_int ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_power
% 0.35/0.58 thf(fact_3815_zero__power,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58 => ( ( power_power_nat @ zero_zero_nat @ N )
% 0.35/0.58 = zero_zero_nat ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_power
% 0.35/0.58 thf(fact_3816_zero__power,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58 => ( ( power_power_real @ zero_zero_real @ N )
% 0.35/0.58 = zero_zero_real ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_power
% 0.35/0.58 thf(fact_3817_zero__power,axiom,
% 0.35/0.58 ! [N: nat] :
% 0.35/0.58 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.58 => ( ( power_power_complex @ zero_zero_complex @ N )
% 0.35/0.58 = zero_zero_complex ) ) ).
% 0.35/0.58
% 0.35/0.58 % zero_power
% 0.35/0.58 thf(fact_3818_power__Suc__le__self,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 0.35/0.58 => ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_Suc_le_self
% 0.35/0.58 thf(fact_3819_power__Suc__le__self,axiom,
% 0.35/0.58 ! [A2: rat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.35/0.58 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_Suc_le_self
% 0.35/0.58 thf(fact_3820_power__Suc__le__self,axiom,
% 0.35/0.58 ! [A2: nat,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.35/0.58 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_Suc_le_self
% 0.35/0.58 thf(fact_3821_power__Suc__le__self,axiom,
% 0.35/0.58 ! [A2: int,N: nat] :
% 0.35/0.58 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.58 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.35/0.58 => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 0.35/0.58
% 0.35/0.58 % power_Suc_le_self
% 0.35/0.58 thf(fact_3822_power__Suc__less__one,axiom,
% 0.35/0.58 ! [A2: real,N: nat] :
% 0.35/0.58 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.58 => ( ( ord_less_real @ A2 @ one_one_real )
% 0.35/0.58 => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_Suc_less_one
% 0.35/0.59 thf(fact_3823_power__Suc__less__one,axiom,
% 0.35/0.59 ! [A2: rat,N: nat] :
% 0.35/0.59 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.35/0.59 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_Suc_less_one
% 0.35/0.59 thf(fact_3824_power__Suc__less__one,axiom,
% 0.35/0.59 ! [A2: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.35/0.59 => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_Suc_less_one
% 0.35/0.59 thf(fact_3825_power__Suc__less__one,axiom,
% 0.35/0.59 ! [A2: int,N: nat] :
% 0.35/0.59 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.35/0.59 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_Suc_less_one
% 0.35/0.59 thf(fact_3826_power__strict__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: real] :
% 0.35/0.59 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ A2 @ one_one_real )
% 0.35/0.59 => ( ord_less_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_decreasing
% 0.35/0.59 thf(fact_3827_power__strict__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: rat] :
% 0.35/0.59 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.35/0.59 => ( ord_less_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_decreasing
% 0.35/0.59 thf(fact_3828_power__strict__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: nat] :
% 0.35/0.59 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.35/0.59 => ( ord_less_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_decreasing
% 0.35/0.59 thf(fact_3829_power__strict__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: int] :
% 0.35/0.59 ( ( ord_less_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.35/0.59 => ( ord_less_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_decreasing
% 0.35/0.59 thf(fact_3830_power__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: real] :
% 0.35/0.59 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_decreasing
% 0.35/0.59 thf(fact_3831_power__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: rat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_decreasing
% 0.35/0.59 thf(fact_3832_power__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.35/0.59 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_decreasing
% 0.35/0.59 thf(fact_3833_power__decreasing,axiom,
% 0.35/0.59 ! [N: nat,N6: nat,A2: int] :
% 0.35/0.59 ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.35/0.59 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_decreasing
% 0.35/0.59 thf(fact_3834_power__le__imp__le__exp,axiom,
% 0.35/0.59 ! [A2: real,M: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
% 0.35/0.59 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_le_imp_le_exp
% 0.35/0.59 thf(fact_3835_power__le__imp__le__exp,axiom,
% 0.35/0.59 ! [A2: rat,M: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
% 0.35/0.59 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_le_imp_le_exp
% 0.35/0.59 thf(fact_3836_power__le__imp__le__exp,axiom,
% 0.35/0.59 ! [A2: nat,M: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 0.35/0.59 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_le_imp_le_exp
% 0.35/0.59 thf(fact_3837_power__le__imp__le__exp,axiom,
% 0.35/0.59 ! [A2: int,M: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 0.35/0.59 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_le_imp_le_exp
% 0.35/0.59 thf(fact_3838_power__eq__imp__eq__base,axiom,
% 0.35/0.59 ! [A2: real,N: nat,B3: real] :
% 0.35/0.59 ( ( ( power_power_real @ A2 @ N )
% 0.35/0.59 = ( power_power_real @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_imp_eq_base
% 0.35/0.59 thf(fact_3839_power__eq__imp__eq__base,axiom,
% 0.35/0.59 ! [A2: rat,N: nat,B3: rat] :
% 0.35/0.59 ( ( ( power_power_rat @ A2 @ N )
% 0.35/0.59 = ( power_power_rat @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_imp_eq_base
% 0.35/0.59 thf(fact_3840_power__eq__imp__eq__base,axiom,
% 0.35/0.59 ! [A2: nat,N: nat,B3: nat] :
% 0.35/0.59 ( ( ( power_power_nat @ A2 @ N )
% 0.35/0.59 = ( power_power_nat @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_imp_eq_base
% 0.35/0.59 thf(fact_3841_power__eq__imp__eq__base,axiom,
% 0.35/0.59 ! [A2: int,N: nat,B3: int] :
% 0.35/0.59 ( ( ( power_power_int @ A2 @ N )
% 0.35/0.59 = ( power_power_int @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_imp_eq_base
% 0.35/0.59 thf(fact_3842_power__eq__iff__eq__base,axiom,
% 0.35/0.59 ! [N: nat,A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.59 => ( ( ( power_power_real @ A2 @ N )
% 0.35/0.59 = ( power_power_real @ B3 @ N ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_iff_eq_base
% 0.35/0.59 thf(fact_3843_power__eq__iff__eq__base,axiom,
% 0.35/0.59 ! [N: nat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.59 => ( ( ( power_power_rat @ A2 @ N )
% 0.35/0.59 = ( power_power_rat @ B3 @ N ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_iff_eq_base
% 0.35/0.59 thf(fact_3844_power__eq__iff__eq__base,axiom,
% 0.35/0.59 ! [N: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.59 => ( ( ( power_power_nat @ A2 @ N )
% 0.35/0.59 = ( power_power_nat @ B3 @ N ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_iff_eq_base
% 0.35/0.59 thf(fact_3845_power__eq__iff__eq__base,axiom,
% 0.35/0.59 ! [N: nat,A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.59 => ( ( ( power_power_int @ A2 @ N )
% 0.35/0.59 = ( power_power_int @ B3 @ N ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_eq_iff_eq_base
% 0.35/0.59 thf(fact_3846_self__le__power,axiom,
% 0.35/0.59 ! [A2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % self_le_power
% 0.35/0.59 thf(fact_3847_self__le__power,axiom,
% 0.35/0.59 ! [A2: rat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % self_le_power
% 0.35/0.59 thf(fact_3848_self__le__power,axiom,
% 0.35/0.59 ! [A2: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % self_le_power
% 0.35/0.59 thf(fact_3849_self__le__power,axiom,
% 0.35/0.59 ! [A2: int,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % self_le_power
% 0.35/0.59 thf(fact_3850_one__less__power,axiom,
% 0.35/0.59 ! [A2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_power
% 0.35/0.59 thf(fact_3851_one__less__power,axiom,
% 0.35/0.59 ! [A2: rat,N: nat] :
% 0.35/0.59 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_power
% 0.35/0.59 thf(fact_3852_one__less__power,axiom,
% 0.35/0.59 ! [A2: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_power
% 0.35/0.59 thf(fact_3853_one__less__power,axiom,
% 0.35/0.59 ! [A2: int,N: nat] :
% 0.35/0.59 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_power
% 0.35/0.59 thf(fact_3854_power__diff,axiom,
% 0.35/0.59 ! [A2: rat,N: nat,M: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_rat )
% 0.35/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59 => ( ( power_power_rat @ A2 @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59 = ( divide_divide_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_diff
% 0.35/0.59 thf(fact_3855_power__diff,axiom,
% 0.35/0.59 ! [A2: int,N: nat,M: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_int )
% 0.35/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59 => ( ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59 = ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_diff
% 0.35/0.59 thf(fact_3856_power__diff,axiom,
% 0.35/0.59 ! [A2: nat,N: nat,M: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_nat )
% 0.35/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59 => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59 = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_diff
% 0.35/0.59 thf(fact_3857_power__diff,axiom,
% 0.35/0.59 ! [A2: real,N: nat,M: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_real )
% 0.35/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59 => ( ( power_power_real @ A2 @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59 = ( divide_divide_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_diff
% 0.35/0.59 thf(fact_3858_power__diff,axiom,
% 0.35/0.59 ! [A2: complex,N: nat,M: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_complex )
% 0.35/0.59 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.59 => ( ( power_power_complex @ A2 @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_diff
% 0.35/0.59 thf(fact_3859_power__strict__mono,axiom,
% 0.35/0.59 ! [A2: real,B3: real,N: nat] :
% 0.35/0.59 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_mono
% 0.35/0.59 thf(fact_3860_power__strict__mono,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,N: nat] :
% 0.35/0.59 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_mono
% 0.35/0.59 thf(fact_3861_power__strict__mono,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_mono
% 0.35/0.59 thf(fact_3862_power__strict__mono,axiom,
% 0.35/0.59 ! [A2: int,B3: int,N: nat] :
% 0.35/0.59 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % power_strict_mono
% 0.35/0.59 thf(fact_3863_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.59 ! [K: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.59 thf(fact_3864_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.59 ! [K: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.59 thf(fact_3865_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.59 ! [K: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.59 thf(fact_3866_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.59 ! [K: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.59 thf(fact_3867_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.35/0.59 ! [K: nat,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.59 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 0.35/0.59 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % neg_one_power_add_eq_neg_one_power_diff
% 0.35/0.59 thf(fact_3868_add__shift,axiom,
% 0.35/0.59 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.59 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 0.35/0.59 = Z )
% 0.35/0.59 = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.35/0.59 = ( some_nat @ Z ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_shift
% 0.35/0.59 thf(fact_3869_lemma__interval,axiom,
% 0.35/0.59 ! [A2: real,X2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_real @ A2 @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ X2 @ B3 )
% 0.35/0.59 => ? [D6: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.59 & ! [Y5: real] :
% 0.35/0.59 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
% 0.35/0.59 => ( ( ord_less_eq_real @ A2 @ Y5 )
% 0.35/0.59 & ( ord_less_eq_real @ Y5 @ B3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % lemma_interval
% 0.35/0.59 thf(fact_3870_realpow__pos__nth__unique,axiom,
% 0.35/0.59 ! [N: nat,A2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ? [X5: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.35/0.59 & ( ( power_power_real @ X5 @ N )
% 0.35/0.59 = A2 )
% 0.35/0.59 & ! [Y5: real] :
% 0.35/0.59 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 0.35/0.59 & ( ( power_power_real @ Y5 @ N )
% 0.35/0.59 = A2 ) )
% 0.35/0.59 => ( Y5 = X5 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % realpow_pos_nth_unique
% 0.35/0.59 thf(fact_3871_realpow__pos__nth,axiom,
% 0.35/0.59 ! [N: nat,A2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ? [R3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.59 & ( ( power_power_real @ R3 @ N )
% 0.35/0.59 = A2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % realpow_pos_nth
% 0.35/0.59 thf(fact_3872_lemma__interval__lt,axiom,
% 0.35/0.59 ! [A2: real,X2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_real @ A2 @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ X2 @ B3 )
% 0.35/0.59 => ? [D6: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.59 & ! [Y5: real] :
% 0.35/0.59 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
% 0.35/0.59 => ( ( ord_less_real @ A2 @ Y5 )
% 0.35/0.59 & ( ord_less_real @ Y5 @ B3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % lemma_interval_lt
% 0.35/0.59 thf(fact_3873_realpow__pos__nth2,axiom,
% 0.35/0.59 ! [A2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ? [R3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.59 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 0.35/0.59 = A2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % realpow_pos_nth2
% 0.35/0.59 thf(fact_3874_add__def,axiom,
% 0.35/0.59 ( vEBT_VEBT_add
% 0.35/0.59 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_def
% 0.35/0.59 thf(fact_3875_nth__enumerate__eq,axiom,
% 0.35/0.59 ! [M: nat,Xs: list_int,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
% 0.35/0.59 = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_enumerate_eq
% 0.35/0.59 thf(fact_3876_nth__enumerate__eq,axiom,
% 0.35/0.59 ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
% 0.35/0.59 = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_enumerate_eq
% 0.35/0.59 thf(fact_3877_nth__enumerate__eq,axiom,
% 0.35/0.59 ! [M: nat,Xs: list_o,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 0.35/0.59 => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
% 0.35/0.59 = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_enumerate_eq
% 0.35/0.59 thf(fact_3878_nth__enumerate__eq,axiom,
% 0.35/0.59 ! [M: nat,Xs: list_nat,N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 0.35/0.59 => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
% 0.35/0.59 = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_enumerate_eq
% 0.35/0.59 thf(fact_3879_zle__add1__eq__le,axiom,
% 0.35/0.59 ! [W2: int,Z: int] :
% 0.35/0.59 ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.35/0.59 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 0.35/0.59
% 0.35/0.59 % zle_add1_eq_le
% 0.35/0.59 thf(fact_3880_real__0__less__add__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_0_less_add_iff
% 0.35/0.59 thf(fact_3881_real__add__less__0__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_add_less_0_iff
% 0.35/0.59 thf(fact_3882_real__add__le__0__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_add_le_0_iff
% 0.35/0.59 thf(fact_3883_real__0__le__add__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_0_le_add_iff
% 0.35/0.59 thf(fact_3884_int__ge__induct,axiom,
% 0.35/0.59 ! [K: int,I: int,P: int > $o] :
% 0.35/0.59 ( ( ord_less_eq_int @ K @ I )
% 0.35/0.59 => ( ( P @ K )
% 0.35/0.59 => ( ! [I2: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ K @ I2 )
% 0.35/0.59 => ( ( P @ I2 )
% 0.35/0.59 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.59 => ( P @ I ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % int_ge_induct
% 0.35/0.59 thf(fact_3885_zless__add1__eq,axiom,
% 0.35/0.59 ! [W2: int,Z: int] :
% 0.35/0.59 ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
% 0.35/0.59 = ( ( ord_less_int @ W2 @ Z )
% 0.35/0.59 | ( W2 = Z ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zless_add1_eq
% 0.35/0.59 thf(fact_3886_int__gr__induct,axiom,
% 0.35/0.59 ! [K: int,I: int,P: int > $o] :
% 0.35/0.59 ( ( ord_less_int @ K @ I )
% 0.35/0.59 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.35/0.59 => ( ! [I2: int] :
% 0.35/0.59 ( ( ord_less_int @ K @ I2 )
% 0.35/0.59 => ( ( P @ I2 )
% 0.35/0.59 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.59 => ( P @ I ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % int_gr_induct
% 0.35/0.59 thf(fact_3887_zle__iff__zadd,axiom,
% 0.35/0.59 ( ord_less_eq_int
% 0.35/0.59 = ( ^ [W3: int,Z2: int] :
% 0.35/0.59 ? [N2: nat] :
% 0.35/0.59 ( Z2
% 0.35/0.59 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zle_iff_zadd
% 0.35/0.59 thf(fact_3888_zless__iff__Suc__zadd,axiom,
% 0.35/0.59 ( ord_less_int
% 0.35/0.59 = ( ^ [W3: int,Z2: int] :
% 0.35/0.59 ? [N2: nat] :
% 0.35/0.59 ( Z2
% 0.35/0.59 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zless_iff_Suc_zadd
% 0.35/0.59 thf(fact_3889_odd__less__0__iff,axiom,
% 0.35/0.59 ! [Z: int] :
% 0.35/0.59 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 0.35/0.59 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.59
% 0.35/0.59 % odd_less_0_iff
% 0.35/0.59 thf(fact_3890_add1__zle__eq,axiom,
% 0.35/0.59 ! [W2: int,Z: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
% 0.35/0.59 = ( ord_less_int @ W2 @ Z ) ) ).
% 0.35/0.59
% 0.35/0.59 % add1_zle_eq
% 0.35/0.59 thf(fact_3891_zless__imp__add1__zle,axiom,
% 0.35/0.59 ! [W2: int,Z: int] :
% 0.35/0.59 ( ( ord_less_int @ W2 @ Z )
% 0.35/0.59 => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% 0.35/0.59
% 0.35/0.59 % zless_imp_add1_zle
% 0.35/0.59 thf(fact_3892_int__induct,axiom,
% 0.35/0.59 ! [P: int > $o,K: int,I: int] :
% 0.35/0.59 ( ( P @ K )
% 0.35/0.59 => ( ! [I2: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ K @ I2 )
% 0.35/0.59 => ( ( P @ I2 )
% 0.35/0.59 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.59 => ( ! [I2: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ I2 @ K )
% 0.35/0.59 => ( ( P @ I2 )
% 0.35/0.59 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 0.35/0.59 => ( P @ I ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % int_induct
% 0.35/0.59 thf(fact_3893_nat__less__real__le,axiom,
% 0.35/0.59 ( ord_less_nat
% 0.35/0.59 = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nat_less_real_le
% 0.35/0.59 thf(fact_3894_nat__le__real__less,axiom,
% 0.35/0.59 ( ord_less_eq_nat
% 0.35/0.59 = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nat_le_real_less
% 0.35/0.59 thf(fact_3895_le__imp__0__less,axiom,
% 0.35/0.59 ! [Z: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.59 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % le_imp_0_less
% 0.35/0.59 thf(fact_3896_div__pos__neg__trivial,axiom,
% 0.35/0.59 ! [K: int,L: int] :
% 0.35/0.59 ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.59 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ K @ L )
% 0.35/0.59 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_pos_neg_trivial
% 0.35/0.59 thf(fact_3897_div__pos__geq,axiom,
% 0.35/0.59 ! [L: int,K: int] :
% 0.35/0.59 ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.59 => ( ( ord_less_eq_int @ L @ K )
% 0.35/0.59 => ( ( divide_divide_int @ K @ L )
% 0.35/0.59 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_pos_geq
% 0.35/0.59 thf(fact_3898_sin__bound__lemma,axiom,
% 0.35/0.59 ! [X2: real,Y3: real,U: real,V: real] :
% 0.35/0.59 ( ( X2 = Y3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 0.35/0.59 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sin_bound_lemma
% 0.35/0.59 thf(fact_3899_Euclid__induct,axiom,
% 0.35/0.59 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 0.35/0.59 ( ! [A: nat,B: nat] :
% 0.35/0.59 ( ( P @ A @ B )
% 0.35/0.59 = ( P @ B @ A ) )
% 0.35/0.59 => ( ! [A: nat] : ( P @ A @ zero_zero_nat )
% 0.35/0.59 => ( ! [A: nat,B: nat] :
% 0.35/0.59 ( ( P @ A @ B )
% 0.35/0.59 => ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
% 0.35/0.59 => ( P @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Euclid_induct
% 0.35/0.59 thf(fact_3900_add__0__iff,axiom,
% 0.35/0.59 ! [B3: real,A2: real] :
% 0.35/0.59 ( ( B3
% 0.35/0.59 = ( plus_plus_real @ B3 @ A2 ) )
% 0.35/0.59 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_0_iff
% 0.35/0.59 thf(fact_3901_add__0__iff,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat] :
% 0.35/0.59 ( ( B3
% 0.35/0.59 = ( plus_plus_rat @ B3 @ A2 ) )
% 0.35/0.59 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_0_iff
% 0.35/0.59 thf(fact_3902_add__0__iff,axiom,
% 0.35/0.59 ! [B3: nat,A2: nat] :
% 0.35/0.59 ( ( B3
% 0.35/0.59 = ( plus_plus_nat @ B3 @ A2 ) )
% 0.35/0.59 = ( A2 = zero_zero_nat ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_0_iff
% 0.35/0.59 thf(fact_3903_add__0__iff,axiom,
% 0.35/0.59 ! [B3: int,A2: int] :
% 0.35/0.59 ( ( B3
% 0.35/0.59 = ( plus_plus_int @ B3 @ A2 ) )
% 0.35/0.59 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.59
% 0.35/0.59 % add_0_iff
% 0.35/0.59 thf(fact_3904_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.35/0.59 ! [X2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ge_one_minus_x_over_n_power_n
% 0.35/0.59 thf(fact_3905_frac__unique__iff,axiom,
% 0.35/0.59 ! [X2: real,A2: real] :
% 0.35/0.59 ( ( ( archim2898591450579166408c_real @ X2 )
% 0.35/0.59 = A2 )
% 0.35/0.59 = ( ( member_real @ ( minus_minus_real @ X2 @ A2 ) @ ring_1_Ints_real )
% 0.35/0.59 & ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 & ( ord_less_real @ A2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_unique_iff
% 0.35/0.59 thf(fact_3906_frac__unique__iff,axiom,
% 0.35/0.59 ! [X2: rat,A2: rat] :
% 0.35/0.59 ( ( ( archimedean_frac_rat @ X2 )
% 0.35/0.59 = A2 )
% 0.35/0.59 = ( ( member_rat @ ( minus_minus_rat @ X2 @ A2 ) @ ring_1_Ints_rat )
% 0.35/0.59 & ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 & ( ord_less_rat @ A2 @ one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_unique_iff
% 0.35/0.59 thf(fact_3907_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_int,Ys2: list_int] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3908_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3909_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_int,Ys2: list_o] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( product_Pair_int_o @ ( nth_int @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3910_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_int,Ys2: list_nat] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3911_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3912_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3913_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3914_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3915_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_o,Ys2: list_int] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3916_nth__zip,axiom,
% 0.35/0.59 ! [I: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
% 0.35/0.59 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.59 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.35/0.59 => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys2 ) @ I )
% 0.35/0.59 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nth_zip
% 0.35/0.59 thf(fact_3917_exp__less__mono,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.59 => ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_less_mono
% 0.35/0.59 thf(fact_3918_exp__less__cancel__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_less_cancel_iff
% 0.35/0.59 thf(fact_3919_exp__le__cancel__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_le_cancel_iff
% 0.35/0.59 thf(fact_3920_exp__zero,axiom,
% 0.35/0.59 ( ( exp_complex @ zero_zero_complex )
% 0.35/0.59 = one_one_complex ) ).
% 0.35/0.59
% 0.35/0.59 % exp_zero
% 0.35/0.59 thf(fact_3921_exp__zero,axiom,
% 0.35/0.59 ( ( exp_real @ zero_zero_real )
% 0.35/0.59 = one_one_real ) ).
% 0.35/0.59
% 0.35/0.59 % exp_zero
% 0.35/0.59 thf(fact_3922_frac__eq__0__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ( archim2898591450579166408c_real @ X2 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_eq_0_iff
% 0.35/0.59 thf(fact_3923_frac__eq__0__iff,axiom,
% 0.35/0.59 ! [X2: rat] :
% 0.35/0.59 ( ( ( archimedean_frac_rat @ X2 )
% 0.35/0.59 = zero_zero_rat )
% 0.35/0.59 = ( member_rat @ X2 @ ring_1_Ints_rat ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_eq_0_iff
% 0.35/0.59 thf(fact_3924_exp__less__one__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_less_one_iff
% 0.35/0.59 thf(fact_3925_one__less__exp__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
% 0.35/0.59 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_exp_iff
% 0.35/0.59 thf(fact_3926_exp__le__one__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_le_one_iff
% 0.35/0.59 thf(fact_3927_one__le__exp__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_le_exp_iff
% 0.35/0.59 thf(fact_3928_frac__gt__0__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) )
% 0.35/0.59 = ( ~ ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_gt_0_iff
% 0.35/0.59 thf(fact_3929_frac__gt__0__iff,axiom,
% 0.35/0.59 ! [X2: rat] :
% 0.35/0.59 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) )
% 0.35/0.59 = ( ~ ( member_rat @ X2 @ ring_1_Ints_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_gt_0_iff
% 0.35/0.59 thf(fact_3930_exp__not__eq__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( exp_real @ X2 )
% 0.35/0.59 != zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % exp_not_eq_zero
% 0.35/0.59 thf(fact_3931_exp__less__cancel,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 0.35/0.59 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_less_cancel
% 0.35/0.59 thf(fact_3932_Ints__0,axiom,
% 0.35/0.59 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 0.35/0.59
% 0.35/0.59 % Ints_0
% 0.35/0.59 thf(fact_3933_Ints__0,axiom,
% 0.35/0.59 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 0.35/0.59
% 0.35/0.59 % Ints_0
% 0.35/0.59 thf(fact_3934_Ints__0,axiom,
% 0.35/0.59 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 0.35/0.59
% 0.35/0.59 % Ints_0
% 0.35/0.59 thf(fact_3935_not__exp__less__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ~ ( ord_less_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % not_exp_less_zero
% 0.35/0.59 thf(fact_3936_exp__gt__zero,axiom,
% 0.35/0.59 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_gt_zero
% 0.35/0.59 thf(fact_3937_exp__total,axiom,
% 0.35/0.59 ! [Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ? [X5: real] :
% 0.35/0.59 ( ( exp_real @ X5 )
% 0.35/0.59 = Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_total
% 0.35/0.59 thf(fact_3938_exp__ge__zero,axiom,
% 0.35/0.59 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ge_zero
% 0.35/0.59 thf(fact_3939_not__exp__le__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % not_exp_le_zero
% 0.35/0.59 thf(fact_3940_exp__ge__add__one__self,axiom,
% 0.35/0.59 ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ge_add_one_self
% 0.35/0.59 thf(fact_3941_exp__gt__one,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_gt_one
% 0.35/0.59 thf(fact_3942_exp__ge__add__one__self__aux,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ge_add_one_self_aux
% 0.35/0.59 thf(fact_3943_Ints__double__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( ( plus_plus_real @ A2 @ A2 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( A2 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_double_eq_0_iff
% 0.35/0.59 thf(fact_3944_Ints__double__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( ( plus_plus_rat @ A2 @ A2 )
% 0.35/0.59 = zero_zero_rat )
% 0.35/0.59 = ( A2 = zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_double_eq_0_iff
% 0.35/0.59 thf(fact_3945_Ints__double__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( member_int @ A2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( ( plus_plus_int @ A2 @ A2 )
% 0.35/0.59 = zero_zero_int )
% 0.35/0.59 = ( A2 = zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_double_eq_0_iff
% 0.35/0.59 thf(fact_3946_lemma__exp__total,axiom,
% 0.35/0.59 ! [Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 0.35/0.59 => ? [X5: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.35/0.59 & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 0.35/0.59 & ( ( exp_real @ X5 )
% 0.35/0.59 = Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % lemma_exp_total
% 0.35/0.59 thf(fact_3947_Ints__odd__nonzero,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( member_complex @ A2 @ ring_1_Ints_complex )
% 0.35/0.59 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A2 ) @ A2 )
% 0.35/0.59 != zero_zero_complex ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_nonzero
% 0.35/0.59 thf(fact_3948_Ints__odd__nonzero,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 )
% 0.35/0.59 != zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_nonzero
% 0.35/0.59 thf(fact_3949_Ints__odd__nonzero,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 )
% 0.35/0.59 != zero_zero_rat ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_nonzero
% 0.35/0.59 thf(fact_3950_Ints__odd__nonzero,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( member_int @ A2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 )
% 0.35/0.59 != zero_zero_int ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_nonzero
% 0.35/0.59 thf(fact_3951_exp__divide__power__eq,axiom,
% 0.35/0.59 ! [N: nat,X2: complex] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 0.35/0.59 = ( exp_complex @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_divide_power_eq
% 0.35/0.59 thf(fact_3952_exp__divide__power__eq,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 0.35/0.59 = ( exp_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_divide_power_eq
% 0.35/0.59 thf(fact_3953_Ints__odd__less__0,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_less_0
% 0.35/0.59 thf(fact_3954_Ints__odd__less__0,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 ) @ zero_zero_rat )
% 0.35/0.59 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_less_0
% 0.35/0.59 thf(fact_3955_Ints__odd__less__0,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( member_int @ A2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 ) @ zero_zero_int )
% 0.35/0.59 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_odd_less_0
% 0.35/0.59 thf(fact_3956_Ints__nonzero__abs__ge1,axiom,
% 0.35/0.59 ! [X2: code_integer] :
% 0.35/0.59 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 0.35/0.59 => ( ( X2 != zero_z3403309356797280102nteger )
% 0.35/0.59 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_ge1
% 0.35/0.59 thf(fact_3957_Ints__nonzero__abs__ge1,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( member_real @ X2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( X2 != zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_ge1
% 0.35/0.59 thf(fact_3958_Ints__nonzero__abs__ge1,axiom,
% 0.35/0.59 ! [X2: rat] :
% 0.35/0.59 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( X2 != zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_ge1
% 0.35/0.59 thf(fact_3959_Ints__nonzero__abs__ge1,axiom,
% 0.35/0.59 ! [X2: int] :
% 0.35/0.59 ( ( member_int @ X2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( X2 != zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_ge1
% 0.35/0.59 thf(fact_3960_Ints__nonzero__abs__less1,axiom,
% 0.35/0.59 ! [X2: code_integer] :
% 0.35/0.59 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 0.35/0.59 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer )
% 0.35/0.59 => ( X2 = zero_z3403309356797280102nteger ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_less1
% 0.35/0.59 thf(fact_3961_Ints__nonzero__abs__less1,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( member_real @ X2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.59 => ( X2 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_less1
% 0.35/0.59 thf(fact_3962_Ints__nonzero__abs__less1,axiom,
% 0.35/0.59 ! [X2: rat] :
% 0.35/0.59 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat )
% 0.35/0.59 => ( X2 = zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_less1
% 0.35/0.59 thf(fact_3963_Ints__nonzero__abs__less1,axiom,
% 0.35/0.59 ! [X2: int] :
% 0.35/0.59 ( ( member_int @ X2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int )
% 0.35/0.59 => ( X2 = zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_nonzero_abs_less1
% 0.35/0.59 thf(fact_3964_Ints__eq__abs__less1,axiom,
% 0.35/0.59 ! [X2: code_integer,Y3: code_integer] :
% 0.35/0.59 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 0.35/0.59 => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
% 0.35/0.59 => ( ( X2 = Y3 )
% 0.35/0.59 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_eq_abs_less1
% 0.35/0.59 thf(fact_3965_Ints__eq__abs__less1,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( member_real @ X2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( member_real @ Y3 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( X2 = Y3 )
% 0.35/0.59 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_eq_abs_less1
% 0.35/0.59 thf(fact_3966_Ints__eq__abs__less1,axiom,
% 0.35/0.59 ! [X2: rat,Y3: rat] :
% 0.35/0.59 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( X2 = Y3 )
% 0.35/0.59 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_eq_abs_less1
% 0.35/0.59 thf(fact_3967_Ints__eq__abs__less1,axiom,
% 0.35/0.59 ! [X2: int,Y3: int] :
% 0.35/0.59 ( ( member_int @ X2 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( member_int @ Y3 @ ring_1_Ints_int )
% 0.35/0.59 => ( ( X2 = Y3 )
% 0.35/0.59 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Ints_eq_abs_less1
% 0.35/0.59 thf(fact_3968_frac__neg,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ( member_real @ X2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.59 = zero_zero_real ) )
% 0.35/0.59 & ( ~ ( member_real @ X2 @ ring_1_Ints_real )
% 0.35/0.59 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.59 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X2 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_neg
% 0.35/0.59 thf(fact_3969_frac__neg,axiom,
% 0.35/0.59 ! [X2: rat] :
% 0.35/0.59 ( ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
% 0.35/0.59 = zero_zero_rat ) )
% 0.35/0.59 & ( ~ ( member_rat @ X2 @ ring_1_Ints_rat )
% 0.35/0.59 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
% 0.35/0.59 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X2 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % frac_neg
% 0.35/0.59 thf(fact_3970_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X2 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ge_one_plus_x_over_n_power_n
% 0.35/0.59 thf(fact_3971_arcosh__1,axiom,
% 0.35/0.59 ( ( arcosh_real @ one_one_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % arcosh_1
% 0.35/0.59 thf(fact_3972_artanh__0,axiom,
% 0.35/0.59 ( ( artanh_real @ zero_zero_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % artanh_0
% 0.35/0.59 thf(fact_3973_arsinh__0,axiom,
% 0.35/0.59 ( ( arsinh_real @ zero_zero_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % arsinh_0
% 0.35/0.59 thf(fact_3974_artanh__minus__real,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.59 => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.59 = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % artanh_minus_real
% 0.35/0.59 thf(fact_3975_ln__one,axiom,
% 0.35/0.59 ( ( ln_ln_real @ one_one_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % ln_one
% 0.35/0.59 thf(fact_3976_sinh__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ( sinh_real @ X2 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( member_real @ ( exp_real @ X2 ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_zero_iff
% 0.35/0.59 thf(fact_3977_sinh__zero__iff,axiom,
% 0.35/0.59 ! [X2: complex] :
% 0.35/0.59 ( ( ( sinh_complex @ X2 )
% 0.35/0.59 = zero_zero_complex )
% 0.35/0.59 = ( member_complex @ ( exp_complex @ X2 ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_zero_iff
% 0.35/0.59 thf(fact_3978_Gcd__remove0__nat,axiom,
% 0.35/0.59 ! [M5: set_nat] :
% 0.35/0.59 ( ( finite_finite_nat @ M5 )
% 0.35/0.59 => ( ( gcd_Gcd_nat @ M5 )
% 0.35/0.59 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Gcd_remove0_nat
% 0.35/0.59 thf(fact_3979_log__of__power__le,axiom,
% 0.35/0.59 ! [M: nat,B3: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.59 => ( ord_less_eq_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_of_power_le
% 0.35/0.59 thf(fact_3980_sinh__real__less__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_less_iff
% 0.35/0.59 thf(fact_3981_sinh__real__le__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_le_iff
% 0.35/0.59 thf(fact_3982_ln__less__cancel__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_less_cancel_iff
% 0.35/0.59 thf(fact_3983_ln__inj__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ( ln_ln_real @ X2 )
% 0.35/0.59 = ( ln_ln_real @ Y3 ) )
% 0.35/0.59 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_inj_iff
% 0.35/0.59 thf(fact_3984_sinh__real__pos__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 0.35/0.59 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_pos_iff
% 0.35/0.59 thf(fact_3985_sinh__real__neg__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_neg_iff
% 0.35/0.59 thf(fact_3986_sinh__real__nonpos__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_nonpos_iff
% 0.35/0.59 thf(fact_3987_sinh__real__nonneg__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_real_nonneg_iff
% 0.35/0.59 thf(fact_3988_sinh__0,axiom,
% 0.35/0.59 ( ( sinh_real @ zero_zero_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % sinh_0
% 0.35/0.59 thf(fact_3989_ln__le__cancel__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_le_cancel_iff
% 0.35/0.59 thf(fact_3990_ln__less__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_less_zero_iff
% 0.35/0.59 thf(fact_3991_ln__gt__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 = ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_gt_zero_iff
% 0.35/0.59 thf(fact_3992_ln__eq__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ( ln_ln_real @ X2 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( X2 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_eq_zero_iff
% 0.35/0.59 thf(fact_3993_Gcd__empty,axiom,
% 0.35/0.59 ( ( gcd_Gcd_nat @ bot_bot_set_nat )
% 0.35/0.59 = zero_zero_nat ) ).
% 0.35/0.59
% 0.35/0.59 % Gcd_empty
% 0.35/0.59 thf(fact_3994_Gcd__empty,axiom,
% 0.35/0.59 ( ( gcd_Gcd_int @ bot_bot_set_int )
% 0.35/0.59 = zero_zero_int ) ).
% 0.35/0.59
% 0.35/0.59 % Gcd_empty
% 0.35/0.59 thf(fact_3995_exp__ln,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( exp_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 = X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ln
% 0.35/0.59 thf(fact_3996_exp__ln__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ( exp_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 = X2 )
% 0.35/0.59 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % exp_ln_iff
% 0.35/0.59 thf(fact_3997_ln__ge__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_ge_zero_iff
% 0.35/0.59 thf(fact_3998_ln__le__zero__iff,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_le_zero_iff
% 0.35/0.59 thf(fact_3999_zero__less__log__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
% 0.35/0.59 = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_less_log_cancel_iff
% 0.35/0.59 thf(fact_4000_log__less__zero__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_less_zero_cancel_iff
% 0.35/0.59 thf(fact_4001_one__less__log__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ ( log @ A2 @ X2 ) )
% 0.35/0.59 = ( ord_less_real @ A2 @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_less_log_cancel_iff
% 0.35/0.59 thf(fact_4002_log__less__one__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ A2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_less_one_cancel_iff
% 0.35/0.59 thf(fact_4003_log__less__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_less_cancel_iff
% 0.35/0.59 thf(fact_4004_log__eq__one,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( A2 != one_one_real )
% 0.35/0.59 => ( ( log @ A2 @ A2 )
% 0.35/0.59 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_eq_one
% 0.35/0.59 thf(fact_4005_Gcd__0__iff,axiom,
% 0.35/0.59 ! [A3: set_nat] :
% 0.35/0.59 ( ( ( gcd_Gcd_nat @ A3 )
% 0.35/0.59 = zero_zero_nat )
% 0.35/0.59 = ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Gcd_0_iff
% 0.35/0.59 thf(fact_4006_Gcd__0__iff,axiom,
% 0.35/0.59 ! [A3: set_int] :
% 0.35/0.59 ( ( ( gcd_Gcd_int @ A3 )
% 0.35/0.59 = zero_zero_int )
% 0.35/0.59 = ( ord_less_eq_set_int @ A3 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Gcd_0_iff
% 0.35/0.59 thf(fact_4007_log__le__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_le_cancel_iff
% 0.35/0.59 thf(fact_4008_log__le__one__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ A2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_le_one_cancel_iff
% 0.35/0.59 thf(fact_4009_one__le__log__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A2 @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ A2 @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % one_le_log_cancel_iff
% 0.35/0.59 thf(fact_4010_log__le__zero__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_le_zero_cancel_iff
% 0.35/0.59 thf(fact_4011_zero__le__log__cancel__iff,axiom,
% 0.35/0.59 ! [A2: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_le_log_cancel_iff
% 0.35/0.59 thf(fact_4012_log__pow__cancel,axiom,
% 0.35/0.59 ! [A2: real,B3: nat] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( A2 != one_one_real )
% 0.35/0.59 => ( ( log @ A2 @ ( power_power_real @ A2 @ B3 ) )
% 0.35/0.59 = ( semiri5074537144036343181t_real @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_pow_cancel
% 0.35/0.59 thf(fact_4013_ln__less__self,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_less_self
% 0.35/0.59 thf(fact_4014_ln__bound,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_bound
% 0.35/0.59 thf(fact_4015_ln__gt__zero__imp__gt__one,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_gt_zero_imp_gt_one
% 0.35/0.59 thf(fact_4016_ln__less__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.59 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_less_zero
% 0.35/0.59 thf(fact_4017_ln__gt__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.59 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_gt_zero
% 0.35/0.59 thf(fact_4018_ln__ge__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_ge_zero
% 0.35/0.59 thf(fact_4019_ln__ge__zero__imp__ge__one,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_ge_zero_imp_ge_one
% 0.35/0.59 thf(fact_4020_ln__add__one__self__le__self,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_add_one_self_le_self
% 0.35/0.59 thf(fact_4021_ln__eq__minus__one,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ( ln_ln_real @ X2 )
% 0.35/0.59 = ( minus_minus_real @ X2 @ one_one_real ) )
% 0.35/0.59 => ( X2 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_eq_minus_one
% 0.35/0.59 thf(fact_4022_ln__div,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.35/0.59 = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_div
% 0.35/0.59 thf(fact_4023_log__base__change,axiom,
% 0.35/0.59 ! [A2: real,B3: real,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( A2 != one_one_real )
% 0.35/0.59 => ( ( log @ B3 @ X2 )
% 0.35/0.59 = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_base_change
% 0.35/0.59 thf(fact_4024_ln__ge__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X2 ) )
% 0.35/0.59 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_ge_iff
% 0.35/0.59 thf(fact_4025_ln__x__over__x__mono,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.59 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_x_over_x_mono
% 0.35/0.59 thf(fact_4026_less__log__of__power,axiom,
% 0.35/0.59 ! [B3: real,N: nat,M: real] :
% 0.35/0.59 ( ( ord_less_real @ ( power_power_real @ B3 @ N ) @ M )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.59 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % less_log_of_power
% 0.35/0.59 thf(fact_4027_log__of__power__eq,axiom,
% 0.35/0.59 ! [M: nat,B3: real,N: nat] :
% 0.35/0.59 ( ( ( semiri5074537144036343181t_real @ M )
% 0.35/0.59 = ( power_power_real @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.59 => ( ( semiri5074537144036343181t_real @ N )
% 0.35/0.59 = ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_of_power_eq
% 0.35/0.59 thf(fact_4028_ln__le__minus__one,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_le_minus_one
% 0.35/0.59 thf(fact_4029_ln__diff__le,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_diff_le
% 0.35/0.59 thf(fact_4030_ln__add__one__self__le__self2,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_add_one_self_le_self2
% 0.35/0.59 thf(fact_4031_log__divide,axiom,
% 0.35/0.59 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( A2 != one_one_real )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.59 => ( ( log @ A2 @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.35/0.59 = ( minus_minus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_divide
% 0.35/0.59 thf(fact_4032_le__log__of__power,axiom,
% 0.35/0.59 ! [B3: real,N: nat,M: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( power_power_real @ B3 @ N ) @ M )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.59 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % le_log_of_power
% 0.35/0.59 thf(fact_4033_log__base__pow,axiom,
% 0.35/0.59 ! [A2: real,N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( log @ ( power_power_real @ A2 @ N ) @ X2 )
% 0.35/0.59 = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_base_pow
% 0.35/0.59 thf(fact_4034_ln__one__minus__pos__upper__bound,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ln_one_minus_pos_upper_bound
% 0.35/0.59 thf(fact_4035_log__of__power__less,axiom,
% 0.35/0.59 ! [M: nat,B3: real,N: nat] :
% 0.35/0.59 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.59 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.59 => ( ord_less_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_of_power_less
% 0.35/0.59 thf(fact_4036_log__root,axiom,
% 0.35/0.59 ! [N: nat,A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( log @ B3 @ ( root @ N @ A2 ) )
% 0.35/0.59 = ( divide_divide_real @ ( log @ B3 @ A2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % log_root
% 0.35/0.59 thf(fact_4037_decr__lemma,axiom,
% 0.35/0.59 ! [D: int,X2: int,Z: int] :
% 0.35/0.59 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.59 => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 0.35/0.59
% 0.35/0.59 % decr_lemma
% 0.35/0.59 thf(fact_4038_incr__lemma,axiom,
% 0.35/0.59 ! [D: int,Z: int,X2: int] :
% 0.35/0.59 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.59 => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % incr_lemma
% 0.35/0.59 thf(fact_4039_Bernoulli__inequality,axiom,
% 0.35/0.59 ! [X2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % Bernoulli_inequality
% 0.35/0.59 thf(fact_4040_linear__plus__1__le__power,axiom,
% 0.35/0.59 ! [X2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % linear_plus_1_le_power
% 0.35/0.59 thf(fact_4041_find__Some__iff,axiom,
% 0.35/0.59 ! [P: int > $o,Xs: list_int,X2: int] :
% 0.35/0.59 ( ( ( find_int @ P @ Xs )
% 0.35/0.59 = ( some_int @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_int @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_int @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4042_find__Some__iff,axiom,
% 0.35/0.59 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
% 0.35/0.59 ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.35/0.59 = ( some_P7363390416028606310at_nat @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4043_find__Some__iff,axiom,
% 0.35/0.59 ! [P: num > $o,Xs: list_num,X2: num] :
% 0.35/0.59 ( ( ( find_num @ P @ Xs )
% 0.35/0.59 = ( some_num @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_num @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_num @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4044_find__Some__iff,axiom,
% 0.35/0.59 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.35/0.59 ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.35/0.59 = ( some_VEBT_VEBT @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4045_find__Some__iff,axiom,
% 0.35/0.59 ! [P: $o > $o,Xs: list_o,X2: $o] :
% 0.35/0.59 ( ( ( find_o @ P @ Xs )
% 0.35/0.59 = ( some_o @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_o @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_o @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4046_find__Some__iff,axiom,
% 0.35/0.59 ! [P: nat > $o,Xs: list_nat,X2: nat] :
% 0.35/0.59 ( ( ( find_nat @ P @ Xs )
% 0.35/0.59 = ( some_nat @ X2 ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_nat @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_nat @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff
% 0.35/0.59 thf(fact_4047_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: int,P: int > $o,Xs: list_int] :
% 0.35/0.59 ( ( ( some_int @ X2 )
% 0.35/0.59 = ( find_int @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_int @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_int @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4048_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.35/0.59 ( ( ( some_P7363390416028606310at_nat @ X2 )
% 0.35/0.59 = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4049_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: num,P: num > $o,Xs: list_num] :
% 0.35/0.59 ( ( ( some_num @ X2 )
% 0.35/0.59 = ( find_num @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_num @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_num @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4050_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.35/0.59 ( ( ( some_VEBT_VEBT @ X2 )
% 0.35/0.59 = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4051_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: $o,P: $o > $o,Xs: list_o] :
% 0.35/0.59 ( ( ( some_o @ X2 )
% 0.35/0.59 = ( find_o @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_o @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_o @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4052_find__Some__iff2,axiom,
% 0.35/0.59 ! [X2: nat,P: nat > $o,Xs: list_nat] :
% 0.35/0.59 ( ( ( some_nat @ X2 )
% 0.35/0.59 = ( find_nat @ P @ Xs ) )
% 0.35/0.59 = ( ? [I4: nat] :
% 0.35/0.59 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 0.35/0.59 & ( P @ ( nth_nat @ Xs @ I4 ) )
% 0.35/0.59 & ( X2
% 0.35/0.59 = ( nth_nat @ Xs @ I4 ) )
% 0.35/0.59 & ! [J3: nat] :
% 0.35/0.59 ( ( ord_less_nat @ J3 @ I4 )
% 0.35/0.59 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % find_Some_iff2
% 0.35/0.59 thf(fact_4053_mult__cancel__right,axiom,
% 0.35/0.59 ! [A2: complex,C: complex,B3: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ A2 @ C )
% 0.35/0.59 = ( times_times_complex @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right
% 0.35/0.59 thf(fact_4054_mult__cancel__right,axiom,
% 0.35/0.59 ! [A2: real,C: real,B3: real] :
% 0.35/0.59 ( ( ( times_times_real @ A2 @ C )
% 0.35/0.59 = ( times_times_real @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right
% 0.35/0.59 thf(fact_4055_mult__cancel__right,axiom,
% 0.35/0.59 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ A2 @ C )
% 0.35/0.59 = ( times_times_rat @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right
% 0.35/0.59 thf(fact_4056_mult__cancel__right,axiom,
% 0.35/0.59 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.59 ( ( ( times_times_nat @ A2 @ C )
% 0.35/0.59 = ( times_times_nat @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_nat )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right
% 0.35/0.59 thf(fact_4057_mult__cancel__right,axiom,
% 0.35/0.59 ! [A2: int,C: int,B3: int] :
% 0.35/0.59 ( ( ( times_times_int @ A2 @ C )
% 0.35/0.59 = ( times_times_int @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right
% 0.35/0.59 thf(fact_4058_mult__cancel__left,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ C @ A2 )
% 0.35/0.59 = ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left
% 0.35/0.59 thf(fact_4059_mult__cancel__left,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( ( times_times_real @ C @ A2 )
% 0.35/0.59 = ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left
% 0.35/0.59 thf(fact_4060_mult__cancel__left,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ C @ A2 )
% 0.35/0.59 = ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left
% 0.35/0.59 thf(fact_4061_mult__cancel__left,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( times_times_nat @ C @ A2 )
% 0.35/0.59 = ( times_times_nat @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_nat )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left
% 0.35/0.59 thf(fact_4062_mult__cancel__left,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( ( times_times_int @ C @ A2 )
% 0.35/0.59 = ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left
% 0.35/0.59 thf(fact_4063_mult__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ A2 @ B3 )
% 0.35/0.59 = zero_zero_complex )
% 0.35/0.59 = ( ( A2 = zero_zero_complex )
% 0.35/0.59 | ( B3 = zero_zero_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_eq_0_iff
% 0.35/0.59 thf(fact_4064_mult__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ( times_times_real @ A2 @ B3 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( ( A2 = zero_zero_real )
% 0.35/0.59 | ( B3 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_eq_0_iff
% 0.35/0.59 thf(fact_4065_mult__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ A2 @ B3 )
% 0.35/0.59 = zero_zero_rat )
% 0.35/0.59 = ( ( A2 = zero_zero_rat )
% 0.35/0.59 | ( B3 = zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_eq_0_iff
% 0.35/0.59 thf(fact_4066_mult__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( times_times_nat @ A2 @ B3 )
% 0.35/0.59 = zero_zero_nat )
% 0.35/0.59 = ( ( A2 = zero_zero_nat )
% 0.35/0.59 | ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_eq_0_iff
% 0.35/0.59 thf(fact_4067_mult__eq__0__iff,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ( times_times_int @ A2 @ B3 )
% 0.35/0.59 = zero_zero_int )
% 0.35/0.59 = ( ( A2 = zero_zero_int )
% 0.35/0.59 | ( B3 = zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_eq_0_iff
% 0.35/0.59 thf(fact_4068_mult__zero__right,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ A2 @ zero_zero_complex )
% 0.35/0.59 = zero_zero_complex ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_right
% 0.35/0.59 thf(fact_4069_mult__zero__right,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ A2 @ zero_zero_real )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_right
% 0.35/0.59 thf(fact_4070_mult__zero__right,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 = zero_zero_rat ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_right
% 0.35/0.59 thf(fact_4071_mult__zero__right,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 0.35/0.59 = zero_zero_nat ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_right
% 0.35/0.59 thf(fact_4072_mult__zero__right,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ A2 @ zero_zero_int )
% 0.35/0.59 = zero_zero_int ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_right
% 0.35/0.59 thf(fact_4073_mult__zero__left,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ zero_zero_complex @ A2 )
% 0.35/0.59 = zero_zero_complex ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_left
% 0.35/0.59 thf(fact_4074_mult__zero__left,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ zero_zero_real @ A2 )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_left
% 0.35/0.59 thf(fact_4075_mult__zero__left,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 = zero_zero_rat ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_left
% 0.35/0.59 thf(fact_4076_mult__zero__left,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 = zero_zero_nat ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_left
% 0.35/0.59 thf(fact_4077_mult__zero__left,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ zero_zero_int @ A2 )
% 0.35/0.59 = zero_zero_int ) ).
% 0.35/0.59
% 0.35/0.59 % mult_zero_left
% 0.35/0.59 thf(fact_4078_mult_Oright__neutral,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ A2 @ one_one_complex )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.right_neutral
% 0.35/0.59 thf(fact_4079_mult_Oright__neutral,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ A2 @ one_one_real )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.right_neutral
% 0.35/0.59 thf(fact_4080_mult_Oright__neutral,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ A2 @ one_one_rat )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.right_neutral
% 0.35/0.59 thf(fact_4081_mult_Oright__neutral,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.right_neutral
% 0.35/0.59 thf(fact_4082_mult_Oright__neutral,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ A2 @ one_one_int )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.right_neutral
% 0.35/0.59 thf(fact_4083_mult__1,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ one_one_complex @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult_1
% 0.35/0.59 thf(fact_4084_mult__1,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ one_one_real @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult_1
% 0.35/0.59 thf(fact_4085_mult__1,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ one_one_rat @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult_1
% 0.35/0.59 thf(fact_4086_mult__1,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult_1
% 0.35/0.59 thf(fact_4087_mult__1,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ one_one_int @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult_1
% 0.35/0.59 thf(fact_4088_of__nat__mult,axiom,
% 0.35/0.59 ! [M: nat,N: nat] :
% 0.35/0.59 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 0.35/0.59 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % of_nat_mult
% 0.35/0.59 thf(fact_4089_of__nat__mult,axiom,
% 0.35/0.59 ! [M: nat,N: nat] :
% 0.35/0.59 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 0.35/0.59 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % of_nat_mult
% 0.35/0.59 thf(fact_4090_of__nat__mult,axiom,
% 0.35/0.59 ! [M: nat,N: nat] :
% 0.35/0.59 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 0.35/0.59 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % of_nat_mult
% 0.35/0.59 thf(fact_4091_of__nat__mult,axiom,
% 0.35/0.59 ! [M: nat,N: nat] :
% 0.35/0.59 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 0.35/0.59 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % of_nat_mult
% 0.35/0.59 thf(fact_4092_of__nat__mult,axiom,
% 0.35/0.59 ! [M: nat,N: nat] :
% 0.35/0.59 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 0.35/0.59 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % of_nat_mult
% 0.35/0.59 thf(fact_4093_sum__squares__eq__zero__iff,axiom,
% 0.35/0.59 ! [X2: real,Y3: real] :
% 0.35/0.59 ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( ( X2 = zero_zero_real )
% 0.35/0.59 & ( Y3 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sum_squares_eq_zero_iff
% 0.35/0.59 thf(fact_4094_sum__squares__eq__zero__iff,axiom,
% 0.35/0.59 ! [X2: rat,Y3: rat] :
% 0.35/0.59 ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.35/0.59 = zero_zero_rat )
% 0.35/0.59 = ( ( X2 = zero_zero_rat )
% 0.35/0.59 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sum_squares_eq_zero_iff
% 0.35/0.59 thf(fact_4095_sum__squares__eq__zero__iff,axiom,
% 0.35/0.59 ! [X2: int,Y3: int] :
% 0.35/0.59 ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.35/0.59 = zero_zero_int )
% 0.35/0.59 = ( ( X2 = zero_zero_int )
% 0.35/0.59 & ( Y3 = zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % sum_squares_eq_zero_iff
% 0.35/0.59 thf(fact_4096_mult__cancel__right2,axiom,
% 0.35/0.59 ! [A2: complex,C: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ A2 @ C )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( A2 = one_one_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right2
% 0.35/0.59 thf(fact_4097_mult__cancel__right2,axiom,
% 0.35/0.59 ! [A2: real,C: real] :
% 0.35/0.59 ( ( ( times_times_real @ A2 @ C )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( A2 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right2
% 0.35/0.59 thf(fact_4098_mult__cancel__right2,axiom,
% 0.35/0.59 ! [A2: rat,C: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ A2 @ C )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( A2 = one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right2
% 0.35/0.59 thf(fact_4099_mult__cancel__right2,axiom,
% 0.35/0.59 ! [A2: int,C: int] :
% 0.35/0.59 ( ( ( times_times_int @ A2 @ C )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( A2 = one_one_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right2
% 0.35/0.59 thf(fact_4100_mult__cancel__right1,axiom,
% 0.35/0.59 ! [C: complex,B3: complex] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_complex @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( B3 = one_one_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right1
% 0.35/0.59 thf(fact_4101_mult__cancel__right1,axiom,
% 0.35/0.59 ! [C: real,B3: real] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_real @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( B3 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right1
% 0.35/0.59 thf(fact_4102_mult__cancel__right1,axiom,
% 0.35/0.59 ! [C: rat,B3: rat] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_rat @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( B3 = one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right1
% 0.35/0.59 thf(fact_4103_mult__cancel__right1,axiom,
% 0.35/0.59 ! [C: int,B3: int] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_int @ B3 @ C ) )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( B3 = one_one_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_right1
% 0.35/0.59 thf(fact_4104_mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: complex,A2: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ C @ A2 )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( A2 = one_one_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left2
% 0.35/0.59 thf(fact_4105_mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: real,A2: real] :
% 0.35/0.59 ( ( ( times_times_real @ C @ A2 )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( A2 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left2
% 0.35/0.59 thf(fact_4106_mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: rat,A2: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ C @ A2 )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( A2 = one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left2
% 0.35/0.59 thf(fact_4107_mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: int,A2: int] :
% 0.35/0.59 ( ( ( times_times_int @ C @ A2 )
% 0.35/0.59 = C )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( A2 = one_one_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left2
% 0.35/0.59 thf(fact_4108_mult__cancel__left1,axiom,
% 0.35/0.59 ! [C: complex,B3: complex] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_complex )
% 0.35/0.59 | ( B3 = one_one_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left1
% 0.35/0.59 thf(fact_4109_mult__cancel__left1,axiom,
% 0.35/0.59 ! [C: real,B3: real] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_real )
% 0.35/0.59 | ( B3 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left1
% 0.35/0.59 thf(fact_4110_mult__cancel__left1,axiom,
% 0.35/0.59 ! [C: rat,B3: rat] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_rat )
% 0.35/0.59 | ( B3 = one_one_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left1
% 0.35/0.59 thf(fact_4111_mult__cancel__left1,axiom,
% 0.35/0.59 ! [C: int,B3: int] :
% 0.35/0.59 ( ( C
% 0.35/0.59 = ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = ( ( C = zero_zero_int )
% 0.35/0.59 | ( B3 = one_one_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_cancel_left1
% 0.35/0.59 thf(fact_4112_div__mult__mult1,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( C != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_int @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult1
% 0.35/0.59 thf(fact_4113_div__mult__mult1,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( C != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult1
% 0.35/0.59 thf(fact_4114_div__mult__mult2,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( C != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_int @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult2
% 0.35/0.59 thf(fact_4115_div__mult__mult2,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( C != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult2
% 0.35/0.59 thf(fact_4116_div__mult__mult1__if,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( ( C = zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = zero_zero_int ) )
% 0.35/0.59 & ( ( C != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult1_if
% 0.35/0.59 thf(fact_4117_div__mult__mult1__if,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( C = zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 0.35/0.59 = zero_zero_nat ) )
% 0.35/0.59 & ( ( C != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_mult1_if
% 0.35/0.59 thf(fact_4118_nonzero__mult__div__cancel__left,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( A2 != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ A2 )
% 0.35/0.59 = B3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_left
% 0.35/0.59 thf(fact_4119_nonzero__mult__div__cancel__left,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( A2 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ A2 )
% 0.35/0.59 = B3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_left
% 0.35/0.59 thf(fact_4120_nonzero__mult__div__cancel__left,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ A2 )
% 0.35/0.59 = B3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_left
% 0.35/0.59 thf(fact_4121_nonzero__mult__div__cancel__left,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( A2 != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ A2 )
% 0.35/0.59 = B3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_left
% 0.35/0.59 thf(fact_4122_nonzero__mult__div__cancel__left,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( A2 != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ A2 )
% 0.35/0.59 = B3 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_left
% 0.35/0.59 thf(fact_4123_nonzero__mult__div__cancel__right,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat] :
% 0.35/0.59 ( ( B3 != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ B3 )
% 0.35/0.59 = A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_right
% 0.35/0.59 thf(fact_4124_nonzero__mult__div__cancel__right,axiom,
% 0.35/0.59 ! [B3: int,A2: int] :
% 0.35/0.59 ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
% 0.35/0.59 = A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_right
% 0.35/0.59 thf(fact_4125_nonzero__mult__div__cancel__right,axiom,
% 0.35/0.59 ! [B3: nat,A2: nat] :
% 0.35/0.59 ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.59 = A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_right
% 0.35/0.59 thf(fact_4126_nonzero__mult__div__cancel__right,axiom,
% 0.35/0.59 ! [B3: real,A2: real] :
% 0.35/0.59 ( ( B3 != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ B3 )
% 0.35/0.59 = A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_right
% 0.35/0.59 thf(fact_4127_nonzero__mult__div__cancel__right,axiom,
% 0.35/0.59 ! [B3: complex,A2: complex] :
% 0.35/0.59 ( ( B3 != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ B3 )
% 0.35/0.59 = A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_div_cancel_right
% 0.35/0.59 thf(fact_4128_mult__divide__mult__cancel__left__if,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( C = zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = zero_zero_rat ) )
% 0.35/0.59 & ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_divide_mult_cancel_left_if
% 0.35/0.59 thf(fact_4129_mult__divide__mult__cancel__left__if,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( ( C = zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = zero_zero_real ) )
% 0.35/0.59 & ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_divide_mult_cancel_left_if
% 0.35/0.59 thf(fact_4130_mult__divide__mult__cancel__left__if,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( ( C = zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = zero_zero_complex ) )
% 0.35/0.59 & ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_divide_mult_cancel_left_if
% 0.35/0.59 thf(fact_4131_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4132_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4133_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4134_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left2
% 0.35/0.59 thf(fact_4135_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left2
% 0.35/0.59 thf(fact_4136_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ B3 @ C ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_left2
% 0.35/0.59 thf(fact_4137_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4138_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.59 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4139_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4140_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right2
% 0.35/0.59 thf(fact_4141_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right2
% 0.35/0.59 thf(fact_4142_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_mult_divide_mult_cancel_right2
% 0.35/0.59 thf(fact_4143_not__real__square__gt__zero,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
% 0.35/0.59 = ( X2 = zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % not_real_square_gt_zero
% 0.35/0.59 thf(fact_4144_real__root__Suc__0,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
% 0.35/0.59 = X2 ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_Suc_0
% 0.35/0.59 thf(fact_4145_real__root__eq__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ( root @ N @ X2 )
% 0.35/0.59 = ( root @ N @ Y3 ) )
% 0.35/0.59 = ( X2 = Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_eq_iff
% 0.35/0.59 thf(fact_4146_root__0,axiom,
% 0.35/0.59 ! [X2: real] :
% 0.35/0.59 ( ( root @ zero_zero_nat @ X2 )
% 0.35/0.59 = zero_zero_real ) ).
% 0.35/0.59
% 0.35/0.59 % root_0
% 0.35/0.59 thf(fact_4147_div__mult__self1,axiom,
% 0.35/0.59 ! [B3: int,A2: int,C: int] :
% 0.35/0.59 ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B3 ) ) @ B3 )
% 0.35/0.59 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self1
% 0.35/0.59 thf(fact_4148_div__mult__self1,axiom,
% 0.35/0.59 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.59 ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
% 0.35/0.59 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self1
% 0.35/0.59 thf(fact_4149_div__mult__self2,axiom,
% 0.35/0.59 ! [B3: int,A2: int,C: int] :
% 0.35/0.59 ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B3 @ C ) ) @ B3 )
% 0.35/0.59 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self2
% 0.35/0.59 thf(fact_4150_div__mult__self2,axiom,
% 0.35/0.59 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.59 ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
% 0.35/0.59 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self2
% 0.35/0.59 thf(fact_4151_div__mult__self3,axiom,
% 0.35/0.59 ! [B3: int,C: int,A2: int] :
% 0.35/0.59 ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A2 ) @ B3 )
% 0.35/0.59 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self3
% 0.35/0.59 thf(fact_4152_div__mult__self3,axiom,
% 0.35/0.59 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.59 ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A2 ) @ B3 )
% 0.35/0.59 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self3
% 0.35/0.59 thf(fact_4153_div__mult__self4,axiom,
% 0.35/0.59 ! [B3: int,C: int,A2: int] :
% 0.35/0.59 ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A2 ) @ B3 )
% 0.35/0.59 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self4
% 0.35/0.59 thf(fact_4154_div__mult__self4,axiom,
% 0.35/0.59 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.59 ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A2 ) @ B3 )
% 0.35/0.59 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % div_mult_self4
% 0.35/0.59 thf(fact_4155_nonzero__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat] :
% 0.35/0.59 ( ( B3 != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ B3 @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.59 = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4156_nonzero__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [B3: real,A2: real] :
% 0.35/0.59 ( ( B3 != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ B3 @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.59 = ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4157_nonzero__divide__mult__cancel__right,axiom,
% 0.35/0.59 ! [B3: complex,A2: complex] :
% 0.35/0.59 ( ( B3 != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ B3 @ ( times_times_complex @ A2 @ B3 ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ one_one_complex @ A2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_right
% 0.35/0.59 thf(fact_4158_nonzero__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( A2 != zero_zero_rat )
% 0.35/0.59 => ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.59 = ( divide_divide_rat @ one_one_rat @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4159_nonzero__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( A2 != zero_zero_real )
% 0.35/0.59 => ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.59 = ( divide_divide_real @ one_one_real @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4160_nonzero__divide__mult__cancel__left,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( A2 != zero_zero_complex )
% 0.35/0.59 => ( ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ A2 @ B3 ) )
% 0.35/0.59 = ( divide1717551699836669952omplex @ one_one_complex @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % nonzero_divide_mult_cancel_left
% 0.35/0.59 thf(fact_4161_real__root__eq__0__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ( root @ N @ X2 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 = ( X2 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_eq_0_iff
% 0.35/0.59 thf(fact_4162_real__root__less__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_less_iff
% 0.35/0.59 thf(fact_4163_real__root__le__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_le_iff
% 0.35/0.59 thf(fact_4164_real__root__eq__1__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ( root @ N @ X2 )
% 0.35/0.59 = one_one_real )
% 0.35/0.59 = ( X2 = one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_eq_1_iff
% 0.35/0.59 thf(fact_4165_real__root__one,axiom,
% 0.35/0.59 ! [N: nat] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( root @ N @ one_one_real )
% 0.35/0.59 = one_one_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_one
% 0.35/0.59 thf(fact_4166_real__root__lt__0__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_lt_0_iff
% 0.35/0.59 thf(fact_4167_real__root__gt__0__iff,axiom,
% 0.35/0.59 ! [N: nat,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_gt_0_iff
% 0.35/0.59 thf(fact_4168_real__root__le__0__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_le_0_iff
% 0.35/0.59 thf(fact_4169_real__root__ge__0__iff,axiom,
% 0.35/0.59 ! [N: nat,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_ge_0_iff
% 0.35/0.59 thf(fact_4170_real__root__lt__1__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_lt_1_iff
% 0.35/0.59 thf(fact_4171_real__root__gt__1__iff,axiom,
% 0.35/0.59 ! [N: nat,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_gt_1_iff
% 0.35/0.59 thf(fact_4172_real__root__le__1__iff,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
% 0.35/0.59 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_le_1_iff
% 0.35/0.59 thf(fact_4173_real__root__ge__1__iff,axiom,
% 0.35/0.59 ! [N: nat,Y3: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 0.35/0.59 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_ge_1_iff
% 0.35/0.59 thf(fact_4174_real__root__pow__pos2,axiom,
% 0.35/0.59 ! [N: nat,X2: real] :
% 0.35/0.59 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 0.35/0.59 = X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_pow_pos2
% 0.35/0.59 thf(fact_4175_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex,C: complex] :
% 0.35/0.59 ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ab_semigroup_mult_class.mult_ac(1)
% 0.35/0.59 thf(fact_4176_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real] :
% 0.35/0.59 ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ab_semigroup_mult_class.mult_ac(1)
% 0.35/0.59 thf(fact_4177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.59 ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ab_semigroup_mult_class.mult_ac(1)
% 0.35/0.59 thf(fact_4178_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.59 ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ab_semigroup_mult_class.mult_ac(1)
% 0.35/0.59 thf(fact_4179_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int] :
% 0.35/0.59 ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % ab_semigroup_mult_class.mult_ac(1)
% 0.35/0.59 thf(fact_4180_mult_Oassoc,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex,C: complex] :
% 0.35/0.59 ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.assoc
% 0.35/0.59 thf(fact_4181_mult_Oassoc,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real] :
% 0.35/0.59 ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.assoc
% 0.35/0.59 thf(fact_4182_mult_Oassoc,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.59 ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.assoc
% 0.35/0.59 thf(fact_4183_mult_Oassoc,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.59 ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.assoc
% 0.35/0.59 thf(fact_4184_mult_Oassoc,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int] :
% 0.35/0.59 ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
% 0.35/0.59 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.assoc
% 0.35/0.59 thf(fact_4185_mult_Ocommute,axiom,
% 0.35/0.59 ( times_times_complex
% 0.35/0.59 = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.commute
% 0.35/0.59 thf(fact_4186_mult_Ocommute,axiom,
% 0.35/0.59 ( times_times_real
% 0.35/0.59 = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.commute
% 0.35/0.59 thf(fact_4187_mult_Ocommute,axiom,
% 0.35/0.59 ( times_times_rat
% 0.35/0.59 = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.commute
% 0.35/0.59 thf(fact_4188_mult_Ocommute,axiom,
% 0.35/0.59 ( times_times_nat
% 0.35/0.59 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.commute
% 0.35/0.59 thf(fact_4189_mult_Ocommute,axiom,
% 0.35/0.59 ( times_times_int
% 0.35/0.59 = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.commute
% 0.35/0.59 thf(fact_4190_mult_Oleft__commute,axiom,
% 0.35/0.59 ! [B3: complex,A2: complex,C: complex] :
% 0.35/0.59 ( ( times_times_complex @ B3 @ ( times_times_complex @ A2 @ C ) )
% 0.35/0.59 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.left_commute
% 0.35/0.59 thf(fact_4191_mult_Oleft__commute,axiom,
% 0.35/0.59 ! [B3: real,A2: real,C: real] :
% 0.35/0.59 ( ( times_times_real @ B3 @ ( times_times_real @ A2 @ C ) )
% 0.35/0.59 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.left_commute
% 0.35/0.59 thf(fact_4192_mult_Oleft__commute,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.59 ( ( times_times_rat @ B3 @ ( times_times_rat @ A2 @ C ) )
% 0.35/0.59 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.left_commute
% 0.35/0.59 thf(fact_4193_mult_Oleft__commute,axiom,
% 0.35/0.59 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.59 ( ( times_times_nat @ B3 @ ( times_times_nat @ A2 @ C ) )
% 0.35/0.59 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.left_commute
% 0.35/0.59 thf(fact_4194_mult_Oleft__commute,axiom,
% 0.35/0.59 ! [B3: int,A2: int,C: int] :
% 0.35/0.59 ( ( times_times_int @ B3 @ ( times_times_int @ A2 @ C ) )
% 0.35/0.59 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult.left_commute
% 0.35/0.59 thf(fact_4195_mult__right__cancel,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( ( times_times_complex @ A2 @ C )
% 0.35/0.59 = ( times_times_complex @ B3 @ C ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_cancel
% 0.35/0.59 thf(fact_4196_mult__right__cancel,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( ( times_times_real @ A2 @ C )
% 0.35/0.59 = ( times_times_real @ B3 @ C ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_cancel
% 0.35/0.59 thf(fact_4197_mult__right__cancel,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( ( times_times_rat @ A2 @ C )
% 0.35/0.59 = ( times_times_rat @ B3 @ C ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_cancel
% 0.35/0.59 thf(fact_4198_mult__right__cancel,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( C != zero_zero_nat )
% 0.35/0.59 => ( ( ( times_times_nat @ A2 @ C )
% 0.35/0.59 = ( times_times_nat @ B3 @ C ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_cancel
% 0.35/0.59 thf(fact_4199_mult__right__cancel,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( C != zero_zero_int )
% 0.35/0.59 => ( ( ( times_times_int @ A2 @ C )
% 0.35/0.59 = ( times_times_int @ B3 @ C ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_cancel
% 0.35/0.59 thf(fact_4200_mult__left__cancel,axiom,
% 0.35/0.59 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.59 ( ( C != zero_zero_complex )
% 0.35/0.59 => ( ( ( times_times_complex @ C @ A2 )
% 0.35/0.59 = ( times_times_complex @ C @ B3 ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_cancel
% 0.35/0.59 thf(fact_4201_mult__left__cancel,axiom,
% 0.35/0.59 ! [C: real,A2: real,B3: real] :
% 0.35/0.59 ( ( C != zero_zero_real )
% 0.35/0.59 => ( ( ( times_times_real @ C @ A2 )
% 0.35/0.59 = ( times_times_real @ C @ B3 ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_cancel
% 0.35/0.59 thf(fact_4202_mult__left__cancel,axiom,
% 0.35/0.59 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.59 ( ( C != zero_zero_rat )
% 0.35/0.59 => ( ( ( times_times_rat @ C @ A2 )
% 0.35/0.59 = ( times_times_rat @ C @ B3 ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_cancel
% 0.35/0.59 thf(fact_4203_mult__left__cancel,axiom,
% 0.35/0.59 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.59 ( ( C != zero_zero_nat )
% 0.35/0.59 => ( ( ( times_times_nat @ C @ A2 )
% 0.35/0.59 = ( times_times_nat @ C @ B3 ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_cancel
% 0.35/0.59 thf(fact_4204_mult__left__cancel,axiom,
% 0.35/0.59 ! [C: int,A2: int,B3: int] :
% 0.35/0.59 ( ( C != zero_zero_int )
% 0.35/0.59 => ( ( ( times_times_int @ C @ A2 )
% 0.35/0.59 = ( times_times_int @ C @ B3 ) )
% 0.35/0.59 = ( A2 = B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_cancel
% 0.35/0.59 thf(fact_4205_no__zero__divisors,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( A2 != zero_zero_complex )
% 0.35/0.59 => ( ( B3 != zero_zero_complex )
% 0.35/0.59 => ( ( times_times_complex @ A2 @ B3 )
% 0.35/0.59 != zero_zero_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % no_zero_divisors
% 0.35/0.59 thf(fact_4206_no__zero__divisors,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( A2 != zero_zero_real )
% 0.35/0.59 => ( ( B3 != zero_zero_real )
% 0.35/0.59 => ( ( times_times_real @ A2 @ B3 )
% 0.35/0.59 != zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % no_zero_divisors
% 0.35/0.59 thf(fact_4207_no__zero__divisors,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( A2 != zero_zero_rat )
% 0.35/0.59 => ( ( B3 != zero_zero_rat )
% 0.35/0.59 => ( ( times_times_rat @ A2 @ B3 )
% 0.35/0.59 != zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % no_zero_divisors
% 0.35/0.59 thf(fact_4208_no__zero__divisors,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( A2 != zero_zero_nat )
% 0.35/0.59 => ( ( B3 != zero_zero_nat )
% 0.35/0.59 => ( ( times_times_nat @ A2 @ B3 )
% 0.35/0.59 != zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % no_zero_divisors
% 0.35/0.59 thf(fact_4209_no__zero__divisors,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( A2 != zero_zero_int )
% 0.35/0.59 => ( ( B3 != zero_zero_int )
% 0.35/0.59 => ( ( times_times_int @ A2 @ B3 )
% 0.35/0.59 != zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % no_zero_divisors
% 0.35/0.59 thf(fact_4210_divisors__zero,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ A2 @ B3 )
% 0.35/0.59 = zero_zero_complex )
% 0.35/0.59 => ( ( A2 = zero_zero_complex )
% 0.35/0.59 | ( B3 = zero_zero_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % divisors_zero
% 0.35/0.59 thf(fact_4211_divisors__zero,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ( times_times_real @ A2 @ B3 )
% 0.35/0.59 = zero_zero_real )
% 0.35/0.59 => ( ( A2 = zero_zero_real )
% 0.35/0.59 | ( B3 = zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % divisors_zero
% 0.35/0.59 thf(fact_4212_divisors__zero,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ A2 @ B3 )
% 0.35/0.59 = zero_zero_rat )
% 0.35/0.59 => ( ( A2 = zero_zero_rat )
% 0.35/0.59 | ( B3 = zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % divisors_zero
% 0.35/0.59 thf(fact_4213_divisors__zero,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( times_times_nat @ A2 @ B3 )
% 0.35/0.59 = zero_zero_nat )
% 0.35/0.59 => ( ( A2 = zero_zero_nat )
% 0.35/0.59 | ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % divisors_zero
% 0.35/0.59 thf(fact_4214_divisors__zero,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ( times_times_int @ A2 @ B3 )
% 0.35/0.59 = zero_zero_int )
% 0.35/0.59 => ( ( A2 = zero_zero_int )
% 0.35/0.59 | ( B3 = zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % divisors_zero
% 0.35/0.59 thf(fact_4215_mult__not__zero,axiom,
% 0.35/0.59 ! [A2: complex,B3: complex] :
% 0.35/0.59 ( ( ( times_times_complex @ A2 @ B3 )
% 0.35/0.59 != zero_zero_complex )
% 0.35/0.59 => ( ( A2 != zero_zero_complex )
% 0.35/0.59 & ( B3 != zero_zero_complex ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_not_zero
% 0.35/0.59 thf(fact_4216_mult__not__zero,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ( times_times_real @ A2 @ B3 )
% 0.35/0.59 != zero_zero_real )
% 0.35/0.59 => ( ( A2 != zero_zero_real )
% 0.35/0.59 & ( B3 != zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_not_zero
% 0.35/0.59 thf(fact_4217_mult__not__zero,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( times_times_rat @ A2 @ B3 )
% 0.35/0.59 != zero_zero_rat )
% 0.35/0.59 => ( ( A2 != zero_zero_rat )
% 0.35/0.59 & ( B3 != zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_not_zero
% 0.35/0.59 thf(fact_4218_mult__not__zero,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( times_times_nat @ A2 @ B3 )
% 0.35/0.59 != zero_zero_nat )
% 0.35/0.59 => ( ( A2 != zero_zero_nat )
% 0.35/0.59 & ( B3 != zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_not_zero
% 0.35/0.59 thf(fact_4219_mult__not__zero,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ( times_times_int @ A2 @ B3 )
% 0.35/0.59 != zero_zero_int )
% 0.35/0.59 => ( ( A2 != zero_zero_int )
% 0.35/0.59 & ( B3 != zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_not_zero
% 0.35/0.59 thf(fact_4220_comm__monoid__mult__class_Omult__1,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ one_one_complex @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % comm_monoid_mult_class.mult_1
% 0.35/0.59 thf(fact_4221_comm__monoid__mult__class_Omult__1,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ one_one_real @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % comm_monoid_mult_class.mult_1
% 0.35/0.59 thf(fact_4222_comm__monoid__mult__class_Omult__1,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ one_one_rat @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % comm_monoid_mult_class.mult_1
% 0.35/0.59 thf(fact_4223_comm__monoid__mult__class_Omult__1,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ one_one_nat @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % comm_monoid_mult_class.mult_1
% 0.35/0.59 thf(fact_4224_comm__monoid__mult__class_Omult__1,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ one_one_int @ A2 )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % comm_monoid_mult_class.mult_1
% 0.35/0.59 thf(fact_4225_mult_Ocomm__neutral,axiom,
% 0.35/0.59 ! [A2: complex] :
% 0.35/0.59 ( ( times_times_complex @ A2 @ one_one_complex )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.comm_neutral
% 0.35/0.59 thf(fact_4226_mult_Ocomm__neutral,axiom,
% 0.35/0.59 ! [A2: real] :
% 0.35/0.59 ( ( times_times_real @ A2 @ one_one_real )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.comm_neutral
% 0.35/0.59 thf(fact_4227_mult_Ocomm__neutral,axiom,
% 0.35/0.59 ! [A2: rat] :
% 0.35/0.59 ( ( times_times_rat @ A2 @ one_one_rat )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.comm_neutral
% 0.35/0.59 thf(fact_4228_mult_Ocomm__neutral,axiom,
% 0.35/0.59 ! [A2: nat] :
% 0.35/0.59 ( ( times_times_nat @ A2 @ one_one_nat )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.comm_neutral
% 0.35/0.59 thf(fact_4229_mult_Ocomm__neutral,axiom,
% 0.35/0.59 ! [A2: int] :
% 0.35/0.59 ( ( times_times_int @ A2 @ one_one_int )
% 0.35/0.59 = A2 ) ).
% 0.35/0.59
% 0.35/0.59 % mult.comm_neutral
% 0.35/0.59 thf(fact_4230_mult__of__nat__commute,axiom,
% 0.35/0.59 ! [X2: nat,Y3: complex] :
% 0.35/0.59 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y3 )
% 0.35/0.59 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_of_nat_commute
% 0.35/0.59 thf(fact_4231_mult__of__nat__commute,axiom,
% 0.35/0.59 ! [X2: nat,Y3: rat] :
% 0.35/0.59 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y3 )
% 0.35/0.59 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_of_nat_commute
% 0.35/0.59 thf(fact_4232_mult__of__nat__commute,axiom,
% 0.35/0.59 ! [X2: nat,Y3: nat] :
% 0.35/0.59 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y3 )
% 0.35/0.59 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_of_nat_commute
% 0.35/0.59 thf(fact_4233_mult__of__nat__commute,axiom,
% 0.35/0.59 ! [X2: nat,Y3: int] :
% 0.35/0.59 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y3 )
% 0.35/0.59 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_of_nat_commute
% 0.35/0.59 thf(fact_4234_mult__of__nat__commute,axiom,
% 0.35/0.59 ! [X2: nat,Y3: real] :
% 0.35/0.59 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y3 )
% 0.35/0.59 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_of_nat_commute
% 0.35/0.59 thf(fact_4235_real__root__pos__pos__le,axiom,
% 0.35/0.59 ! [X2: real,N: nat] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % real_root_pos_pos_le
% 0.35/0.59 thf(fact_4236_mult__mono,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono
% 0.35/0.59 thf(fact_4237_mult__mono,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono
% 0.35/0.59 thf(fact_4238_mult__mono,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono
% 0.35/0.59 thf(fact_4239_mult__mono,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono
% 0.35/0.59 thf(fact_4240_mult__mono_H,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono'
% 0.35/0.59 thf(fact_4241_mult__mono_H,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono'
% 0.35/0.59 thf(fact_4242_mult__mono_H,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono'
% 0.35/0.59 thf(fact_4243_mult__mono_H,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_mono'
% 0.35/0.59 thf(fact_4244_zero__le__square,axiom,
% 0.35/0.59 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_le_square
% 0.35/0.59 thf(fact_4245_zero__le__square,axiom,
% 0.35/0.59 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_le_square
% 0.35/0.59 thf(fact_4246_zero__le__square,axiom,
% 0.35/0.59 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_le_square
% 0.35/0.59 thf(fact_4247_split__mult__pos__le,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 0.35/0.59 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
% 0.35/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_pos_le
% 0.35/0.59 thf(fact_4248_split__mult__pos__le,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 0.35/0.59 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
% 0.35/0.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_pos_le
% 0.35/0.59 thf(fact_4249_split__mult__pos__le,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
% 0.35/0.59 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.59 & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
% 0.35/0.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_pos_le
% 0.35/0.59 thf(fact_4250_mult__left__mono__neg,axiom,
% 0.35/0.59 ! [B3: real,A2: real,C: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono_neg
% 0.35/0.59 thf(fact_4251_mult__left__mono__neg,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono_neg
% 0.35/0.59 thf(fact_4252_mult__left__mono__neg,axiom,
% 0.35/0.59 ! [B3: int,A2: int,C: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono_neg
% 0.35/0.59 thf(fact_4253_mult__nonpos__nonpos,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonpos
% 0.35/0.59 thf(fact_4254_mult__nonpos__nonpos,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonpos
% 0.35/0.59 thf(fact_4255_mult__nonpos__nonpos,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.59 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonpos
% 0.35/0.59 thf(fact_4256_mult__left__mono,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono
% 0.35/0.59 thf(fact_4257_mult__left__mono,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono
% 0.35/0.59 thf(fact_4258_mult__left__mono,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono
% 0.35/0.59 thf(fact_4259_mult__left__mono,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_left_mono
% 0.35/0.59 thf(fact_4260_mult__right__mono__neg,axiom,
% 0.35/0.59 ! [B3: real,A2: real,C: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono_neg
% 0.35/0.59 thf(fact_4261_mult__right__mono__neg,axiom,
% 0.35/0.59 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono_neg
% 0.35/0.59 thf(fact_4262_mult__right__mono__neg,axiom,
% 0.35/0.59 ! [B3: int,A2: int,C: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono_neg
% 0.35/0.59 thf(fact_4263_mult__right__mono,axiom,
% 0.35/0.59 ! [A2: real,B3: real,C: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono
% 0.35/0.59 thf(fact_4264_mult__right__mono,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono
% 0.35/0.59 thf(fact_4265_mult__right__mono,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono
% 0.35/0.59 thf(fact_4266_mult__right__mono,axiom,
% 0.35/0.59 ! [A2: int,B3: int,C: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_right_mono
% 0.35/0.59 thf(fact_4267_mult__le__0__iff,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
% 0.35/0.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 0.35/0.59 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_le_0_iff
% 0.35/0.59 thf(fact_4268_mult__le__0__iff,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
% 0.35/0.59 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 0.35/0.59 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_le_0_iff
% 0.35/0.59 thf(fact_4269_mult__le__0__iff,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
% 0.35/0.59 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
% 0.35/0.59 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.59 & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_le_0_iff
% 0.35/0.59 thf(fact_4270_split__mult__neg__le,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 0.35/0.59 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_neg_le
% 0.35/0.59 thf(fact_4271_split__mult__neg__le,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 0.35/0.59 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_neg_le
% 0.35/0.59 thf(fact_4272_split__mult__neg__le,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 & ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
% 0.35/0.59 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.35/0.59 & ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_neg_le
% 0.35/0.59 thf(fact_4273_split__mult__neg__le,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
% 0.35/0.59 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.59 & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ).
% 0.35/0.59
% 0.35/0.59 % split_mult_neg_le
% 0.35/0.59 thf(fact_4274_mult__nonneg__nonneg,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.59 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonneg
% 0.35/0.59 thf(fact_4275_mult__nonneg__nonneg,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.59 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonneg
% 0.35/0.59 thf(fact_4276_mult__nonneg__nonneg,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.59 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonneg
% 0.35/0.59 thf(fact_4277_mult__nonneg__nonneg,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.59 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonneg
% 0.35/0.59 thf(fact_4278_mult__nonneg__nonpos,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos
% 0.35/0.59 thf(fact_4279_mult__nonneg__nonpos,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos
% 0.35/0.59 thf(fact_4280_mult__nonneg__nonpos,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos
% 0.35/0.59 thf(fact_4281_mult__nonneg__nonpos,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos
% 0.35/0.59 thf(fact_4282_mult__nonpos__nonneg,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonneg
% 0.35/0.59 thf(fact_4283_mult__nonpos__nonneg,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.59 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonneg
% 0.35/0.59 thf(fact_4284_mult__nonpos__nonneg,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 0.35/0.59 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonneg
% 0.35/0.59 thf(fact_4285_mult__nonpos__nonneg,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.59 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonpos_nonneg
% 0.35/0.59 thf(fact_4286_mult__nonneg__nonpos2,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 0.35/0.59 => ( ord_less_eq_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos2
% 0.35/0.59 thf(fact_4287_mult__nonneg__nonpos2,axiom,
% 0.35/0.59 ! [A2: rat,B3: rat] :
% 0.35/0.59 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 0.35/0.59 => ( ord_less_eq_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos2
% 0.35/0.59 thf(fact_4288_mult__nonneg__nonpos2,axiom,
% 0.35/0.59 ! [A2: nat,B3: nat] :
% 0.35/0.59 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 0.35/0.59 => ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos2
% 0.35/0.59 thf(fact_4289_mult__nonneg__nonpos2,axiom,
% 0.35/0.59 ! [A2: int,B3: int] :
% 0.35/0.59 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.59 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.59 => ( ord_less_eq_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % mult_nonneg_nonpos2
% 0.35/0.59 thf(fact_4290_zero__le__mult__iff,axiom,
% 0.35/0.59 ! [A2: real,B3: real] :
% 0.35/0.59 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.59 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.59 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 0.35/0.59 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.59 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.35/0.59
% 0.35/0.59 % zero_le_mult_iff
% 0.35/0.59 thf(fact_4291_zero__le__mult__iff,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 0.35/0.60 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 0.35/0.60 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_le_mult_iff
% 0.35/0.60 thf(fact_4292_zero__le__mult__iff,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.60 & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
% 0.35/0.60 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.60 & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_le_mult_iff
% 0.35/0.60 thf(fact_4293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.35/0.60 thf(fact_4294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.35/0.60 thf(fact_4295_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.35/0.60 thf(fact_4296_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_comm_semiring_class.comm_mult_left_mono
% 0.35/0.60 thf(fact_4297_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.35/0.60 thf(fact_4298_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.35/0.60 thf(fact_4299_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.35/0.60 thf(fact_4300_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.35/0.60 thf(fact_4301_mult__less__cancel__right__disj,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 & ( ord_less_real @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right_disj
% 0.35/0.60 thf(fact_4302_mult__less__cancel__right__disj,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 & ( ord_less_rat @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right_disj
% 0.35/0.60 thf(fact_4303_mult__less__cancel__right__disj,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 & ( ord_less_int @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 & ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right_disj
% 0.35/0.60 thf(fact_4304_mult__strict__right__mono,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono
% 0.35/0.60 thf(fact_4305_mult__strict__right__mono,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono
% 0.35/0.60 thf(fact_4306_mult__strict__right__mono,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono
% 0.35/0.60 thf(fact_4307_mult__strict__right__mono,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono
% 0.35/0.60 thf(fact_4308_mult__strict__right__mono__neg,axiom,
% 0.35/0.60 ! [B3: real,A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono_neg
% 0.35/0.60 thf(fact_4309_mult__strict__right__mono__neg,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono_neg
% 0.35/0.60 thf(fact_4310_mult__strict__right__mono__neg,axiom,
% 0.35/0.60 ! [B3: int,A2: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_right_mono_neg
% 0.35/0.60 thf(fact_4311_mult__less__cancel__left__disj,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 & ( ord_less_real @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_disj
% 0.35/0.60 thf(fact_4312_mult__less__cancel__left__disj,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 & ( ord_less_rat @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_disj
% 0.35/0.60 thf(fact_4313_mult__less__cancel__left__disj,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 & ( ord_less_int @ A2 @ B3 ) )
% 0.35/0.60 | ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 & ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_disj
% 0.35/0.60 thf(fact_4314_mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono
% 0.35/0.60 thf(fact_4315_mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono
% 0.35/0.60 thf(fact_4316_mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono
% 0.35/0.60 thf(fact_4317_mult__strict__left__mono,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono
% 0.35/0.60 thf(fact_4318_mult__strict__left__mono__neg,axiom,
% 0.35/0.60 ! [B3: real,A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono_neg
% 0.35/0.60 thf(fact_4319_mult__strict__left__mono__neg,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono_neg
% 0.35/0.60 thf(fact_4320_mult__strict__left__mono__neg,axiom,
% 0.35/0.60 ! [B3: int,A2: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_left_mono_neg
% 0.35/0.60 thf(fact_4321_mult__less__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_pos
% 0.35/0.60 thf(fact_4322_mult__less__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_pos
% 0.35/0.60 thf(fact_4323_mult__less__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_pos
% 0.35/0.60 thf(fact_4324_mult__less__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_neg
% 0.35/0.60 thf(fact_4325_mult__less__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_neg
% 0.35/0.60 thf(fact_4326_mult__less__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_int @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left_neg
% 0.35/0.60 thf(fact_4327_zero__less__mult__pos2,axiom,
% 0.35/0.60 ! [B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A2 ) )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos2
% 0.35/0.60 thf(fact_4328_zero__less__mult__pos2,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B3 @ A2 ) )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos2
% 0.35/0.60 thf(fact_4329_zero__less__mult__pos2,axiom,
% 0.35/0.60 ! [B3: nat,A2: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A2 ) )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos2
% 0.35/0.60 thf(fact_4330_zero__less__mult__pos2,axiom,
% 0.35/0.60 ! [B3: int,A2: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A2 ) )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos2
% 0.35/0.60 thf(fact_4331_zero__less__mult__pos,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos
% 0.35/0.60 thf(fact_4332_zero__less__mult__pos,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos
% 0.35/0.60 thf(fact_4333_zero__less__mult__pos,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos
% 0.35/0.60 thf(fact_4334_zero__less__mult__pos,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_pos
% 0.35/0.60 thf(fact_4335_zero__less__mult__iff,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 & ( ord_less_real @ zero_zero_real @ B3 ) )
% 0.35/0.60 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.60 & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_iff
% 0.35/0.60 thf(fact_4336_zero__less__mult__iff,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 & ( ord_less_rat @ zero_zero_rat @ B3 ) )
% 0.35/0.60 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.60 & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_iff
% 0.35/0.60 thf(fact_4337_zero__less__mult__iff,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 & ( ord_less_int @ zero_zero_int @ B3 ) )
% 0.35/0.60 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.60 & ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zero_less_mult_iff
% 0.35/0.60 thf(fact_4338_mult__pos__neg2,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg2
% 0.35/0.60 thf(fact_4339_mult__pos__neg2,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg2
% 0.35/0.60 thf(fact_4340_mult__pos__neg2,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg2
% 0.35/0.60 thf(fact_4341_mult__pos__neg2,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg2
% 0.35/0.60 thf(fact_4342_mult__pos__pos,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_pos
% 0.35/0.60 thf(fact_4343_mult__pos__pos,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_pos
% 0.35/0.60 thf(fact_4344_mult__pos__pos,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.60 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_pos
% 0.35/0.60 thf(fact_4345_mult__pos__pos,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.60 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_pos
% 0.35/0.60 thf(fact_4346_mult__pos__neg,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg
% 0.35/0.60 thf(fact_4347_mult__pos__neg,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg
% 0.35/0.60 thf(fact_4348_mult__pos__neg,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg
% 0.35/0.60 thf(fact_4349_mult__pos__neg,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_pos_neg
% 0.35/0.60 thf(fact_4350_mult__neg__pos,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_pos
% 0.35/0.60 thf(fact_4351_mult__neg__pos,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_pos
% 0.35/0.60 thf(fact_4352_mult__neg__pos,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_pos
% 0.35/0.60 thf(fact_4353_mult__neg__pos,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_pos
% 0.35/0.60 thf(fact_4354_mult__less__0__iff,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 & ( ord_less_real @ B3 @ zero_zero_real ) )
% 0.35/0.60 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.60 & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_0_iff
% 0.35/0.60 thf(fact_4355_mult__less__0__iff,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 & ( ord_less_rat @ B3 @ zero_zero_rat ) )
% 0.35/0.60 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.60 & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_0_iff
% 0.35/0.60 thf(fact_4356_mult__less__0__iff,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 & ( ord_less_int @ B3 @ zero_zero_int ) )
% 0.35/0.60 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.60 & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_0_iff
% 0.35/0.60 thf(fact_4357_not__square__less__zero,axiom,
% 0.35/0.60 ! [A2: real] :
% 0.35/0.60 ~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% 0.35/0.60
% 0.35/0.60 % not_square_less_zero
% 0.35/0.60 thf(fact_4358_not__square__less__zero,axiom,
% 0.35/0.60 ! [A2: rat] :
% 0.35/0.60 ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% 0.35/0.60
% 0.35/0.60 % not_square_less_zero
% 0.35/0.60 thf(fact_4359_not__square__less__zero,axiom,
% 0.35/0.60 ! [A2: int] :
% 0.35/0.60 ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% 0.35/0.60
% 0.35/0.60 % not_square_less_zero
% 0.35/0.60 thf(fact_4360_mult__neg__neg,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_neg
% 0.35/0.60 thf(fact_4361_mult__neg__neg,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_neg
% 0.35/0.60 thf(fact_4362_mult__neg__neg,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_neg_neg
% 0.35/0.60 thf(fact_4363_add__scale__eq__noteq,axiom,
% 0.35/0.60 ! [R2: complex,A2: complex,B3: complex,C: complex,D: complex] :
% 0.35/0.60 ( ( R2 != zero_zero_complex )
% 0.35/0.60 => ( ( ( A2 = B3 )
% 0.35/0.60 & ( C != D ) )
% 0.35/0.60 => ( ( plus_plus_complex @ A2 @ ( times_times_complex @ R2 @ C ) )
% 0.35/0.60 != ( plus_plus_complex @ B3 @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_scale_eq_noteq
% 0.35/0.60 thf(fact_4364_add__scale__eq__noteq,axiom,
% 0.35/0.60 ! [R2: real,A2: real,B3: real,C: real,D: real] :
% 0.35/0.60 ( ( R2 != zero_zero_real )
% 0.35/0.60 => ( ( ( A2 = B3 )
% 0.35/0.60 & ( C != D ) )
% 0.35/0.60 => ( ( plus_plus_real @ A2 @ ( times_times_real @ R2 @ C ) )
% 0.35/0.60 != ( plus_plus_real @ B3 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_scale_eq_noteq
% 0.35/0.60 thf(fact_4365_add__scale__eq__noteq,axiom,
% 0.35/0.60 ! [R2: rat,A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.60 ( ( R2 != zero_zero_rat )
% 0.35/0.60 => ( ( ( A2 = B3 )
% 0.35/0.60 & ( C != D ) )
% 0.35/0.60 => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
% 0.35/0.60 != ( plus_plus_rat @ B3 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_scale_eq_noteq
% 0.35/0.60 thf(fact_4366_add__scale__eq__noteq,axiom,
% 0.35/0.60 ! [R2: nat,A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.60 ( ( R2 != zero_zero_nat )
% 0.35/0.60 => ( ( ( A2 = B3 )
% 0.35/0.60 & ( C != D ) )
% 0.35/0.60 => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
% 0.35/0.60 != ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_scale_eq_noteq
% 0.35/0.60 thf(fact_4367_add__scale__eq__noteq,axiom,
% 0.35/0.60 ! [R2: int,A2: int,B3: int,C: int,D: int] :
% 0.35/0.60 ( ( R2 != zero_zero_int )
% 0.35/0.60 => ( ( ( A2 = B3 )
% 0.35/0.60 & ( C != D ) )
% 0.35/0.60 => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
% 0.35/0.60 != ( plus_plus_int @ B3 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_scale_eq_noteq
% 0.35/0.60 thf(fact_4368_less__1__mult,axiom,
% 0.35/0.60 ! [M: real,N: real] :
% 0.35/0.60 ( ( ord_less_real @ one_one_real @ M )
% 0.35/0.60 => ( ( ord_less_real @ one_one_real @ N )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_1_mult
% 0.35/0.60 thf(fact_4369_less__1__mult,axiom,
% 0.35/0.60 ! [M: rat,N: rat] :
% 0.35/0.60 ( ( ord_less_rat @ one_one_rat @ M )
% 0.35/0.60 => ( ( ord_less_rat @ one_one_rat @ N )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_1_mult
% 0.35/0.60 thf(fact_4370_less__1__mult,axiom,
% 0.35/0.60 ! [M: nat,N: nat] :
% 0.35/0.60 ( ( ord_less_nat @ one_one_nat @ M )
% 0.35/0.60 => ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.60 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_1_mult
% 0.35/0.60 thf(fact_4371_less__1__mult,axiom,
% 0.35/0.60 ! [M: int,N: int] :
% 0.35/0.60 ( ( ord_less_int @ one_one_int @ M )
% 0.35/0.60 => ( ( ord_less_int @ one_one_int @ N )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_1_mult
% 0.35/0.60 thf(fact_4372_frac__eq__eq,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( ( divide_divide_rat @ X2 @ Y3 )
% 0.35/0.60 = ( divide_divide_rat @ W2 @ Z ) )
% 0.35/0.60 = ( ( times_times_rat @ X2 @ Z )
% 0.35/0.60 = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_eq_eq
% 0.35/0.60 thf(fact_4373_frac__eq__eq,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real,W2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( ( divide_divide_real @ X2 @ Y3 )
% 0.35/0.60 = ( divide_divide_real @ W2 @ Z ) )
% 0.35/0.60 = ( ( times_times_real @ X2 @ Z )
% 0.35/0.60 = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_eq_eq
% 0.35/0.60 thf(fact_4374_frac__eq__eq,axiom,
% 0.35/0.60 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 0.35/0.60 ( ( Y3 != zero_zero_complex )
% 0.35/0.60 => ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( ( divide1717551699836669952omplex @ X2 @ Y3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ W2 @ Z ) )
% 0.35/0.60 = ( ( times_times_complex @ X2 @ Z )
% 0.35/0.60 = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_eq_eq
% 0.35/0.60 thf(fact_4375_divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ( divide_divide_rat @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_rat )
% 0.35/0.60 => ( B3
% 0.35/0.60 = ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_rat )
% 0.35/0.60 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_eq
% 0.35/0.60 thf(fact_4376_divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ( divide_divide_real @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_real )
% 0.35/0.60 => ( B3
% 0.35/0.60 = ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_real )
% 0.35/0.60 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_eq
% 0.35/0.60 thf(fact_4377_divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: complex,C: complex,A2: complex] :
% 0.35/0.60 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_complex )
% 0.35/0.60 => ( B3
% 0.35/0.60 = ( times_times_complex @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_complex )
% 0.35/0.60 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_eq
% 0.35/0.60 thf(fact_4378_eq__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( times_times_rat @ A2 @ C )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( C = zero_zero_rat )
% 0.35/0.60 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_eq
% 0.35/0.60 thf(fact_4379_eq__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( times_times_real @ A2 @ C )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( C = zero_zero_real )
% 0.35/0.60 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_eq
% 0.35/0.60 thf(fact_4380_eq__divide__eq,axiom,
% 0.35/0.60 ! [A2: complex,B3: complex,C: complex] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.35/0.60 = ( ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( times_times_complex @ A2 @ C )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( C = zero_zero_complex )
% 0.35/0.60 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_eq
% 0.35/0.60 thf(fact_4381_divide__eq__imp,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( B3
% 0.35/0.60 = ( times_times_rat @ A2 @ C ) )
% 0.35/0.60 => ( ( divide_divide_rat @ B3 @ C )
% 0.35/0.60 = A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_imp
% 0.35/0.60 thf(fact_4382_divide__eq__imp,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( B3
% 0.35/0.60 = ( times_times_real @ A2 @ C ) )
% 0.35/0.60 => ( ( divide_divide_real @ B3 @ C )
% 0.35/0.60 = A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_imp
% 0.35/0.60 thf(fact_4383_divide__eq__imp,axiom,
% 0.35/0.60 ! [C: complex,B3: complex,A2: complex] :
% 0.35/0.60 ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( B3
% 0.35/0.60 = ( times_times_complex @ A2 @ C ) )
% 0.35/0.60 => ( ( divide1717551699836669952omplex @ B3 @ C )
% 0.35/0.60 = A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_eq_imp
% 0.35/0.60 thf(fact_4384_eq__divide__imp,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( ( times_times_rat @ A2 @ C )
% 0.35/0.60 = B3 )
% 0.35/0.60 => ( A2
% 0.35/0.60 = ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_imp
% 0.35/0.60 thf(fact_4385_eq__divide__imp,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( ( times_times_real @ A2 @ C )
% 0.35/0.60 = B3 )
% 0.35/0.60 => ( A2
% 0.35/0.60 = ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_imp
% 0.35/0.60 thf(fact_4386_eq__divide__imp,axiom,
% 0.35/0.60 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( ( times_times_complex @ A2 @ C )
% 0.35/0.60 = B3 )
% 0.35/0.60 => ( A2
% 0.35/0.60 = ( divide1717551699836669952omplex @ B3 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_divide_imp
% 0.35/0.60 thf(fact_4387_nonzero__divide__eq__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( ( divide_divide_rat @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( B3
% 0.35/0.60 = ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_divide_eq_eq
% 0.35/0.60 thf(fact_4388_nonzero__divide__eq__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( ( divide_divide_real @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( B3
% 0.35/0.60 = ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_divide_eq_eq
% 0.35/0.60 thf(fact_4389_nonzero__divide__eq__eq,axiom,
% 0.35/0.60 ! [C: complex,B3: complex,A2: complex] :
% 0.35/0.60 ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( B3
% 0.35/0.60 = ( times_times_complex @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_divide_eq_eq
% 0.35/0.60 thf(fact_4390_nonzero__eq__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( A2
% 0.35/0.60 = ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( times_times_rat @ A2 @ C )
% 0.35/0.60 = B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_eq_divide_eq
% 0.35/0.60 thf(fact_4391_nonzero__eq__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( A2
% 0.35/0.60 = ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ( times_times_real @ A2 @ C )
% 0.35/0.60 = B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_eq_divide_eq
% 0.35/0.60 thf(fact_4392_nonzero__eq__divide__eq,axiom,
% 0.35/0.60 ! [C: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( A2
% 0.35/0.60 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.35/0.60 = ( ( times_times_complex @ A2 @ C )
% 0.35/0.60 = B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_eq_divide_eq
% 0.35/0.60 thf(fact_4393_abs__mult__less,axiom,
% 0.35/0.60 ! [A2: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
% 0.35/0.60 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C )
% 0.35/0.60 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B3 ) @ D )
% 0.35/0.60 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_less
% 0.35/0.60 thf(fact_4394_abs__mult__less,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C )
% 0.35/0.60 => ( ( ord_less_real @ ( abs_abs_real @ B3 ) @ D )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_less
% 0.35/0.60 thf(fact_4395_abs__mult__less,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C )
% 0.35/0.60 => ( ( ord_less_rat @ ( abs_abs_rat @ B3 ) @ D )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_less
% 0.35/0.60 thf(fact_4396_abs__mult__less,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
% 0.35/0.60 => ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_less
% 0.35/0.60 thf(fact_4397_zmult__zless__mono2,axiom,
% 0.35/0.60 ! [I: int,J: int,K: int] :
% 0.35/0.60 ( ( ord_less_int @ I @ J )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zmult_zless_mono2
% 0.35/0.60 thf(fact_4398_real__minus__mult__self__le,axiom,
% 0.35/0.60 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_minus_mult_self_le
% 0.35/0.60 thf(fact_4399_Gcd__int__greater__eq__0,axiom,
% 0.35/0.60 ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% 0.35/0.60
% 0.35/0.60 % Gcd_int_greater_eq_0
% 0.35/0.60 thf(fact_4400_real__root__less__mono,axiom,
% 0.35/0.60 ! [N: nat,X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.60 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_less_mono
% 0.35/0.60 thf(fact_4401_real__root__le__mono,axiom,
% 0.35/0.60 ! [N: nat,X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.60 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_le_mono
% 0.35/0.60 thf(fact_4402_real__root__power,axiom,
% 0.35/0.60 ! [N: nat,X2: real,K: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( root @ N @ ( power_power_real @ X2 @ K ) )
% 0.35/0.60 = ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_power
% 0.35/0.60 thf(fact_4403_real__root__abs,axiom,
% 0.35/0.60 ! [N: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( root @ N @ ( abs_abs_real @ X2 ) )
% 0.35/0.60 = ( abs_abs_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_abs
% 0.35/0.60 thf(fact_4404_log__base__root,axiom,
% 0.35/0.60 ! [N: nat,B3: real,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.60 => ( ( log @ ( root @ N @ B3 ) @ X2 )
% 0.35/0.60 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % log_base_root
% 0.35/0.60 thf(fact_4405_mult__less__le__imp__less,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_le_imp_less
% 0.35/0.60 thf(fact_4406_mult__less__le__imp__less,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_le_imp_less
% 0.35/0.60 thf(fact_4407_mult__less__le__imp__less,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_le_imp_less
% 0.35/0.60 thf(fact_4408_mult__less__le__imp__less,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_le_imp_less
% 0.35/0.60 thf(fact_4409_mult__le__less__imp__less,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ C @ D )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_less_imp_less
% 0.35/0.60 thf(fact_4410_mult__le__less__imp__less,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ D )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_less_imp_less
% 0.35/0.60 thf(fact_4411_mult__le__less__imp__less,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ C @ D )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_less_imp_less
% 0.35/0.60 thf(fact_4412_mult__le__less__imp__less,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ C @ D )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_less_imp_less
% 0.35/0.60 thf(fact_4413_mult__right__le__imp__le,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_imp_le
% 0.35/0.60 thf(fact_4414_mult__right__le__imp__le,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_imp_le
% 0.35/0.60 thf(fact_4415_mult__right__le__imp__le,axiom,
% 0.35/0.60 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_imp_le
% 0.35/0.60 thf(fact_4416_mult__right__le__imp__le,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_imp_le
% 0.35/0.60 thf(fact_4417_mult__left__le__imp__le,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_imp_le
% 0.35/0.60 thf(fact_4418_mult__left__le__imp__le,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_imp_le
% 0.35/0.60 thf(fact_4419_mult__left__le__imp__le,axiom,
% 0.35/0.60 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_imp_le
% 0.35/0.60 thf(fact_4420_mult__left__le__imp__le,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_imp_le
% 0.35/0.60 thf(fact_4421_mult__le__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_pos
% 0.35/0.60 thf(fact_4422_mult__le__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_pos
% 0.35/0.60 thf(fact_4423_mult__le__cancel__left__pos,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_pos
% 0.35/0.60 thf(fact_4424_mult__le__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_neg
% 0.35/0.60 thf(fact_4425_mult__le__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_neg
% 0.35/0.60 thf(fact_4426_mult__le__cancel__left__neg,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left_neg
% 0.35/0.60 thf(fact_4427_mult__less__cancel__right,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right
% 0.35/0.60 thf(fact_4428_mult__less__cancel__right,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right
% 0.35/0.60 thf(fact_4429_mult__less__cancel__right,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right
% 0.35/0.60 thf(fact_4430_mult__strict__mono_H,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono'
% 0.35/0.60 thf(fact_4431_mult__strict__mono_H,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono'
% 0.35/0.60 thf(fact_4432_mult__strict__mono_H,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono'
% 0.35/0.60 thf(fact_4433_mult__strict__mono_H,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ C @ D )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono'
% 0.35/0.60 thf(fact_4434_mult__right__less__imp__less,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_less_imp_less
% 0.35/0.60 thf(fact_4435_mult__right__less__imp__less,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_less_imp_less
% 0.35/0.60 thf(fact_4436_mult__right__less__imp__less,axiom,
% 0.35/0.60 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_less_imp_less
% 0.35/0.60 thf(fact_4437_mult__right__less__imp__less,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_less_imp_less
% 0.35/0.60 thf(fact_4438_mult__less__cancel__left,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left
% 0.35/0.60 thf(fact_4439_mult__less__cancel__left,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left
% 0.35/0.60 thf(fact_4440_mult__less__cancel__left,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left
% 0.35/0.60 thf(fact_4441_mult__strict__mono,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ C @ D )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono
% 0.35/0.60 thf(fact_4442_mult__strict__mono,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ D )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono
% 0.35/0.60 thf(fact_4443_mult__strict__mono,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.60 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_nat @ C @ D )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono
% 0.35/0.60 thf(fact_4444_mult__strict__mono,axiom,
% 0.35/0.60 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ C @ D )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_strict_mono
% 0.35/0.60 thf(fact_4445_mult__left__less__imp__less,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_less_imp_less
% 0.35/0.60 thf(fact_4446_mult__left__less__imp__less,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_less_imp_less
% 0.35/0.60 thf(fact_4447_mult__left__less__imp__less,axiom,
% 0.35/0.60 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_less_imp_less
% 0.35/0.60 thf(fact_4448_mult__left__less__imp__less,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_less_imp_less
% 0.35/0.60 thf(fact_4449_mult__le__cancel__right,axiom,
% 0.35/0.60 ! [A2: real,C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right
% 0.35/0.60 thf(fact_4450_mult__le__cancel__right,axiom,
% 0.35/0.60 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right
% 0.35/0.60 thf(fact_4451_mult__le__cancel__right,axiom,
% 0.35/0.60 ! [A2: int,C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right
% 0.35/0.60 thf(fact_4452_mult__le__cancel__left,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left
% 0.35/0.60 thf(fact_4453_mult__le__cancel__left,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left
% 0.35/0.60 thf(fact_4454_mult__le__cancel__left,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ B3 ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left
% 0.35/0.60 thf(fact_4455_sum__squares__le__zero__iff,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.35/0.60 = ( ( X2 = zero_zero_real )
% 0.35/0.60 & ( Y3 = zero_zero_real ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_le_zero_iff
% 0.35/0.60 thf(fact_4456_sum__squares__le__zero__iff,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.35/0.60 = ( ( X2 = zero_zero_rat )
% 0.35/0.60 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_le_zero_iff
% 0.35/0.60 thf(fact_4457_sum__squares__le__zero__iff,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.35/0.60 = ( ( X2 = zero_zero_int )
% 0.35/0.60 & ( Y3 = zero_zero_int ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_le_zero_iff
% 0.35/0.60 thf(fact_4458_sum__squares__ge__zero,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_ge_zero
% 0.35/0.60 thf(fact_4459_sum__squares__ge__zero,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_ge_zero
% 0.35/0.60 thf(fact_4460_sum__squares__ge__zero,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_ge_zero
% 0.35/0.60 thf(fact_4461_mult__left__le__one__le,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_one_le
% 0.35/0.60 thf(fact_4462_mult__left__le__one__le,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_one_le
% 0.35/0.60 thf(fact_4463_mult__left__le__one__le,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.35/0.60 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le_one_le
% 0.35/0.60 thf(fact_4464_mult__right__le__one__le,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_one_le
% 0.35/0.60 thf(fact_4465_mult__right__le__one__le,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_one_le
% 0.35/0.60 thf(fact_4466_mult__right__le__one__le,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.35/0.60 => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_right_le_one_le
% 0.35/0.60 thf(fact_4467_mult__le__one,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ one_one_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ B3 @ one_one_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ one_one_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_one
% 0.35/0.60 thf(fact_4468_mult__le__one,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ B3 @ one_one_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ one_one_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_one
% 0.35/0.60 thf(fact_4469_mult__le__one,axiom,
% 0.35/0.60 ! [A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_nat @ B3 @ one_one_nat )
% 0.35/0.60 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_one
% 0.35/0.60 thf(fact_4470_mult__le__one,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ A2 @ one_one_int )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ B3 @ one_one_int )
% 0.35/0.60 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_one
% 0.35/0.60 thf(fact_4471_mult__left__le,axiom,
% 0.35/0.60 ! [C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ C @ one_one_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le
% 0.35/0.60 thf(fact_4472_mult__left__le,axiom,
% 0.35/0.60 ! [C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le
% 0.35/0.60 thf(fact_4473_mult__left__le,axiom,
% 0.35/0.60 ! [C: nat,A2: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.35/0.60 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le
% 0.35/0.60 thf(fact_4474_mult__left__le,axiom,
% 0.35/0.60 ! [C: int,A2: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ C @ one_one_int )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_left_le
% 0.35/0.60 thf(fact_4475_sum__squares__gt__zero__iff,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.35/0.60 = ( ( X2 != zero_zero_real )
% 0.35/0.60 | ( Y3 != zero_zero_real ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_gt_zero_iff
% 0.35/0.60 thf(fact_4476_sum__squares__gt__zero__iff,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.35/0.60 = ( ( X2 != zero_zero_rat )
% 0.35/0.60 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_gt_zero_iff
% 0.35/0.60 thf(fact_4477_sum__squares__gt__zero__iff,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.35/0.60 = ( ( X2 != zero_zero_int )
% 0.35/0.60 | ( Y3 != zero_zero_int ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % sum_squares_gt_zero_iff
% 0.35/0.60 thf(fact_4478_not__sum__squares__lt__zero,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.35/0.60
% 0.35/0.60 % not_sum_squares_lt_zero
% 0.35/0.60 thf(fact_4479_not__sum__squares__lt__zero,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.35/0.60
% 0.35/0.60 % not_sum_squares_lt_zero
% 0.35/0.60 thf(fact_4480_not__sum__squares__lt__zero,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.35/0.60
% 0.35/0.60 % not_sum_squares_lt_zero
% 0.35/0.60 thf(fact_4481_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.35/0.60 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.60 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.60 => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
% 0.35/0.60 = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.35/0.60 thf(fact_4482_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.35/0.60 thf(fact_4483_divide__strict__left__mono__neg,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_strict_left_mono_neg
% 0.35/0.60 thf(fact_4484_divide__strict__left__mono__neg,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_strict_left_mono_neg
% 0.35/0.60 thf(fact_4485_divide__strict__left__mono,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_strict_left_mono
% 0.35/0.60 thf(fact_4486_divide__strict__left__mono,axiom,
% 0.35/0.60 ! [B3: real,A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_strict_left_mono
% 0.35/0.60 thf(fact_4487_mult__imp__less__div__pos,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 0.35/0.60 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_less_div_pos
% 0.35/0.60 thf(fact_4488_mult__imp__less__div__pos,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 0.35/0.60 => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_less_div_pos
% 0.35/0.60 thf(fact_4489_mult__imp__div__pos__less,axiom,
% 0.35/0.60 ! [Y3: rat,X2: rat,Z: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 0.35/0.60 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_div_pos_less
% 0.35/0.60 thf(fact_4490_mult__imp__div__pos__less,axiom,
% 0.35/0.60 ! [Y3: real,X2: real,Z: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 0.35/0.60 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_div_pos_less
% 0.35/0.60 thf(fact_4491_pos__less__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_less_divide_eq
% 0.35/0.60 thf(fact_4492_pos__less__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_less_divide_eq
% 0.35/0.60 thf(fact_4493_pos__divide__less__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_divide_less_eq
% 0.35/0.60 thf(fact_4494_pos__divide__less__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_divide_less_eq
% 0.35/0.60 thf(fact_4495_neg__less__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_less_divide_eq
% 0.35/0.60 thf(fact_4496_neg__less__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_less_divide_eq
% 0.35/0.60 thf(fact_4497_neg__divide__less__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_divide_less_eq
% 0.35/0.60 thf(fact_4498_neg__divide__less__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_divide_less_eq
% 0.35/0.60 thf(fact_4499_less__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_divide_eq
% 0.35/0.60 thf(fact_4500_less__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_divide_eq
% 0.35/0.60 thf(fact_4501_divide__less__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_less_eq
% 0.35/0.60 thf(fact_4502_divide__less__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_less_eq
% 0.35/0.60 thf(fact_4503_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff1
% 0.35/0.60 thf(fact_4504_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff1
% 0.35/0.60 thf(fact_4505_ordered__ring__class_Ole__add__iff1,axiom,
% 0.35/0.60 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff1
% 0.35/0.60 thf(fact_4506_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff2
% 0.35/0.60 thf(fact_4507_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff2
% 0.35/0.60 thf(fact_4508_ordered__ring__class_Ole__add__iff2,axiom,
% 0.35/0.60 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ordered_ring_class.le_add_iff2
% 0.35/0.60 thf(fact_4509_add__divide__eq__if__simps_I2_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(2)
% 0.35/0.60 thf(fact_4510_add__divide__eq__if__simps_I2_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(2)
% 0.35/0.60 thf(fact_4511_add__divide__eq__if__simps_I2_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(2)
% 0.35/0.60 thf(fact_4512_add__divide__eq__if__simps_I1_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(1)
% 0.35/0.60 thf(fact_4513_add__divide__eq__if__simps_I1_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(1)
% 0.35/0.60 thf(fact_4514_add__divide__eq__if__simps_I1_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(1)
% 0.35/0.60 thf(fact_4515_add__frac__eq,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_eq
% 0.35/0.60 thf(fact_4516_add__frac__eq,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real,W2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_eq
% 0.35/0.60 thf(fact_4517_add__frac__eq,axiom,
% 0.35/0.60 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 0.35/0.60 ( ( Y3 != zero_zero_complex )
% 0.35/0.60 => ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_eq
% 0.35/0.60 thf(fact_4518_add__frac__num,axiom,
% 0.35/0.60 ! [Y3: rat,X2: rat,Z: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_num
% 0.35/0.60 thf(fact_4519_add__frac__num,axiom,
% 0.35/0.60 ! [Y3: real,X2: real,Z: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_num
% 0.35/0.60 thf(fact_4520_add__frac__num,axiom,
% 0.35/0.60 ! [Y3: complex,X2: complex,Z: complex] :
% 0.35/0.60 ( ( Y3 != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ Z )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_frac_num
% 0.35/0.60 thf(fact_4521_add__num__frac,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_num_frac
% 0.35/0.60 thf(fact_4522_add__num__frac,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_num_frac
% 0.35/0.60 thf(fact_4523_add__num__frac,axiom,
% 0.35/0.60 ! [Y3: complex,Z: complex,X2: complex] :
% 0.35/0.60 ( ( Y3 != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_num_frac
% 0.35/0.60 thf(fact_4524_add__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_iff
% 0.35/0.60 thf(fact_4525_add__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_iff
% 0.35/0.60 thf(fact_4526_add__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_iff
% 0.35/0.60 thf(fact_4527_divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_add_eq_iff
% 0.35/0.60 thf(fact_4528_divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_add_eq_iff
% 0.35/0.60 thf(fact_4529_divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_add_eq_iff
% 0.35/0.60 thf(fact_4530_less__add__iff1,axiom,
% 0.35/0.60 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff1
% 0.35/0.60 thf(fact_4531_less__add__iff1,axiom,
% 0.35/0.60 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff1
% 0.35/0.60 thf(fact_4532_less__add__iff1,axiom,
% 0.35/0.60 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff1
% 0.35/0.60 thf(fact_4533_less__add__iff2,axiom,
% 0.35/0.60 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 0.35/0.60 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff2
% 0.35/0.60 thf(fact_4534_less__add__iff2,axiom,
% 0.35/0.60 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff2
% 0.35/0.60 thf(fact_4535_less__add__iff2,axiom,
% 0.35/0.60 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 0.35/0.60 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 0.35/0.60 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_add_iff2
% 0.35/0.60 thf(fact_4536_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(4)
% 0.35/0.60 thf(fact_4537_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(4)
% 0.35/0.60 thf(fact_4538_add__divide__eq__if__simps_I4_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 0.35/0.60 = A2 ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(4)
% 0.35/0.60 thf(fact_4539_diff__frac__eq,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_frac_eq
% 0.35/0.60 thf(fact_4540_diff__frac__eq,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real,W2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_frac_eq
% 0.35/0.60 thf(fact_4541_diff__frac__eq,axiom,
% 0.35/0.60 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 0.35/0.60 ( ( Y3 != zero_zero_complex )
% 0.35/0.60 => ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_frac_eq
% 0.35/0.60 thf(fact_4542_diff__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_divide_eq_iff
% 0.35/0.60 thf(fact_4543_diff__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_divide_eq_iff
% 0.35/0.60 thf(fact_4544_diff__divide__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % diff_divide_eq_iff
% 0.35/0.60 thf(fact_4545_divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_diff_eq_iff
% 0.35/0.60 thf(fact_4546_divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_diff_eq_iff
% 0.35/0.60 thf(fact_4547_divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_diff_eq_iff
% 0.35/0.60 thf(fact_4548_ex__less__of__nat__mult,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 0.35/0.60 => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ex_less_of_nat_mult
% 0.35/0.60 thf(fact_4549_ex__less__of__nat__mult,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ex_less_of_nat_mult
% 0.35/0.60 thf(fact_4550_eq__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( times_times_real @ A2 @ C )
% 0.35/0.60 = ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ( C = zero_zero_real )
% 0.35/0.60 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_minus_divide_eq
% 0.35/0.60 thf(fact_4551_eq__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( times_times_rat @ A2 @ C )
% 0.35/0.60 = ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ( C = zero_zero_rat )
% 0.35/0.60 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_minus_divide_eq
% 0.35/0.60 thf(fact_4552_eq__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: complex,B3: complex,C: complex] :
% 0.35/0.60 ( ( A2
% 0.35/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( times_times_complex @ A2 @ C )
% 0.35/0.60 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 0.35/0.60 & ( ( C = zero_zero_complex )
% 0.35/0.60 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % eq_minus_divide_eq
% 0.35/0.60 thf(fact_4553_minus__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_real )
% 0.35/0.60 => ( ( uminus_uminus_real @ B3 )
% 0.35/0.60 = ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_real )
% 0.35/0.60 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_eq_eq
% 0.35/0.60 thf(fact_4554_minus__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_rat )
% 0.35/0.60 => ( ( uminus_uminus_rat @ B3 )
% 0.35/0.60 = ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_rat )
% 0.35/0.60 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_eq_eq
% 0.35/0.60 thf(fact_4555_minus__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: complex,C: complex,A2: complex] :
% 0.35/0.60 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.35/0.60 = A2 )
% 0.35/0.60 = ( ( ( C != zero_zero_complex )
% 0.35/0.60 => ( ( uminus1482373934393186551omplex @ B3 )
% 0.35/0.60 = ( times_times_complex @ A2 @ C ) ) )
% 0.35/0.60 & ( ( C = zero_zero_complex )
% 0.35/0.60 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_eq_eq
% 0.35/0.60 thf(fact_4556_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: real,A2: real,C: real] :
% 0.35/0.60 ( ( B3 != zero_zero_real )
% 0.35/0.60 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.35/0.60 = C )
% 0.35/0.60 = ( ( uminus_uminus_real @ A2 )
% 0.35/0.60 = ( times_times_real @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq
% 0.35/0.60 thf(fact_4557_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.60 ( ( B3 != zero_zero_rat )
% 0.35/0.60 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 0.35/0.60 = C )
% 0.35/0.60 = ( ( uminus_uminus_rat @ A2 )
% 0.35/0.60 = ( times_times_rat @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq
% 0.35/0.60 thf(fact_4558_nonzero__neg__divide__eq__eq,axiom,
% 0.35/0.60 ! [B3: complex,A2: complex,C: complex] :
% 0.35/0.60 ( ( B3 != zero_zero_complex )
% 0.35/0.60 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 0.35/0.60 = C )
% 0.35/0.60 = ( ( uminus1482373934393186551omplex @ A2 )
% 0.35/0.60 = ( times_times_complex @ C @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq
% 0.35/0.60 thf(fact_4559_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( B3 != zero_zero_real )
% 0.35/0.60 => ( ( C
% 0.35/0.60 = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) ) )
% 0.35/0.60 = ( ( times_times_real @ C @ B3 )
% 0.35/0.60 = ( uminus_uminus_real @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq2
% 0.35/0.60 thf(fact_4560_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( B3 != zero_zero_rat )
% 0.35/0.60 => ( ( C
% 0.35/0.60 = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) ) )
% 0.35/0.60 = ( ( times_times_rat @ C @ B3 )
% 0.35/0.60 = ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq2
% 0.35/0.60 thf(fact_4561_nonzero__neg__divide__eq__eq2,axiom,
% 0.35/0.60 ! [B3: complex,C: complex,A2: complex] :
% 0.35/0.60 ( ( B3 != zero_zero_complex )
% 0.35/0.60 => ( ( C
% 0.35/0.60 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.60 = ( ( times_times_complex @ C @ B3 )
% 0.35/0.60 = ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % nonzero_neg_divide_eq_eq2
% 0.35/0.60 thf(fact_4562_power__gt1__lemma,axiom,
% 0.35/0.60 ! [A2: real,N: nat] :
% 0.35/0.60 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_gt1_lemma
% 0.35/0.60 thf(fact_4563_power__gt1__lemma,axiom,
% 0.35/0.60 ! [A2: rat,N: nat] :
% 0.35/0.60 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_gt1_lemma
% 0.35/0.60 thf(fact_4564_power__gt1__lemma,axiom,
% 0.35/0.60 ! [A2: nat,N: nat] :
% 0.35/0.60 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.60 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_gt1_lemma
% 0.35/0.60 thf(fact_4565_power__gt1__lemma,axiom,
% 0.35/0.60 ! [A2: int,N: nat] :
% 0.35/0.60 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_gt1_lemma
% 0.35/0.60 thf(fact_4566_power__less__power__Suc,axiom,
% 0.35/0.60 ! [A2: real,N: nat] :
% 0.35/0.60 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.60 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_less_power_Suc
% 0.35/0.60 thf(fact_4567_power__less__power__Suc,axiom,
% 0.35/0.60 ! [A2: rat,N: nat] :
% 0.35/0.60 ( ( ord_less_rat @ one_one_rat @ A2 )
% 0.35/0.60 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_less_power_Suc
% 0.35/0.60 thf(fact_4568_power__less__power__Suc,axiom,
% 0.35/0.60 ! [A2: nat,N: nat] :
% 0.35/0.60 ( ( ord_less_nat @ one_one_nat @ A2 )
% 0.35/0.60 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_less_power_Suc
% 0.35/0.60 thf(fact_4569_power__less__power__Suc,axiom,
% 0.35/0.60 ! [A2: int,N: nat] :
% 0.35/0.60 ( ( ord_less_int @ one_one_int @ A2 )
% 0.35/0.60 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_less_power_Suc
% 0.35/0.60 thf(fact_4570_abs__mult__pos,axiom,
% 0.35/0.60 ! [X2: code_integer,Y3: code_integer] :
% 0.35/0.60 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 0.35/0.60 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X2 )
% 0.35/0.60 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_pos
% 0.35/0.60 thf(fact_4571_abs__mult__pos,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X2 )
% 0.35/0.60 = ( abs_abs_real @ ( times_times_real @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_pos
% 0.35/0.60 thf(fact_4572_abs__mult__pos,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.60 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X2 )
% 0.35/0.60 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_pos
% 0.35/0.60 thf(fact_4573_abs__mult__pos,axiom,
% 0.35/0.60 ! [X2: int,Y3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.60 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X2 )
% 0.35/0.60 = ( abs_abs_int @ ( times_times_int @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_mult_pos
% 0.35/0.60 thf(fact_4574_abs__eq__mult,axiom,
% 0.35/0.60 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.60 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.60 | ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
% 0.35/0.60 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.60 | ( ord_le3102999989581377725nteger @ B3 @ zero_z3403309356797280102nteger ) ) )
% 0.35/0.60 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B3 ) )
% 0.35/0.60 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_eq_mult
% 0.35/0.60 thf(fact_4575_abs__eq__mult,axiom,
% 0.35/0.60 ! [A2: real,B3: real] :
% 0.35/0.60 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.60 | ( ord_less_eq_real @ A2 @ zero_zero_real ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.60 | ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
% 0.35/0.60 => ( ( abs_abs_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 = ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_eq_mult
% 0.35/0.60 thf(fact_4576_abs__eq__mult,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 | ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.60 | ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
% 0.35/0.60 => ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_eq_mult
% 0.35/0.60 thf(fact_4577_abs__eq__mult,axiom,
% 0.35/0.60 ! [A2: int,B3: int] :
% 0.35/0.60 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.60 | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.60 | ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
% 0.35/0.60 => ( ( abs_abs_int @ ( times_times_int @ A2 @ B3 ) )
% 0.35/0.60 = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % abs_eq_mult
% 0.35/0.60 thf(fact_4578_ln__mult,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
% 0.35/0.60 = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ln_mult
% 0.35/0.60 thf(fact_4579_reals__Archimedean3,axiom,
% 0.35/0.60 ! [X2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ! [Y5: real] :
% 0.35/0.60 ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % reals_Archimedean3
% 0.35/0.60 thf(fact_4580_pos__zmult__eq__1__iff,axiom,
% 0.35/0.60 ! [M: int,N: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.60 => ( ( ( times_times_int @ M @ N )
% 0.35/0.60 = one_one_int )
% 0.35/0.60 = ( ( M = one_one_int )
% 0.35/0.60 & ( N = one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_zmult_eq_1_iff
% 0.35/0.60 thf(fact_4581_minusinfinity,axiom,
% 0.35/0.60 ! [D: int,P1: int > $o,P: int > $o] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60 => ( ! [X5: int,K2: int] :
% 0.35/0.60 ( ( P1 @ X5 )
% 0.35/0.60 = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.60 => ( ? [Z5: int] :
% 0.35/0.60 ! [X5: int] :
% 0.35/0.60 ( ( ord_less_int @ X5 @ Z5 )
% 0.35/0.60 => ( ( P @ X5 )
% 0.35/0.60 = ( P1 @ X5 ) ) )
% 0.35/0.60 => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.35/0.60 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minusinfinity
% 0.35/0.60 thf(fact_4582_plusinfinity,axiom,
% 0.35/0.60 ! [D: int,P4: int > $o,P: int > $o] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60 => ( ! [X5: int,K2: int] :
% 0.35/0.60 ( ( P4 @ X5 )
% 0.35/0.60 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.60 => ( ? [Z5: int] :
% 0.35/0.60 ! [X5: int] :
% 0.35/0.60 ( ( ord_less_int @ Z5 @ X5 )
% 0.35/0.60 => ( ( P @ X5 )
% 0.35/0.60 = ( P4 @ X5 ) ) )
% 0.35/0.60 => ( ? [X_12: int] : ( P4 @ X_12 )
% 0.35/0.60 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % plusinfinity
% 0.35/0.60 thf(fact_4583_zdiv__zmult2__eq,axiom,
% 0.35/0.60 ! [C: int,A2: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zdiv_zmult2_eq
% 0.35/0.60 thf(fact_4584_find__None__iff2,axiom,
% 0.35/0.60 ! [P: real > $o,Xs: list_real] :
% 0.35/0.60 ( ( none_real
% 0.35/0.60 = ( find_real @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: real] :
% 0.35/0.60 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4585_find__None__iff2,axiom,
% 0.35/0.60 ! [P: $o > $o,Xs: list_o] :
% 0.35/0.60 ( ( none_o
% 0.35/0.60 = ( find_o @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: $o] :
% 0.35/0.60 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4586_find__None__iff2,axiom,
% 0.35/0.60 ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.35/0.60 ( ( none_set_nat
% 0.35/0.60 = ( find_set_nat @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: set_nat] :
% 0.35/0.60 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4587_find__None__iff2,axiom,
% 0.35/0.60 ! [P: int > $o,Xs: list_int] :
% 0.35/0.60 ( ( none_int
% 0.35/0.60 = ( find_int @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: int] :
% 0.35/0.60 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4588_find__None__iff2,axiom,
% 0.35/0.60 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.35/0.60 ( ( none_VEBT_VEBT
% 0.35/0.60 = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: vEBT_VEBT] :
% 0.35/0.60 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4589_find__None__iff2,axiom,
% 0.35/0.60 ! [P: nat > $o,Xs: list_nat] :
% 0.35/0.60 ( ( none_nat
% 0.35/0.60 = ( find_nat @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: nat] :
% 0.35/0.60 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4590_find__None__iff2,axiom,
% 0.35/0.60 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.35/0.60 ( ( none_P5556105721700978146at_nat
% 0.35/0.60 = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: product_prod_nat_nat] :
% 0.35/0.60 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4591_find__None__iff2,axiom,
% 0.35/0.60 ! [P: num > $o,Xs: list_num] :
% 0.35/0.60 ( ( none_num
% 0.35/0.60 = ( find_num @ P @ Xs ) )
% 0.35/0.60 = ( ~ ? [X: num] :
% 0.35/0.60 ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff2
% 0.35/0.60 thf(fact_4592_find__None__iff,axiom,
% 0.35/0.60 ! [P: real > $o,Xs: list_real] :
% 0.35/0.60 ( ( ( find_real @ P @ Xs )
% 0.35/0.60 = none_real )
% 0.35/0.60 = ( ~ ? [X: real] :
% 0.35/0.60 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4593_find__None__iff,axiom,
% 0.35/0.60 ! [P: $o > $o,Xs: list_o] :
% 0.35/0.60 ( ( ( find_o @ P @ Xs )
% 0.35/0.60 = none_o )
% 0.35/0.60 = ( ~ ? [X: $o] :
% 0.35/0.60 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4594_find__None__iff,axiom,
% 0.35/0.60 ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.35/0.60 ( ( ( find_set_nat @ P @ Xs )
% 0.35/0.60 = none_set_nat )
% 0.35/0.60 = ( ~ ? [X: set_nat] :
% 0.35/0.60 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4595_find__None__iff,axiom,
% 0.35/0.60 ! [P: int > $o,Xs: list_int] :
% 0.35/0.60 ( ( ( find_int @ P @ Xs )
% 0.35/0.60 = none_int )
% 0.35/0.60 = ( ~ ? [X: int] :
% 0.35/0.60 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4596_find__None__iff,axiom,
% 0.35/0.60 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.35/0.60 ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.35/0.60 = none_VEBT_VEBT )
% 0.35/0.60 = ( ~ ? [X: vEBT_VEBT] :
% 0.35/0.60 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4597_find__None__iff,axiom,
% 0.35/0.60 ! [P: nat > $o,Xs: list_nat] :
% 0.35/0.60 ( ( ( find_nat @ P @ Xs )
% 0.35/0.60 = none_nat )
% 0.35/0.60 = ( ~ ? [X: nat] :
% 0.35/0.60 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4598_find__None__iff,axiom,
% 0.35/0.60 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.35/0.60 ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.35/0.60 = none_P5556105721700978146at_nat )
% 0.35/0.60 = ( ~ ? [X: product_prod_nat_nat] :
% 0.35/0.60 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4599_find__None__iff,axiom,
% 0.35/0.60 ! [P: num > $o,Xs: list_num] :
% 0.35/0.60 ( ( ( find_num @ P @ Xs )
% 0.35/0.60 = none_num )
% 0.35/0.60 = ( ~ ? [X: num] :
% 0.35/0.60 ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 0.35/0.60 & ( P @ X ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % find_None_iff
% 0.35/0.60 thf(fact_4600_real__root__gt__zero,axiom,
% 0.35/0.60 ! [N: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_gt_zero
% 0.35/0.60 thf(fact_4601_real__root__strict__decreasing,axiom,
% 0.35/0.60 ! [N: nat,N6: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_nat @ N @ N6 )
% 0.35/0.60 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.60 => ( ord_less_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_strict_decreasing
% 0.35/0.60 thf(fact_4602_root__abs__power,axiom,
% 0.35/0.60 ! [N: nat,Y3: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 0.35/0.60 = ( abs_abs_real @ Y3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % root_abs_power
% 0.35/0.60 thf(fact_4603_mult__less__cancel__right2,axiom,
% 0.35/0.60 ! [A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ one_one_real ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right2
% 0.35/0.60 thf(fact_4604_mult__less__cancel__right2,axiom,
% 0.35/0.60 ! [A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right2
% 0.35/0.60 thf(fact_4605_mult__less__cancel__right2,axiom,
% 0.35/0.60 ! [A2: int,C: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ one_one_int ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right2
% 0.35/0.60 thf(fact_4606_mult__less__cancel__right1,axiom,
% 0.35/0.60 ! [C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right1
% 0.35/0.60 thf(fact_4607_mult__less__cancel__right1,axiom,
% 0.35/0.60 ! [C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right1
% 0.35/0.60 thf(fact_4608_mult__less__cancel__right1,axiom,
% 0.35/0.60 ! [C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ C @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_right1
% 0.35/0.60 thf(fact_4609_mult__less__cancel__left2,axiom,
% 0.35/0.60 ! [C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ A2 @ one_one_real ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left2
% 0.35/0.60 thf(fact_4610_mult__less__cancel__left2,axiom,
% 0.35/0.60 ! [C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left2
% 0.35/0.60 thf(fact_4611_mult__less__cancel__left2,axiom,
% 0.35/0.60 ! [C: int,A2: int] :
% 0.35/0.60 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ A2 @ one_one_int ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left2
% 0.35/0.60 thf(fact_4612_mult__less__cancel__left1,axiom,
% 0.35/0.60 ! [C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ one_one_real @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left1
% 0.35/0.60 thf(fact_4613_mult__less__cancel__left1,axiom,
% 0.35/0.60 ! [C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ one_one_rat @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left1
% 0.35/0.60 thf(fact_4614_mult__less__cancel__left1,axiom,
% 0.35/0.60 ! [C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_int @ C @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_int @ one_one_int @ B3 ) )
% 0.35/0.60 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_less_cancel_left1
% 0.35/0.60 thf(fact_4615_mult__le__cancel__right2,axiom,
% 0.35/0.60 ! [A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ one_one_real ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right2
% 0.35/0.60 thf(fact_4616_mult__le__cancel__right2,axiom,
% 0.35/0.60 ! [A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right2
% 0.35/0.60 thf(fact_4617_mult__le__cancel__right2,axiom,
% 0.35/0.60 ! [A2: int,C: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right2
% 0.35/0.60 thf(fact_4618_mult__le__cancel__right1,axiom,
% 0.35/0.60 ! [C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ one_one_real @ B3 ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right1
% 0.35/0.60 thf(fact_4619_mult__le__cancel__right1,axiom,
% 0.35/0.60 ! [C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right1
% 0.35/0.60 thf(fact_4620_mult__le__cancel__right1,axiom,
% 0.35/0.60 ! [C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ C @ ( times_times_int @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ one_one_int @ B3 ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_right1
% 0.35/0.60 thf(fact_4621_mult__le__cancel__left2,axiom,
% 0.35/0.60 ! [C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ one_one_real ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left2
% 0.35/0.60 thf(fact_4622_mult__le__cancel__left2,axiom,
% 0.35/0.60 ! [C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left2
% 0.35/0.60 thf(fact_4623_mult__le__cancel__left2,axiom,
% 0.35/0.60 ! [C: int,A2: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left2
% 0.35/0.60 thf(fact_4624_mult__le__cancel__left1,axiom,
% 0.35/0.60 ! [C: real,B3: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ one_one_real @ B3 ) )
% 0.35/0.60 & ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left1
% 0.35/0.60 thf(fact_4625_mult__le__cancel__left1,axiom,
% 0.35/0.60 ! [C: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
% 0.35/0.60 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left1
% 0.35/0.60 thf(fact_4626_mult__le__cancel__left1,axiom,
% 0.35/0.60 ! [C: int,B3: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B3 ) )
% 0.35/0.60 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.35/0.60 => ( ord_less_eq_int @ one_one_int @ B3 ) )
% 0.35/0.60 & ( ( ord_less_int @ C @ zero_zero_int )
% 0.35/0.60 => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_le_cancel_left1
% 0.35/0.60 thf(fact_4627_field__le__mult__one__interval,axiom,
% 0.35/0.60 ! [X2: real,Y3: real] :
% 0.35/0.60 ( ! [Z4: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ Z4 )
% 0.35/0.60 => ( ( ord_less_real @ Z4 @ one_one_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y3 ) ) )
% 0.35/0.60 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.60
% 0.35/0.60 % field_le_mult_one_interval
% 0.35/0.60 thf(fact_4628_field__le__mult__one__interval,axiom,
% 0.35/0.60 ! [X2: rat,Y3: rat] :
% 0.35/0.60 ( ! [Z4: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 0.35/0.60 => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X2 ) @ Y3 ) ) )
% 0.35/0.60 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.35/0.60
% 0.35/0.60 % field_le_mult_one_interval
% 0.35/0.60 thf(fact_4629_divide__left__mono__neg,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_left_mono_neg
% 0.35/0.60 thf(fact_4630_divide__left__mono__neg,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_left_mono_neg
% 0.35/0.60 thf(fact_4631_mult__imp__le__div__pos,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 0.35/0.60 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_le_div_pos
% 0.35/0.60 thf(fact_4632_mult__imp__le__div__pos,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 0.35/0.60 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_le_div_pos
% 0.35/0.60 thf(fact_4633_mult__imp__div__pos__le,axiom,
% 0.35/0.60 ! [Y3: real,X2: real,Z: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 0.35/0.60 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_div_pos_le
% 0.35/0.60 thf(fact_4634_mult__imp__div__pos__le,axiom,
% 0.35/0.60 ! [Y3: rat,X2: rat,Z: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 0.35/0.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % mult_imp_div_pos_le
% 0.35/0.60 thf(fact_4635_pos__le__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_le_divide_eq
% 0.35/0.60 thf(fact_4636_pos__le__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_le_divide_eq
% 0.35/0.60 thf(fact_4637_pos__divide__le__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_divide_le_eq
% 0.35/0.60 thf(fact_4638_pos__divide__le__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_divide_le_eq
% 0.35/0.60 thf(fact_4639_neg__le__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_le_divide_eq
% 0.35/0.60 thf(fact_4640_neg__le__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_le_divide_eq
% 0.35/0.60 thf(fact_4641_neg__divide__le__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_divide_le_eq
% 0.35/0.60 thf(fact_4642_neg__divide__le__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_divide_le_eq
% 0.35/0.60 thf(fact_4643_divide__left__mono,axiom,
% 0.35/0.60 ! [B3: real,A2: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_left_mono
% 0.35/0.60 thf(fact_4644_divide__left__mono,axiom,
% 0.35/0.60 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 0.35/0.60 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_left_mono
% 0.35/0.60 thf(fact_4645_le__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % le_divide_eq
% 0.35/0.60 thf(fact_4646_le__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % le_divide_eq
% 0.35/0.60 thf(fact_4647_divide__le__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_le_eq
% 0.35/0.60 thf(fact_4648_divide__le__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % divide_le_eq
% 0.35/0.60 thf(fact_4649_convex__bound__le,axiom,
% 0.35/0.60 ! [X2: real,A2: real,Y3: real,U: real,V: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.35/0.60 => ( ( ( plus_plus_real @ U @ V )
% 0.35/0.60 = one_one_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_le
% 0.35/0.60 thf(fact_4650_convex__bound__le,axiom,
% 0.35/0.60 ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.35/0.60 => ( ( ( plus_plus_rat @ U @ V )
% 0.35/0.60 = one_one_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_le
% 0.35/0.60 thf(fact_4651_convex__bound__le,axiom,
% 0.35/0.60 ! [X2: int,A2: int,Y3: int,U: int,V: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.35/0.60 => ( ( ( plus_plus_int @ U @ V )
% 0.35/0.60 = one_one_int )
% 0.35/0.60 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_le
% 0.35/0.60 thf(fact_4652_frac__le__eq,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real,W2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 0.35/0.60 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_le_eq
% 0.35/0.60 thf(fact_4653_frac__le__eq,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 0.35/0.60 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_le_eq
% 0.35/0.60 thf(fact_4654_frac__less__eq,axiom,
% 0.35/0.60 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 0.35/0.60 ( ( Y3 != zero_zero_rat )
% 0.35/0.60 => ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 0.35/0.60 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_less_eq
% 0.35/0.60 thf(fact_4655_frac__less__eq,axiom,
% 0.35/0.60 ! [Y3: real,Z: real,X2: real,W2: real] :
% 0.35/0.60 ( ( Y3 != zero_zero_real )
% 0.35/0.60 => ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 0.35/0.60 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % frac_less_eq
% 0.35/0.60 thf(fact_4656_power__Suc__less,axiom,
% 0.35/0.60 ! [A2: real,N: nat] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ A2 @ one_one_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_Suc_less
% 0.35/0.60 thf(fact_4657_power__Suc__less,axiom,
% 0.35/0.60 ! [A2: rat,N: nat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_Suc_less
% 0.35/0.60 thf(fact_4658_power__Suc__less,axiom,
% 0.35/0.60 ! [A2: nat,N: nat] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 0.35/0.60 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 0.35/0.60 => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_Suc_less
% 0.35/0.60 thf(fact_4659_power__Suc__less,axiom,
% 0.35/0.60 ! [A2: int,N: nat] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ A2 @ one_one_int )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_Suc_less
% 0.35/0.60 thf(fact_4660_pos__minus__divide__less__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_minus_divide_less_eq
% 0.35/0.60 thf(fact_4661_pos__minus__divide__less__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_minus_divide_less_eq
% 0.35/0.60 thf(fact_4662_pos__less__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_less_minus_divide_eq
% 0.35/0.60 thf(fact_4663_pos__less__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_less_minus_divide_eq
% 0.35/0.60 thf(fact_4664_neg__minus__divide__less__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_minus_divide_less_eq
% 0.35/0.60 thf(fact_4665_neg__minus__divide__less__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_minus_divide_less_eq
% 0.35/0.60 thf(fact_4666_neg__less__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_less_minus_divide_eq
% 0.35/0.60 thf(fact_4667_neg__less__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_less_minus_divide_eq
% 0.35/0.60 thf(fact_4668_minus__divide__less__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_less_eq
% 0.35/0.60 thf(fact_4669_minus__divide__less__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_less_eq
% 0.35/0.60 thf(fact_4670_less__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_minus_divide_eq
% 0.35/0.60 thf(fact_4671_less__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % less_minus_divide_eq
% 0.35/0.60 thf(fact_4672_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(3)
% 0.35/0.60 thf(fact_4673_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(3)
% 0.35/0.60 thf(fact_4674_add__divide__eq__if__simps_I3_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = B3 ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(3)
% 0.35/0.60 thf(fact_4675_minus__divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_add_eq_iff
% 0.35/0.60 thf(fact_4676_minus__divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_add_eq_iff
% 0.35/0.60 thf(fact_4677_minus__divide__add__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_add_eq_iff
% 0.35/0.60 thf(fact_4678_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(6)
% 0.35/0.60 thf(fact_4679_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(6)
% 0.35/0.60 thf(fact_4680_add__divide__eq__if__simps_I6_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(6)
% 0.35/0.60 thf(fact_4681_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.60 ! [Z: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ( Z = zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(5)
% 0.35/0.60 thf(fact_4682_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.60 ! [Z: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ( Z = zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(5)
% 0.35/0.60 thf(fact_4683_add__divide__eq__if__simps_I5_J,axiom,
% 0.35/0.60 ! [Z: complex,A2: complex,B3: complex] :
% 0.35/0.60 ( ( ( Z = zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 0.35/0.60 & ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % add_divide_eq_if_simps(5)
% 0.35/0.60 thf(fact_4684_minus__divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.60 ( ( Z != zero_zero_real )
% 0.35/0.60 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_diff_eq_iff
% 0.35/0.60 thf(fact_4685_minus__divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.60 ( ( Z != zero_zero_rat )
% 0.35/0.60 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_diff_eq_iff
% 0.35/0.60 thf(fact_4686_minus__divide__diff__eq__iff,axiom,
% 0.35/0.60 ! [Z: complex,X2: complex,Y3: complex] :
% 0.35/0.60 ( ( Z != zero_zero_complex )
% 0.35/0.60 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 0.35/0.60 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_diff_eq_iff
% 0.35/0.60 thf(fact_4687_zmult__zless__mono2__lemma,axiom,
% 0.35/0.60 ! [I: int,J: int,K: nat] :
% 0.35/0.60 ( ( ord_less_int @ I @ J )
% 0.35/0.60 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.60 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zmult_zless_mono2_lemma
% 0.35/0.60 thf(fact_4688_ln__realpow,axiom,
% 0.35/0.60 ! [X2: real,N: nat] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( ln_ln_real @ ( power_power_real @ X2 @ N ) )
% 0.35/0.60 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % ln_realpow
% 0.35/0.60 thf(fact_4689_q__pos__lemma,axiom,
% 0.35/0.60 ! [B7: int,Q5: int,R4: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.35/0.60 => ( ( ord_less_int @ R4 @ B7 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % q_pos_lemma
% 0.35/0.60 thf(fact_4690_zdiv__mono2__lemma,axiom,
% 0.35/0.60 ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 0.35/0.60 ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
% 0.35/0.60 = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.35/0.60 => ( ( ord_less_int @ R4 @ B7 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60 => ( ( ord_less_eq_int @ B7 @ B3 )
% 0.35/0.60 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zdiv_mono2_lemma
% 0.35/0.60 thf(fact_4691_zdiv__mono2__neg__lemma,axiom,
% 0.35/0.60 ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 0.35/0.60 ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
% 0.35/0.60 = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 0.35/0.60 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_int @ R2 @ B3 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.35/0.60 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 0.35/0.60 => ( ( ord_less_eq_int @ B7 @ B3 )
% 0.35/0.60 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % zdiv_mono2_neg_lemma
% 0.35/0.60 thf(fact_4692_unique__quotient__lemma,axiom,
% 0.35/0.60 ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 0.35/0.60 => ( ( ord_less_int @ R4 @ B3 )
% 0.35/0.60 => ( ( ord_less_int @ R2 @ B3 )
% 0.35/0.60 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % unique_quotient_lemma
% 0.35/0.60 thf(fact_4693_unique__quotient__lemma__neg,axiom,
% 0.35/0.60 ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
% 0.35/0.60 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.60 => ( ( ord_less_int @ B3 @ R2 )
% 0.35/0.60 => ( ( ord_less_int @ B3 @ R4 )
% 0.35/0.60 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % unique_quotient_lemma_neg
% 0.35/0.60 thf(fact_4694_incr__mult__lemma,axiom,
% 0.35/0.60 ! [D: int,P: int > $o,K: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60 => ( ! [X5: int] :
% 0.35/0.60 ( ( P @ X5 )
% 0.35/0.60 => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.60 => ! [X4: int] :
% 0.35/0.60 ( ( P @ X4 )
% 0.35/0.60 => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % incr_mult_lemma
% 0.35/0.60 thf(fact_4695_decr__mult__lemma,axiom,
% 0.35/0.60 ! [D: int,P: int > $o,K: int] :
% 0.35/0.60 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.60 => ( ! [X5: int] :
% 0.35/0.60 ( ( P @ X5 )
% 0.35/0.60 => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.60 => ! [X4: int] :
% 0.35/0.60 ( ( P @ X4 )
% 0.35/0.60 => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % decr_mult_lemma
% 0.35/0.60 thf(fact_4696_real__root__pos__pos,axiom,
% 0.35/0.60 ! [N: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_pos_pos
% 0.35/0.60 thf(fact_4697_real__root__strict__increasing,axiom,
% 0.35/0.60 ! [N: nat,N6: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_nat @ N @ N6 )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.60 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_strict_increasing
% 0.35/0.60 thf(fact_4698_real__root__decreasing,axiom,
% 0.35/0.60 ! [N: nat,N6: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.60 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.60 => ( ord_less_eq_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_decreasing
% 0.35/0.60 thf(fact_4699_real__root__pow__pos,axiom,
% 0.35/0.60 ! [N: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 0.35/0.60 = X2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_pow_pos
% 0.35/0.60 thf(fact_4700_real__root__pos__unique,axiom,
% 0.35/0.60 ! [N: nat,Y3: real,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.60 => ( ( ( power_power_real @ Y3 @ N )
% 0.35/0.60 = X2 )
% 0.35/0.60 => ( ( root @ N @ X2 )
% 0.35/0.60 = Y3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_pos_unique
% 0.35/0.60 thf(fact_4701_real__root__power__cancel,axiom,
% 0.35/0.60 ! [N: nat,X2: real] :
% 0.35/0.60 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.60 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 0.35/0.60 = X2 ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % real_root_power_cancel
% 0.35/0.60 thf(fact_4702_convex__bound__lt,axiom,
% 0.35/0.60 ! [X2: real,A2: real,Y3: real,U: real,V: real] :
% 0.35/0.60 ( ( ord_less_real @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_real @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.35/0.60 => ( ( ( plus_plus_real @ U @ V )
% 0.35/0.60 = one_one_real )
% 0.35/0.60 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_lt
% 0.35/0.60 thf(fact_4703_convex__bound__lt,axiom,
% 0.35/0.60 ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
% 0.35/0.60 ( ( ord_less_rat @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_rat @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.35/0.60 => ( ( ( plus_plus_rat @ U @ V )
% 0.35/0.60 = one_one_rat )
% 0.35/0.60 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_lt
% 0.35/0.60 thf(fact_4704_convex__bound__lt,axiom,
% 0.35/0.60 ! [X2: int,A2: int,Y3: int,U: int,V: int] :
% 0.35/0.60 ( ( ord_less_int @ X2 @ A2 )
% 0.35/0.60 => ( ( ord_less_int @ Y3 @ A2 )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.35/0.60 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.35/0.60 => ( ( ( plus_plus_int @ U @ V )
% 0.35/0.60 = one_one_int )
% 0.35/0.60 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % convex_bound_lt
% 0.35/0.60 thf(fact_4705_le__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: real,B3: real,C: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % le_minus_divide_eq
% 0.35/0.60 thf(fact_4706_le__minus__divide__eq,axiom,
% 0.35/0.60 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % le_minus_divide_eq
% 0.35/0.60 thf(fact_4707_minus__divide__le__eq,axiom,
% 0.35/0.60 ! [B3: real,C: real,A2: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_le_eq
% 0.35/0.60 thf(fact_4708_minus__divide__le__eq,axiom,
% 0.35/0.60 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 0.35/0.60 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % minus_divide_le_eq
% 0.35/0.60 thf(fact_4709_neg__le__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_le_minus_divide_eq
% 0.35/0.60 thf(fact_4710_neg__le__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_le_minus_divide_eq
% 0.35/0.60 thf(fact_4711_neg__minus__divide__le__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_minus_divide_le_eq
% 0.35/0.60 thf(fact_4712_neg__minus__divide__le__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % neg_minus_divide_le_eq
% 0.35/0.60 thf(fact_4713_pos__le__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: real,A2: real,B3: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_le_minus_divide_eq
% 0.35/0.60 thf(fact_4714_pos__le__minus__divide__eq,axiom,
% 0.35/0.60 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 0.35/0.60 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_le_minus_divide_eq
% 0.35/0.60 thf(fact_4715_pos__minus__divide__le__eq,axiom,
% 0.35/0.60 ! [C: real,B3: real,A2: real] :
% 0.35/0.60 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.60 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_minus_divide_le_eq
% 0.35/0.60 thf(fact_4716_pos__minus__divide__le__eq,axiom,
% 0.35/0.60 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.60 ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.60 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 0.35/0.60 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % pos_minus_divide_le_eq
% 0.35/0.60 thf(fact_4717_scaling__mono,axiom,
% 0.35/0.60 ! [U: real,V: real,R2: real,S2: real] :
% 0.35/0.60 ( ( ord_less_eq_real @ U @ V )
% 0.35/0.60 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.35/0.60 => ( ( ord_less_eq_real @ R2 @ S2 )
% 0.35/0.60 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % scaling_mono
% 0.35/0.60 thf(fact_4718_scaling__mono,axiom,
% 0.35/0.60 ! [U: rat,V: rat,R2: rat,S2: rat] :
% 0.35/0.60 ( ( ord_less_eq_rat @ U @ V )
% 0.35/0.60 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.35/0.60 => ( ( ord_less_eq_rat @ R2 @ S2 )
% 0.35/0.60 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % scaling_mono
% 0.35/0.60 thf(fact_4719_power__eq__if,axiom,
% 0.35/0.60 ( power_power_complex
% 0.35/0.60 = ( ^ [P5: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.60
% 0.35/0.60 % power_eq_if
% 0.35/0.60 thf(fact_4720_power__eq__if,axiom,
% 0.35/0.61 ( power_power_real
% 0.35/0.61 = ( ^ [P5: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_eq_if
% 0.35/0.61 thf(fact_4721_power__eq__if,axiom,
% 0.35/0.61 ( power_power_rat
% 0.35/0.61 = ( ^ [P5: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_eq_if
% 0.35/0.61 thf(fact_4722_power__eq__if,axiom,
% 0.35/0.61 ( power_power_nat
% 0.35/0.61 = ( ^ [P5: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_eq_if
% 0.35/0.61 thf(fact_4723_power__eq__if,axiom,
% 0.35/0.61 ( power_power_int
% 0.35/0.61 = ( ^ [P5: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_eq_if
% 0.35/0.61 thf(fact_4724_power__minus__mult,axiom,
% 0.35/0.61 ! [N: nat,A2: complex] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_complex @ ( power_power_complex @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 0.35/0.61 = ( power_power_complex @ A2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_mult
% 0.35/0.61 thf(fact_4725_power__minus__mult,axiom,
% 0.35/0.61 ! [N: nat,A2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_real @ ( power_power_real @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 0.35/0.61 = ( power_power_real @ A2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_mult
% 0.35/0.61 thf(fact_4726_power__minus__mult,axiom,
% 0.35/0.61 ! [N: nat,A2: rat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 0.35/0.61 = ( power_power_rat @ A2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_mult
% 0.35/0.61 thf(fact_4727_power__minus__mult,axiom,
% 0.35/0.61 ! [N: nat,A2: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 0.35/0.61 = ( power_power_nat @ A2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_mult
% 0.35/0.61 thf(fact_4728_power__minus__mult,axiom,
% 0.35/0.61 ! [N: nat,A2: int] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 0.35/0.61 = ( power_power_int @ A2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_mult
% 0.35/0.61 thf(fact_4729_real__archimedian__rdiv__eq__0,axiom,
% 0.35/0.61 ! [X2: real,C: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.35/0.61 => ( ! [M4: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 0.35/0.61 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
% 0.35/0.61 => ( X2 = zero_zero_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % real_archimedian_rdiv_eq_0
% 0.35/0.61 thf(fact_4730_log__eq__div__ln__mult__log,axiom,
% 0.35/0.61 ! [A2: real,B3: real,X2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( A2 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( B3 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( log @ A2 @ X2 )
% 0.35/0.61 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( ln_ln_real @ A2 ) ) @ ( log @ B3 @ X2 ) ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_eq_div_ln_mult_log
% 0.35/0.61 thf(fact_4731_log__mult,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( A2 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.61 => ( ( log @ A2 @ ( times_times_real @ X2 @ Y3 ) )
% 0.35/0.61 = ( plus_plus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_mult
% 0.35/0.61 thf(fact_4732_split__zdiv,axiom,
% 0.35/0.61 ! [P: int > $o,N: int,K: int] :
% 0.35/0.61 ( ( P @ ( divide_divide_int @ N @ K ) )
% 0.35/0.61 = ( ( ( K = zero_zero_int )
% 0.35/0.61 => ( P @ zero_zero_int ) )
% 0.35/0.61 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.61 => ! [I4: int,J3: int] :
% 0.35/0.61 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.61 & ( ord_less_int @ J3 @ K )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.61 => ( P @ I4 ) ) )
% 0.35/0.61 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.61 => ! [I4: int,J3: int] :
% 0.35/0.61 ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.61 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.61 => ( P @ I4 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_zdiv
% 0.35/0.61 thf(fact_4733_int__div__neg__eq,axiom,
% 0.35/0.61 ! [A2: int,B3: int,Q3: int,R2: int] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.61 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.61 => ( ( ord_less_int @ B3 @ R2 )
% 0.35/0.61 => ( ( divide_divide_int @ A2 @ B3 )
% 0.35/0.61 = Q3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % int_div_neg_eq
% 0.35/0.61 thf(fact_4734_int__div__pos__eq,axiom,
% 0.35/0.61 ! [A2: int,B3: int,Q3: int,R2: int] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.61 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.61 => ( ( ord_less_int @ R2 @ B3 )
% 0.35/0.61 => ( ( divide_divide_int @ A2 @ B3 )
% 0.35/0.61 = Q3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % int_div_pos_eq
% 0.35/0.61 thf(fact_4735_log__nat__power,axiom,
% 0.35/0.61 ! [X2: real,B3: real,N: nat] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( log @ B3 @ ( power_power_real @ X2 @ N ) )
% 0.35/0.61 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_nat_power
% 0.35/0.61 thf(fact_4736_real__root__increasing,axiom,
% 0.35/0.61 ! [N: nat,N6: nat,X2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ N6 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.61 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % real_root_increasing
% 0.35/0.61 thf(fact_4737_ln__root,axiom,
% 0.35/0.61 ! [N: nat,B3: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( ln_ln_real @ ( root @ N @ B3 ) )
% 0.35/0.61 = ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ln_root
% 0.35/0.61 thf(fact_4738_mult__le__cancel__iff2,axiom,
% 0.35/0.61 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.35/0.61 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y3 ) )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff2
% 0.35/0.61 thf(fact_4739_mult__le__cancel__iff2,axiom,
% 0.35/0.61 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.35/0.61 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y3 ) )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff2
% 0.35/0.61 thf(fact_4740_mult__le__cancel__iff2,axiom,
% 0.35/0.61 ! [Z: int,X2: int,Y3: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.61 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y3 ) )
% 0.35/0.61 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff2
% 0.35/0.61 thf(fact_4741_mult__le__cancel__iff1,axiom,
% 0.35/0.61 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.35/0.61 => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff1
% 0.35/0.61 thf(fact_4742_mult__le__cancel__iff1,axiom,
% 0.35/0.61 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.35/0.61 => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff1
% 0.35/0.61 thf(fact_4743_mult__le__cancel__iff1,axiom,
% 0.35/0.61 ! [Z: int,X2: int,Y3: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.61 => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel_iff1
% 0.35/0.61 thf(fact_4744_mult__less__iff1,axiom,
% 0.35/0.61 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.35/0.61 => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_iff1
% 0.35/0.61 thf(fact_4745_mult__less__iff1,axiom,
% 0.35/0.61 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 0.35/0.61 => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_iff1
% 0.35/0.61 thf(fact_4746_mult__less__iff1,axiom,
% 0.35/0.61 ! [Z: int,X2: int,Y3: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.61 => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 0.35/0.61 = ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_iff1
% 0.35/0.61 thf(fact_4747_arctan__add,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.61 => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 0.35/0.61 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_add
% 0.35/0.61 thf(fact_4748_root__powr__inverse,axiom,
% 0.35/0.61 ! [N: nat,X2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( root @ N @ X2 )
% 0.35/0.61 = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % root_powr_inverse
% 0.35/0.61 thf(fact_4749_eucl__rel__int__iff,axiom,
% 0.35/0.61 ! [K: int,L: int,Q3: int,R2: int] :
% 0.35/0.61 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.35/0.61 = ( ( K
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 0.35/0.61 & ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.61 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.61 & ( ord_less_int @ R2 @ L ) ) )
% 0.35/0.61 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.61 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.61 => ( ( ord_less_int @ L @ R2 )
% 0.35/0.61 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.35/0.61 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.61 => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eucl_rel_int_iff
% 0.35/0.61 thf(fact_4750_log__minus__eq__powr,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( B3 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( minus_minus_real @ ( log @ B3 @ X2 ) @ Y3 )
% 0.35/0.61 = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_minus_eq_powr
% 0.35/0.61 thf(fact_4751_split__root,axiom,
% 0.35/0.61 ! [P: real > $o,N: nat,X2: real] :
% 0.35/0.61 ( ( P @ ( root @ N @ X2 ) )
% 0.35/0.61 = ( ( ( N = zero_zero_nat )
% 0.35/0.61 => ( P @ zero_zero_real ) )
% 0.35/0.61 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ! [Y: real] :
% 0.35/0.61 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.35/0.61 = X2 )
% 0.35/0.61 => ( P @ Y ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_root
% 0.35/0.61 thf(fact_4752_mult__is__0,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ M @ N )
% 0.35/0.61 = zero_zero_nat )
% 0.35/0.61 = ( ( M = zero_zero_nat )
% 0.35/0.61 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_is_0
% 0.35/0.61 thf(fact_4753_mult__0__right,axiom,
% 0.35/0.61 ! [M: nat] :
% 0.35/0.61 ( ( times_times_nat @ M @ zero_zero_nat )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mult_0_right
% 0.35/0.61 thf(fact_4754_mult__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ K @ M )
% 0.35/0.61 = ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ( M = N )
% 0.35/0.61 | ( K = zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_cancel1
% 0.35/0.61 thf(fact_4755_mult__cancel2,axiom,
% 0.35/0.61 ! [M: nat,K: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ M @ K )
% 0.35/0.61 = ( times_times_nat @ N @ K ) )
% 0.35/0.61 = ( ( M = N )
% 0.35/0.61 | ( K = zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_cancel2
% 0.35/0.61 thf(fact_4756_sgn__0,axiom,
% 0.35/0.61 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 0.35/0.61 = zero_z3403309356797280102nteger ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0
% 0.35/0.61 thf(fact_4757_sgn__0,axiom,
% 0.35/0.61 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.35/0.61 = zero_zero_complex ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0
% 0.35/0.61 thf(fact_4758_sgn__0,axiom,
% 0.35/0.61 ( ( sgn_sgn_real @ zero_zero_real )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0
% 0.35/0.61 thf(fact_4759_sgn__0,axiom,
% 0.35/0.61 ( ( sgn_sgn_rat @ zero_zero_rat )
% 0.35/0.61 = zero_zero_rat ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0
% 0.35/0.61 thf(fact_4760_sgn__0,axiom,
% 0.35/0.61 ( ( sgn_sgn_int @ zero_zero_int )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0
% 0.35/0.61 thf(fact_4761_powr__0,axiom,
% 0.35/0.61 ! [Z: real] :
% 0.35/0.61 ( ( powr_real @ zero_zero_real @ Z )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % powr_0
% 0.35/0.61 thf(fact_4762_powr__eq__0__iff,axiom,
% 0.35/0.61 ! [W2: real,Z: real] :
% 0.35/0.61 ( ( ( powr_real @ W2 @ Z )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 = ( W2 = zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_eq_0_iff
% 0.35/0.61 thf(fact_4763_nat__mult__eq__1__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ M @ N )
% 0.35/0.61 = one_one_nat )
% 0.35/0.61 = ( ( M = one_one_nat )
% 0.35/0.61 & ( N = one_one_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_eq_1_iff
% 0.35/0.61 thf(fact_4764_nat__1__eq__mult__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( one_one_nat
% 0.35/0.61 = ( times_times_nat @ M @ N ) )
% 0.35/0.61 = ( ( M = one_one_nat )
% 0.35/0.61 & ( N = one_one_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_1_eq_mult_iff
% 0.35/0.61 thf(fact_4765_sgn__less,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
% 0.35/0.61 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_less
% 0.35/0.61 thf(fact_4766_sgn__less,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ord_less_real @ ( sgn_sgn_real @ A2 ) @ zero_zero_real )
% 0.35/0.61 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_less
% 0.35/0.61 thf(fact_4767_sgn__less,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ ( sgn_sgn_rat @ A2 ) @ zero_zero_rat )
% 0.35/0.61 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_less
% 0.35/0.61 thf(fact_4768_sgn__less,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ord_less_int @ ( sgn_sgn_int @ A2 ) @ zero_zero_int )
% 0.35/0.61 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_less
% 0.35/0.61 thf(fact_4769_sgn__greater,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A2 ) )
% 0.35/0.61 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_greater
% 0.35/0.61 thf(fact_4770_sgn__greater,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A2 ) )
% 0.35/0.61 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_greater
% 0.35/0.61 thf(fact_4771_sgn__greater,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A2 ) )
% 0.35/0.61 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_greater
% 0.35/0.61 thf(fact_4772_sgn__greater,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A2 ) )
% 0.35/0.61 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_greater
% 0.35/0.61 thf(fact_4773_one__eq__mult__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ( suc @ zero_zero_nat )
% 0.35/0.61 = ( times_times_nat @ M @ N ) )
% 0.35/0.61 = ( ( M
% 0.35/0.61 = ( suc @ zero_zero_nat ) )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_eq_mult_iff
% 0.35/0.61 thf(fact_4774_mult__eq__1__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ M @ N )
% 0.35/0.61 = ( suc @ zero_zero_nat ) )
% 0.35/0.61 = ( ( M
% 0.35/0.61 = ( suc @ zero_zero_nat ) )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_eq_1_iff
% 0.35/0.61 thf(fact_4775_mult__less__cancel2,axiom,
% 0.35/0.61 ! [M: nat,K: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.35/0.61 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_cancel2
% 0.35/0.61 thf(fact_4776_nat__0__less__mult__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 0.35/0.61 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.61 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_0_less_mult_iff
% 0.35/0.61 thf(fact_4777_powr__zero__eq__one,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ( X2 = zero_zero_real )
% 0.35/0.61 => ( ( powr_real @ X2 @ zero_zero_real )
% 0.35/0.61 = zero_zero_real ) )
% 0.35/0.61 & ( ( X2 != zero_zero_real )
% 0.35/0.61 => ( ( powr_real @ X2 @ zero_zero_real )
% 0.35/0.61 = one_one_real ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_zero_eq_one
% 0.35/0.61 thf(fact_4778_mult__Suc__right,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( times_times_nat @ M @ ( suc @ N ) )
% 0.35/0.61 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_Suc_right
% 0.35/0.61 thf(fact_4779_powr__gt__zero,axiom,
% 0.35/0.61 ! [X2: real,A2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A2 ) )
% 0.35/0.61 = ( X2 != zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_gt_zero
% 0.35/0.61 thf(fact_4780_powr__nonneg__iff,axiom,
% 0.35/0.61 ! [A2: real,X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real )
% 0.35/0.61 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_nonneg_iff
% 0.35/0.61 thf(fact_4781_powr__less__cancel__iff,axiom,
% 0.35/0.61 ! [X2: real,A2: real,B3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 0.35/0.61 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_cancel_iff
% 0.35/0.61 thf(fact_4782_zero__less__arctan__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_less_arctan_iff
% 0.35/0.61 thf(fact_4783_arctan__less__zero__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_real @ ( arctan @ X2 ) @ zero_zero_real )
% 0.35/0.61 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_less_zero_iff
% 0.35/0.61 thf(fact_4784_zero__le__arctan__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
% 0.35/0.61 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_le_arctan_iff
% 0.35/0.61 thf(fact_4785_arctan__le__zero__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_le_zero_iff
% 0.35/0.61 thf(fact_4786_sgn__pos,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.61 => ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = one_one_Code_integer ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_pos
% 0.35/0.61 thf(fact_4787_sgn__pos,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = one_one_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_pos
% 0.35/0.61 thf(fact_4788_sgn__pos,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.61 => ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = one_one_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_pos
% 0.35/0.61 thf(fact_4789_sgn__pos,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.61 => ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = one_one_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_pos
% 0.35/0.61 thf(fact_4790_one__le__mult__iff,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 0.35/0.61 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.61 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_le_mult_iff
% 0.35/0.61 thf(fact_4791_mult__le__cancel2,axiom,
% 0.35/0.61 ! [M: nat,K: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 0.35/0.61 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_cancel2
% 0.35/0.61 thf(fact_4792_abs__sgn__eq__1,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( A2 != zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 0.35/0.61 = one_one_Code_integer ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq_1
% 0.35/0.61 thf(fact_4793_abs__sgn__eq__1,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( A2 != zero_zero_real )
% 0.35/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 0.35/0.61 = one_one_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq_1
% 0.35/0.61 thf(fact_4794_abs__sgn__eq__1,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( A2 != zero_zero_rat )
% 0.35/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 0.35/0.61 = one_one_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq_1
% 0.35/0.61 thf(fact_4795_abs__sgn__eq__1,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( A2 != zero_zero_int )
% 0.35/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 0.35/0.61 = one_one_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq_1
% 0.35/0.61 thf(fact_4796_div__mult__self__is__m,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 0.35/0.61 = M ) ) ).
% 0.35/0.61
% 0.35/0.61 % div_mult_self_is_m
% 0.35/0.61 thf(fact_4797_div__mult__self1__is__m,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 0.35/0.61 = M ) ) ).
% 0.35/0.61
% 0.35/0.61 % div_mult_self1_is_m
% 0.35/0.61 thf(fact_4798_powr__eq__one__iff,axiom,
% 0.35/0.61 ! [A2: real,X2: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ A2 )
% 0.35/0.61 => ( ( ( powr_real @ A2 @ X2 )
% 0.35/0.61 = one_one_real )
% 0.35/0.61 = ( X2 = zero_zero_real ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_eq_one_iff
% 0.35/0.61 thf(fact_4799_powr__one__gt__zero__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ( powr_real @ X2 @ one_one_real )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_one_gt_zero_iff
% 0.35/0.61 thf(fact_4800_powr__one,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( powr_real @ X2 @ one_one_real )
% 0.35/0.61 = X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_one
% 0.35/0.61 thf(fact_4801_powr__le__cancel__iff,axiom,
% 0.35/0.61 ! [X2: real,A2: real,B3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 0.35/0.61 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_le_cancel_iff
% 0.35/0.61 thf(fact_4802_sgn__neg,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.61 => ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_neg
% 0.35/0.61 thf(fact_4803_sgn__neg,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.61 => ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_neg
% 0.35/0.61 thf(fact_4804_sgn__neg,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.61 => ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_neg
% 0.35/0.61 thf(fact_4805_sgn__neg,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_neg
% 0.35/0.61 thf(fact_4806_powr__log__cancel,axiom,
% 0.35/0.61 ! [A2: real,X2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( A2 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( powr_real @ A2 @ ( log @ A2 @ X2 ) )
% 0.35/0.61 = X2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_log_cancel
% 0.35/0.61 thf(fact_4807_log__powr__cancel,axiom,
% 0.35/0.61 ! [A2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( A2 != one_one_real )
% 0.35/0.61 => ( ( log @ A2 @ ( powr_real @ A2 @ Y3 ) )
% 0.35/0.61 = Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_powr_cancel
% 0.35/0.61 thf(fact_4808_sgn__0__0,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = zero_z3403309356797280102nteger )
% 0.35/0.61 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0_0
% 0.35/0.61 thf(fact_4809_sgn__0__0,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0_0
% 0.35/0.61 thf(fact_4810_sgn__0__0,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0_0
% 0.35/0.61 thf(fact_4811_sgn__0__0,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = zero_zero_int )
% 0.35/0.61 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_0_0
% 0.35/0.61 thf(fact_4812_sgn__eq__0__iff,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = zero_z3403309356797280102nteger )
% 0.35/0.61 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_eq_0_iff
% 0.35/0.61 thf(fact_4813_sgn__eq__0__iff,axiom,
% 0.35/0.61 ! [A2: complex] :
% 0.35/0.61 ( ( ( sgn_sgn_complex @ A2 )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 = ( A2 = zero_zero_complex ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_eq_0_iff
% 0.35/0.61 thf(fact_4814_sgn__eq__0__iff,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_eq_0_iff
% 0.35/0.61 thf(fact_4815_sgn__eq__0__iff,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_eq_0_iff
% 0.35/0.61 thf(fact_4816_sgn__eq__0__iff,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = zero_zero_int )
% 0.35/0.61 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_eq_0_iff
% 0.35/0.61 thf(fact_4817_Suc__mult__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.35/0.61 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.35/0.61 = ( M = N ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_mult_cancel1
% 0.35/0.61 thf(fact_4818_mult__0,axiom,
% 0.35/0.61 ! [N: nat] :
% 0.35/0.61 ( ( times_times_nat @ zero_zero_nat @ N )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mult_0
% 0.35/0.61 thf(fact_4819_le__cube,axiom,
% 0.35/0.61 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_cube
% 0.35/0.61 thf(fact_4820_le__square,axiom,
% 0.35/0.61 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_square
% 0.35/0.61 thf(fact_4821_mult__le__mono,axiom,
% 0.35/0.61 ! [I: nat,J: nat,K: nat,L: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ( ord_less_eq_nat @ K @ L )
% 0.35/0.61 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_mono
% 0.35/0.61 thf(fact_4822_mult__le__mono1,axiom,
% 0.35/0.61 ! [I: nat,J: nat,K: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_mono1
% 0.35/0.61 thf(fact_4823_mult__le__mono2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,K: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_le_mono2
% 0.35/0.61 thf(fact_4824_arctan__monotone,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_monotone
% 0.35/0.61 thf(fact_4825_arctan__less__iff,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 0.35/0.61 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_less_iff
% 0.35/0.61 thf(fact_4826_arctan__monotone_H,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_monotone'
% 0.35/0.61 thf(fact_4827_arctan__le__iff,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % arctan_le_iff
% 0.35/0.61 thf(fact_4828_add__mult__distrib2,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % add_mult_distrib2
% 0.35/0.61 thf(fact_4829_add__mult__distrib,axiom,
% 0.35/0.61 ! [M: nat,N: nat,K: nat] :
% 0.35/0.61 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % add_mult_distrib
% 0.35/0.61 thf(fact_4830_diff__mult__distrib2,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.61 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % diff_mult_distrib2
% 0.35/0.61 thf(fact_4831_diff__mult__distrib,axiom,
% 0.35/0.61 ! [M: nat,N: nat,K: nat] :
% 0.35/0.61 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 0.35/0.61 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % diff_mult_distrib
% 0.35/0.61 thf(fact_4832_nat__mult__1__right,axiom,
% 0.35/0.61 ! [N: nat] :
% 0.35/0.61 ( ( times_times_nat @ N @ one_one_nat )
% 0.35/0.61 = N ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_1_right
% 0.35/0.61 thf(fact_4833_nat__mult__1,axiom,
% 0.35/0.61 ! [N: nat] :
% 0.35/0.61 ( ( times_times_nat @ one_one_nat @ N )
% 0.35/0.61 = N ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_1
% 0.35/0.61 thf(fact_4834_powr__non__neg,axiom,
% 0.35/0.61 ! [A2: real,X2: real] :
% 0.35/0.61 ~ ( ord_less_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % powr_non_neg
% 0.35/0.61 thf(fact_4835_powr__less__mono2__neg,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_mono2_neg
% 0.35/0.61 thf(fact_4836_powr__ge__pzero,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_ge_pzero
% 0.35/0.61 thf(fact_4837_powr__mono2,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mono2
% 0.35/0.61 thf(fact_4838_powr__less__mono,axiom,
% 0.35/0.61 ! [A2: real,B3: real,X2: real] :
% 0.35/0.61 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_mono
% 0.35/0.61 thf(fact_4839_powr__less__cancel,axiom,
% 0.35/0.61 ! [X2: real,A2: real,B3: real] :
% 0.35/0.61 ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 0.35/0.61 => ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_cancel
% 0.35/0.61 thf(fact_4840_powr__mono,axiom,
% 0.35/0.61 ! [A2: real,B3: real,X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.61 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mono
% 0.35/0.61 thf(fact_4841_sgn__not__eq__imp,axiom,
% 0.35/0.61 ! [B3: int,A2: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ B3 )
% 0.35/0.61 != ( sgn_sgn_int @ A2 ) )
% 0.35/0.61 => ( ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 != zero_zero_int )
% 0.35/0.61 => ( ( ( sgn_sgn_int @ B3 )
% 0.35/0.61 != zero_zero_int )
% 0.35/0.61 => ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = ( uminus_uminus_int @ ( sgn_sgn_int @ B3 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_not_eq_imp
% 0.35/0.61 thf(fact_4842_sgn__not__eq__imp,axiom,
% 0.35/0.61 ! [B3: real,A2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ B3 )
% 0.35/0.61 != ( sgn_sgn_real @ A2 ) )
% 0.35/0.61 => ( ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( ( ( sgn_sgn_real @ B3 )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = ( uminus_uminus_real @ ( sgn_sgn_real @ B3 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_not_eq_imp
% 0.35/0.61 thf(fact_4843_sgn__not__eq__imp,axiom,
% 0.35/0.61 ! [B3: rat,A2: rat] :
% 0.35/0.61 ( ( ( sgn_sgn_rat @ B3 )
% 0.35/0.61 != ( sgn_sgn_rat @ A2 ) )
% 0.35/0.61 => ( ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( ( ( sgn_sgn_rat @ B3 )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B3 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_not_eq_imp
% 0.35/0.61 thf(fact_4844_sgn__not__eq__imp,axiom,
% 0.35/0.61 ! [B3: code_integer,A2: code_integer] :
% 0.35/0.61 ( ( ( sgn_sgn_Code_integer @ B3 )
% 0.35/0.61 != ( sgn_sgn_Code_integer @ A2 ) )
% 0.35/0.61 => ( ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 != zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( ( sgn_sgn_Code_integer @ B3 )
% 0.35/0.61 != zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B3 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_not_eq_imp
% 0.35/0.61 thf(fact_4845_Suc__mult__less__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.35/0.61 = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_mult_less_cancel1
% 0.35/0.61 thf(fact_4846_mult__less__mono1,axiom,
% 0.35/0.61 ! [I: nat,J: nat,K: nat] :
% 0.35/0.61 ( ( ord_less_nat @ I @ J )
% 0.35/0.61 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_mono1
% 0.35/0.61 thf(fact_4847_mult__less__mono2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,K: nat] :
% 0.35/0.61 ( ( ord_less_nat @ I @ J )
% 0.35/0.61 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_less_mono2
% 0.35/0.61 thf(fact_4848_Suc__mult__le__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 0.35/0.61 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_mult_le_cancel1
% 0.35/0.61 thf(fact_4849_mult__Suc,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( times_times_nat @ ( suc @ M ) @ N )
% 0.35/0.61 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_Suc
% 0.35/0.61 thf(fact_4850_less__mult__imp__div__less,axiom,
% 0.35/0.61 ! [M: nat,I: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 0.35/0.61 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_mult_imp_div_less
% 0.35/0.61 thf(fact_4851_mult__eq__self__implies__10,axiom,
% 0.35/0.61 ! [M: nat,N: nat] :
% 0.35/0.61 ( ( M
% 0.35/0.61 = ( times_times_nat @ M @ N ) )
% 0.35/0.61 => ( ( N = one_one_nat )
% 0.35/0.61 | ( M = zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_eq_self_implies_10
% 0.35/0.61 thf(fact_4852_div__times__less__eq__dividend,axiom,
% 0.35/0.61 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 0.35/0.61
% 0.35/0.61 % div_times_less_eq_dividend
% 0.35/0.61 thf(fact_4853_times__div__less__eq__dividend,axiom,
% 0.35/0.61 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 0.35/0.61
% 0.35/0.61 % times_div_less_eq_dividend
% 0.35/0.61 thf(fact_4854_powr__mono2_H,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mono2'
% 0.35/0.61 thf(fact_4855_powr__less__mono2,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_mono2
% 0.35/0.61 thf(fact_4856_gr__one__powr,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.61 => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gr_one_powr
% 0.35/0.61 thf(fact_4857_powr__inj,axiom,
% 0.35/0.61 ! [A2: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( A2 != one_one_real )
% 0.35/0.61 => ( ( ( powr_real @ A2 @ X2 )
% 0.35/0.61 = ( powr_real @ A2 @ Y3 ) )
% 0.35/0.61 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_inj
% 0.35/0.61 thf(fact_4858_ge__one__powr__ge__zero,axiom,
% 0.35/0.61 ! [X2: real,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ge_one_powr_ge_zero
% 0.35/0.61 thf(fact_4859_powr__mono__both,axiom,
% 0.35/0.61 ! [A2: real,B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.61 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ B3 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mono_both
% 0.35/0.61 thf(fact_4860_powr__le1,axiom,
% 0.35/0.61 ! [A2: real,X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ one_one_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_le1
% 0.35/0.61 thf(fact_4861_powr__divide,axiom,
% 0.35/0.61 ! [X2: real,Y3: real,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.61 => ( ( powr_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
% 0.35/0.61 = ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_divide
% 0.35/0.61 thf(fact_4862_powr__mult,axiom,
% 0.35/0.61 ! [X2: real,Y3: real,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.61 => ( ( powr_real @ ( times_times_real @ X2 @ Y3 ) @ A2 )
% 0.35/0.61 = ( times_times_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mult
% 0.35/0.61 thf(fact_4863_sgn__1__pos,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = one_one_Code_integer )
% 0.35/0.61 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_pos
% 0.35/0.61 thf(fact_4864_sgn__1__pos,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = one_one_real )
% 0.35/0.61 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_pos
% 0.35/0.61 thf(fact_4865_sgn__1__pos,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = one_one_rat )
% 0.35/0.61 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_pos
% 0.35/0.61 thf(fact_4866_sgn__1__pos,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = one_one_int )
% 0.35/0.61 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_pos
% 0.35/0.61 thf(fact_4867_sgn__root,axiom,
% 0.35/0.61 ! [N: nat,X2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( sgn_sgn_real @ ( root @ N @ X2 ) )
% 0.35/0.61 = ( sgn_sgn_real @ X2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_root
% 0.35/0.61 thf(fact_4868_abs__sgn__eq,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ( A2 = zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 0.35/0.61 = zero_z3403309356797280102nteger ) )
% 0.35/0.61 & ( ( A2 != zero_z3403309356797280102nteger )
% 0.35/0.61 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 0.35/0.61 = one_one_Code_integer ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq
% 0.35/0.61 thf(fact_4869_abs__sgn__eq,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ( A2 = zero_zero_real )
% 0.35/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 0.35/0.61 = zero_zero_real ) )
% 0.35/0.61 & ( ( A2 != zero_zero_real )
% 0.35/0.61 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 0.35/0.61 = one_one_real ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq
% 0.35/0.61 thf(fact_4870_abs__sgn__eq,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( A2 = zero_zero_rat )
% 0.35/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 0.35/0.61 = zero_zero_rat ) )
% 0.35/0.61 & ( ( A2 != zero_zero_rat )
% 0.35/0.61 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 0.35/0.61 = one_one_rat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq
% 0.35/0.61 thf(fact_4871_abs__sgn__eq,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ( A2 = zero_zero_int )
% 0.35/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 0.35/0.61 = zero_zero_int ) )
% 0.35/0.61 & ( ( A2 != zero_zero_int )
% 0.35/0.61 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 0.35/0.61 = one_one_int ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % abs_sgn_eq
% 0.35/0.61 thf(fact_4872_one__less__mult,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.61 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.61 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_mult
% 0.35/0.61 thf(fact_4873_n__less__m__mult__n,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.61 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % n_less_m_mult_n
% 0.35/0.61 thf(fact_4874_n__less__n__mult__m,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.61 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % n_less_n_mult_m
% 0.35/0.61 thf(fact_4875_div__less__iff__less__mult,axiom,
% 0.35/0.61 ! [Q3: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.35/0.61 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 0.35/0.61 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % div_less_iff_less_mult
% 0.35/0.61 thf(fact_4876_sgn__real__def,axiom,
% 0.35/0.61 ( sgn_sgn_real
% 0.35/0.61 = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_real_def
% 0.35/0.61 thf(fact_4877_sgn__1__neg,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ A2 )
% 0.35/0.61 = ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.61 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_neg
% 0.35/0.61 thf(fact_4878_sgn__1__neg,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ A2 )
% 0.35/0.61 = ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.61 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_neg
% 0.35/0.61 thf(fact_4879_sgn__1__neg,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( sgn_sgn_rat @ A2 )
% 0.35/0.61 = ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.61 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_neg
% 0.35/0.61 thf(fact_4880_sgn__1__neg,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( ( sgn_sgn_Code_integer @ A2 )
% 0.35/0.61 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.61 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_1_neg
% 0.35/0.61 thf(fact_4881_sgn__if,axiom,
% 0.35/0.61 ( sgn_sgn_int
% 0.35/0.61 = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_if
% 0.35/0.61 thf(fact_4882_sgn__if,axiom,
% 0.35/0.61 ( sgn_sgn_real
% 0.35/0.61 = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_if
% 0.35/0.61 thf(fact_4883_sgn__if,axiom,
% 0.35/0.61 ( sgn_sgn_rat
% 0.35/0.61 = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_if
% 0.35/0.61 thf(fact_4884_sgn__if,axiom,
% 0.35/0.61 ( sgn_sgn_Code_integer
% 0.35/0.61 = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_if
% 0.35/0.61 thf(fact_4885_powr__realpow,axiom,
% 0.35/0.61 ! [X2: real,N: nat] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.61 = ( power_power_real @ X2 @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_realpow
% 0.35/0.61 thf(fact_4886_powr__less__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
% 0.35/0.61 = ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_less_iff
% 0.35/0.61 thf(fact_4887_less__powr__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
% 0.35/0.61 = ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_powr_iff
% 0.35/0.61 thf(fact_4888_log__less__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 )
% 0.35/0.61 = ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_less_iff
% 0.35/0.61 thf(fact_4889_less__log__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_log_iff
% 0.35/0.61 thf(fact_4890_div__nat__eqI,axiom,
% 0.35/0.61 ! [N: nat,Q3: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 0.35/0.61 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 0.35/0.61 => ( ( divide_divide_nat @ M @ N )
% 0.35/0.61 = Q3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % div_nat_eqI
% 0.35/0.61 thf(fact_4891_less__eq__div__iff__mult__less__eq,axiom,
% 0.35/0.61 ! [Q3: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 0.35/0.61 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 0.35/0.61 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_eq_div_iff_mult_less_eq
% 0.35/0.61 thf(fact_4892_dividend__less__times__div,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % dividend_less_times_div
% 0.35/0.61 thf(fact_4893_dividend__less__div__times,axiom,
% 0.35/0.61 ! [N: nat,M: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % dividend_less_div_times
% 0.35/0.61 thf(fact_4894_split__div,axiom,
% 0.35/0.61 ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.61 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.35/0.61 = ( ( ( N = zero_zero_nat )
% 0.35/0.61 => ( P @ zero_zero_nat ) )
% 0.35/0.61 & ( ( N != zero_zero_nat )
% 0.35/0.61 => ! [I4: nat,J3: nat] :
% 0.35/0.61 ( ( ord_less_nat @ J3 @ N )
% 0.35/0.61 => ( ( M
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.35/0.61 => ( P @ I4 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_div
% 0.35/0.61 thf(fact_4895_mult__eq__if,axiom,
% 0.35/0.61 ( times_times_nat
% 0.35/0.61 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_eq_if
% 0.35/0.61 thf(fact_4896_powr__neg__one,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.61 = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_neg_one
% 0.35/0.61 thf(fact_4897_powr__mult__base,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y3 ) )
% 0.35/0.61 = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_mult_base
% 0.35/0.61 thf(fact_4898_sgn__power__injE,axiom,
% 0.35/0.61 ! [A2: real,N: nat,X2: real,B3: real] :
% 0.35/0.61 ( ( ( times_times_real @ ( sgn_sgn_real @ A2 ) @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
% 0.35/0.61 = X2 )
% 0.35/0.61 => ( ( X2
% 0.35/0.61 = ( times_times_real @ ( sgn_sgn_real @ B3 ) @ ( power_power_real @ ( abs_abs_real @ B3 ) @ N ) ) )
% 0.35/0.61 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( A2 = B3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_power_injE
% 0.35/0.61 thf(fact_4899_le__log__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) )
% 0.35/0.61 = ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_log_iff
% 0.35/0.61 thf(fact_4900_log__le__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_le_iff
% 0.35/0.61 thf(fact_4901_le__powr__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
% 0.35/0.61 = ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_powr_iff
% 0.35/0.61 thf(fact_4902_powr__le__iff,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
% 0.35/0.61 = ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_le_iff
% 0.35/0.61 thf(fact_4903_split__div_H,axiom,
% 0.35/0.61 ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.61 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 0.35/0.61 = ( ( ( N = zero_zero_nat )
% 0.35/0.61 & ( P @ zero_zero_nat ) )
% 0.35/0.61 | ? [Q6: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M )
% 0.35/0.61 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
% 0.35/0.61 & ( P @ Q6 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_div'
% 0.35/0.61 thf(fact_4904_ln__powr__bound,axiom,
% 0.35/0.61 ! [X2: real,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ A2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ln_powr_bound
% 0.35/0.61 thf(fact_4905_ln__powr__bound2,axiom,
% 0.35/0.61 ! [X2: real,A2: real] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A2 ) @ ( times_times_real @ ( powr_real @ A2 @ A2 ) @ X2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ln_powr_bound2
% 0.35/0.61 thf(fact_4906_add__log__eq__powr,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( B3 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( plus_plus_real @ Y3 @ ( log @ B3 @ X2 ) )
% 0.35/0.61 = ( log @ B3 @ ( times_times_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % add_log_eq_powr
% 0.35/0.61 thf(fact_4907_log__add__eq__powr,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( B3 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( plus_plus_real @ ( log @ B3 @ X2 ) @ Y3 )
% 0.35/0.61 = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % log_add_eq_powr
% 0.35/0.61 thf(fact_4908_sgn__power__root,axiom,
% 0.35/0.61 ! [N: nat,X2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X2 ) ) @ N ) )
% 0.35/0.61 = X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_power_root
% 0.35/0.61 thf(fact_4909_root__sgn__power,axiom,
% 0.35/0.61 ! [N: nat,Y3: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 0.35/0.61 = Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % root_sgn_power
% 0.35/0.61 thf(fact_4910_minus__log__eq__powr,axiom,
% 0.35/0.61 ! [B3: real,X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ( B3 != one_one_real )
% 0.35/0.61 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( minus_minus_real @ Y3 @ ( log @ B3 @ X2 ) )
% 0.35/0.61 = ( log @ B3 @ ( divide_divide_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % minus_log_eq_powr
% 0.35/0.61 thf(fact_4911_powr__def,axiom,
% 0.35/0.61 ( powr_real
% 0.35/0.61 = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % powr_def
% 0.35/0.61 thf(fact_4912_nat__mult__le__cancel__disj,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_le_cancel_disj
% 0.35/0.61 thf(fact_4913_zero__le__sgn__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 0.35/0.61 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_le_sgn_iff
% 0.35/0.61 thf(fact_4914_sgn__le__0__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_le_0_iff
% 0.35/0.61 thf(fact_4915_nat__mult__div__cancel__disj,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ( K = zero_zero_nat )
% 0.35/0.61 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = zero_zero_nat ) )
% 0.35/0.61 & ( ( K != zero_zero_nat )
% 0.35/0.61 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_div_cancel_disj
% 0.35/0.61 thf(fact_4916_nat__mult__less__cancel__disj,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 & ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_less_cancel_disj
% 0.35/0.61 thf(fact_4917_mul__def,axiom,
% 0.35/0.61 ( vEBT_VEBT_mul
% 0.35/0.61 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.35/0.61
% 0.35/0.61 % mul_def
% 0.35/0.61 thf(fact_4918_sgn__zero,axiom,
% 0.35/0.61 ( ( sgn_sgn_complex @ zero_zero_complex )
% 0.35/0.61 = zero_zero_complex ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_zero
% 0.35/0.61 thf(fact_4919_sgn__zero,axiom,
% 0.35/0.61 ( ( sgn_sgn_real @ zero_zero_real )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_zero
% 0.35/0.61 thf(fact_4920_nat__less__add__iff2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_less_add_iff2
% 0.35/0.61 thf(fact_4921_mul__shift,axiom,
% 0.35/0.61 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ X2 @ Y3 )
% 0.35/0.61 = Z )
% 0.35/0.61 = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 0.35/0.61 = ( some_nat @ Z ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mul_shift
% 0.35/0.61 thf(fact_4922_zsgn__def,axiom,
% 0.35/0.61 ( sgn_sgn_int
% 0.35/0.61 = ( ^ [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 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % zsgn_def
% 0.35/0.61 thf(fact_4923_eucl__rel__int__remainderI,axiom,
% 0.35/0.61 ! [R2: int,L: int,K: int,Q3: int] :
% 0.35/0.61 ( ( ( sgn_sgn_int @ R2 )
% 0.35/0.61 = ( sgn_sgn_int @ L ) )
% 0.35/0.61 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.35/0.61 => ( ( K
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 0.35/0.61 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eucl_rel_int_remainderI
% 0.35/0.61 thf(fact_4924_nat__mult__eq__cancel__disj,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ( times_times_nat @ K @ M )
% 0.35/0.61 = ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ( K = zero_zero_nat )
% 0.35/0.61 | ( M = N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_eq_cancel_disj
% 0.35/0.61 thf(fact_4925_sgn__zero__iff,axiom,
% 0.35/0.61 ! [X2: complex] :
% 0.35/0.61 ( ( ( sgn_sgn_complex @ X2 )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 = ( X2 = zero_zero_complex ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_zero_iff
% 0.35/0.61 thf(fact_4926_sgn__zero__iff,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ( sgn_sgn_real @ X2 )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 = ( X2 = zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % sgn_zero_iff
% 0.35/0.61 thf(fact_4927_eucl__rel__int_Osimps,axiom,
% 0.35/0.61 ( eucl_rel_int
% 0.35/0.61 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 0.35/0.61 ( ? [K3: int] :
% 0.35/0.61 ( ( A12 = K3 )
% 0.35/0.61 & ( A23 = zero_zero_int )
% 0.35/0.61 & ( A32
% 0.35/0.61 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.35/0.61 | ? [L2: int,K3: int,Q6: int] :
% 0.35/0.61 ( ( A12 = K3 )
% 0.35/0.61 & ( A23 = L2 )
% 0.35/0.61 & ( A32
% 0.35/0.61 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 0.35/0.61 & ( L2 != zero_zero_int )
% 0.35/0.61 & ( K3
% 0.35/0.61 = ( times_times_int @ Q6 @ L2 ) ) )
% 0.35/0.61 | ? [R5: int,L2: int,K3: int,Q6: int] :
% 0.35/0.61 ( ( A12 = K3 )
% 0.35/0.61 & ( A23 = L2 )
% 0.35/0.61 & ( A32
% 0.35/0.61 = ( product_Pair_int_int @ Q6 @ R5 ) )
% 0.35/0.61 & ( ( sgn_sgn_int @ R5 )
% 0.35/0.61 = ( sgn_sgn_int @ L2 ) )
% 0.35/0.61 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.35/0.61 & ( K3
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eucl_rel_int.simps
% 0.35/0.61 thf(fact_4928_eucl__rel__int_Ocases,axiom,
% 0.35/0.61 ! [A13: int,A24: int,A33: product_prod_int_int] :
% 0.35/0.61 ( ( eucl_rel_int @ A13 @ A24 @ A33 )
% 0.35/0.61 => ( ( ( A24 = zero_zero_int )
% 0.35/0.61 => ( A33
% 0.35/0.61 != ( product_Pair_int_int @ zero_zero_int @ A13 ) ) )
% 0.35/0.61 => ( ! [Q4: int] :
% 0.35/0.61 ( ( A33
% 0.35/0.61 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 0.35/0.61 => ( ( A24 != zero_zero_int )
% 0.35/0.61 => ( A13
% 0.35/0.61 != ( times_times_int @ Q4 @ A24 ) ) ) )
% 0.35/0.61 => ~ ! [R3: int,Q4: int] :
% 0.35/0.61 ( ( A33
% 0.35/0.61 = ( product_Pair_int_int @ Q4 @ R3 ) )
% 0.35/0.61 => ( ( ( sgn_sgn_int @ R3 )
% 0.35/0.61 = ( sgn_sgn_int @ A24 ) )
% 0.35/0.61 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A24 ) )
% 0.35/0.61 => ( A13
% 0.35/0.61 != ( plus_plus_int @ ( times_times_int @ Q4 @ A24 ) @ R3 ) ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eucl_rel_int.cases
% 0.35/0.61 thf(fact_4929_nat__mult__less__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ord_less_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_less_cancel1
% 0.35/0.61 thf(fact_4930_nat__mult__eq__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( ( times_times_nat @ K @ M )
% 0.35/0.61 = ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( M = N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_eq_cancel1
% 0.35/0.61 thf(fact_4931_nat__mult__le__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_le_cancel1
% 0.35/0.61 thf(fact_4932_nat__mult__div__cancel1,axiom,
% 0.35/0.61 ! [K: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.61 = ( divide_divide_nat @ M @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_mult_div_cancel1
% 0.35/0.61 thf(fact_4933_nat__diff__add__eq2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_diff_add_eq2
% 0.35/0.61 thf(fact_4934_nat__diff__add__eq1,axiom,
% 0.35/0.61 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.61 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_diff_add_eq1
% 0.35/0.61 thf(fact_4935_nat__le__add__iff2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_le_add_iff2
% 0.35/0.61 thf(fact_4936_nat__le__add__iff1,axiom,
% 0.35/0.61 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.61 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_le_add_iff1
% 0.35/0.61 thf(fact_4937_nat__eq__add__iff2,axiom,
% 0.35/0.61 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.61 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( M
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_eq_add_iff2
% 0.35/0.61 thf(fact_4938_nat__eq__add__iff1,axiom,
% 0.35/0.61 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.61 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.35/0.61 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.35/0.61 = N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_eq_add_iff1
% 0.35/0.61 thf(fact_4939_nat__less__add__iff1,axiom,
% 0.35/0.61 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.61 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 0.35/0.61 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nat_less_add_iff1
% 0.35/0.61 thf(fact_4940_ceiling__log__eq__powr__iff,axiom,
% 0.35/0.61 ! [X2: real,B3: real,K: nat] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.61 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.61 => ( ( ( archim7802044766580827645g_real @ ( log @ B3 @ X2 ) )
% 0.35/0.61 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.35/0.61 = ( ( ord_less_real @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
% 0.35/0.61 & ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_log_eq_powr_iff
% 0.35/0.61 thf(fact_4941_bezw__0,axiom,
% 0.35/0.61 ! [X2: nat] :
% 0.35/0.61 ( ( bezw @ X2 @ zero_zero_nat )
% 0.35/0.61 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % bezw_0
% 0.35/0.61 thf(fact_4942_length__mul__elem,axiom,
% 0.35/0.61 ! [Xs: list_list_VEBT_VEBT,N: nat] :
% 0.35/0.61 ( ! [X5: list_VEBT_VEBT] :
% 0.35/0.61 ( ( member2936631157270082147T_VEBT @ X5 @ ( set_list_VEBT_VEBT2 @ Xs ) )
% 0.35/0.61 => ( ( size_s6755466524823107622T_VEBT @ X5 )
% 0.35/0.61 = N ) )
% 0.35/0.61 => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
% 0.35/0.61 = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % length_mul_elem
% 0.35/0.61 thf(fact_4943_length__mul__elem,axiom,
% 0.35/0.61 ! [Xs: list_list_o,N: nat] :
% 0.35/0.61 ( ! [X5: list_o] :
% 0.35/0.61 ( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
% 0.35/0.61 => ( ( size_size_list_o @ X5 )
% 0.35/0.61 = N ) )
% 0.35/0.61 => ( ( size_size_list_o @ ( concat_o @ Xs ) )
% 0.35/0.61 = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % length_mul_elem
% 0.35/0.61 thf(fact_4944_length__mul__elem,axiom,
% 0.35/0.61 ! [Xs: list_list_nat,N: nat] :
% 0.35/0.61 ( ! [X5: list_nat] :
% 0.35/0.61 ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
% 0.35/0.61 => ( ( size_size_list_nat @ X5 )
% 0.35/0.61 = N ) )
% 0.35/0.61 => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
% 0.35/0.61 = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % length_mul_elem
% 0.35/0.61 thf(fact_4945_gbinomial__absorption_H,axiom,
% 0.35/0.61 ! [K: nat,A2: rat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_rat @ A2 @ K )
% 0.35/0.61 = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_absorption'
% 0.35/0.61 thf(fact_4946_gbinomial__absorption_H,axiom,
% 0.35/0.61 ! [K: nat,A2: complex] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_complex @ A2 @ K )
% 0.35/0.61 = ( times_times_complex @ ( divide1717551699836669952omplex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_absorption'
% 0.35/0.61 thf(fact_4947_gbinomial__absorption_H,axiom,
% 0.35/0.61 ! [K: nat,A2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_real @ A2 @ K )
% 0.35/0.61 = ( times_times_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_absorption'
% 0.35/0.61 thf(fact_4948_nth__Cons__pos,axiom,
% 0.35/0.61 ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_pos
% 0.35/0.61 thf(fact_4949_nth__Cons__pos,axiom,
% 0.35/0.61 ! [N: nat,X2: int,Xs: list_int] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_pos
% 0.35/0.61 thf(fact_4950_nth__Cons__pos,axiom,
% 0.35/0.61 ! [N: nat,X2: nat,Xs: list_nat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.61 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_pos
% 0.35/0.61 thf(fact_4951_rotate1__length01,axiom,
% 0.35/0.61 ! [Xs: list_VEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
% 0.35/0.61 => ( ( rotate1_VEBT_VEBT @ Xs )
% 0.35/0.61 = Xs ) ) ).
% 0.35/0.61
% 0.35/0.61 % rotate1_length01
% 0.35/0.61 thf(fact_4952_rotate1__length01,axiom,
% 0.35/0.61 ! [Xs: list_o] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
% 0.35/0.61 => ( ( rotate1_o @ Xs )
% 0.35/0.61 = Xs ) ) ).
% 0.35/0.61
% 0.35/0.61 % rotate1_length01
% 0.35/0.61 thf(fact_4953_rotate1__length01,axiom,
% 0.35/0.61 ! [Xs: list_nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
% 0.35/0.61 => ( ( rotate1_nat @ Xs )
% 0.35/0.61 = Xs ) ) ).
% 0.35/0.61
% 0.35/0.61 % rotate1_length01
% 0.35/0.61 thf(fact_4954_remove__def,axiom,
% 0.35/0.61 ( remove6466555014256735590at_nat
% 0.35/0.61 = ( ^ [X: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % remove_def
% 0.35/0.61 thf(fact_4955_remove__def,axiom,
% 0.35/0.61 ( remove_real
% 0.35/0.61 = ( ^ [X: real,A6: set_real] : ( minus_minus_set_real @ A6 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % remove_def
% 0.35/0.61 thf(fact_4956_remove__def,axiom,
% 0.35/0.61 ( remove_o
% 0.35/0.61 = ( ^ [X: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % remove_def
% 0.35/0.61 thf(fact_4957_remove__def,axiom,
% 0.35/0.61 ( remove_int
% 0.35/0.61 = ( ^ [X: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % remove_def
% 0.35/0.61 thf(fact_4958_remove__def,axiom,
% 0.35/0.61 ( remove_nat
% 0.35/0.61 = ( ^ [X: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % remove_def
% 0.35/0.61 thf(fact_4959_member__remove,axiom,
% 0.35/0.61 ! [X2: real,Y3: real,A3: set_real] :
% 0.35/0.61 ( ( member_real @ X2 @ ( remove_real @ Y3 @ A3 ) )
% 0.35/0.61 = ( ( member_real @ X2 @ A3 )
% 0.35/0.61 & ( X2 != Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_remove
% 0.35/0.61 thf(fact_4960_member__remove,axiom,
% 0.35/0.61 ! [X2: $o,Y3: $o,A3: set_o] :
% 0.35/0.61 ( ( member_o @ X2 @ ( remove_o @ Y3 @ A3 ) )
% 0.35/0.61 = ( ( member_o @ X2 @ A3 )
% 0.35/0.61 & ( X2 != Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_remove
% 0.35/0.61 thf(fact_4961_member__remove,axiom,
% 0.35/0.61 ! [X2: set_nat,Y3: set_nat,A3: set_set_nat] :
% 0.35/0.61 ( ( member_set_nat @ X2 @ ( remove_set_nat @ Y3 @ A3 ) )
% 0.35/0.61 = ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.61 & ( X2 != Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_remove
% 0.35/0.61 thf(fact_4962_member__remove,axiom,
% 0.35/0.61 ! [X2: nat,Y3: nat,A3: set_nat] :
% 0.35/0.61 ( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A3 ) )
% 0.35/0.61 = ( ( member_nat @ X2 @ A3 )
% 0.35/0.61 & ( X2 != Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_remove
% 0.35/0.61 thf(fact_4963_member__remove,axiom,
% 0.35/0.61 ! [X2: int,Y3: int,A3: set_int] :
% 0.35/0.61 ( ( member_int @ X2 @ ( remove_int @ Y3 @ A3 ) )
% 0.35/0.61 = ( ( member_int @ X2 @ A3 )
% 0.35/0.61 & ( X2 != Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_remove
% 0.35/0.61 thf(fact_4964_nth__Cons__0,axiom,
% 0.35/0.61 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.61 ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat )
% 0.35/0.61 = X2 ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_0
% 0.35/0.61 thf(fact_4965_nth__Cons__0,axiom,
% 0.35/0.61 ! [X2: int,Xs: list_int] :
% 0.35/0.61 ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
% 0.35/0.61 = X2 ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_0
% 0.35/0.61 thf(fact_4966_nth__Cons__0,axiom,
% 0.35/0.61 ! [X2: nat,Xs: list_nat] :
% 0.35/0.61 ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
% 0.35/0.61 = X2 ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons_0
% 0.35/0.61 thf(fact_4967_gbinomial__0_I2_J,axiom,
% 0.35/0.61 ! [K: nat] :
% 0.35/0.61 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(2)
% 0.35/0.61 thf(fact_4968_gbinomial__0_I2_J,axiom,
% 0.35/0.61 ! [K: nat] :
% 0.35/0.61 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.35/0.61 = zero_zero_rat ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(2)
% 0.35/0.61 thf(fact_4969_gbinomial__0_I2_J,axiom,
% 0.35/0.61 ! [K: nat] :
% 0.35/0.61 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(2)
% 0.35/0.61 thf(fact_4970_gbinomial__0_I2_J,axiom,
% 0.35/0.61 ! [K: nat] :
% 0.35/0.61 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(2)
% 0.35/0.61 thf(fact_4971_ceiling__zero,axiom,
% 0.35/0.61 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_zero
% 0.35/0.61 thf(fact_4972_ceiling__zero,axiom,
% 0.35/0.61 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_zero
% 0.35/0.61 thf(fact_4973_gbinomial__0_I1_J,axiom,
% 0.35/0.61 ! [A2: complex] :
% 0.35/0.61 ( ( gbinomial_complex @ A2 @ zero_zero_nat )
% 0.35/0.61 = one_one_complex ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(1)
% 0.35/0.61 thf(fact_4974_gbinomial__0_I1_J,axiom,
% 0.35/0.61 ! [A2: real] :
% 0.35/0.61 ( ( gbinomial_real @ A2 @ zero_zero_nat )
% 0.35/0.61 = one_one_real ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(1)
% 0.35/0.61 thf(fact_4975_gbinomial__0_I1_J,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( gbinomial_rat @ A2 @ zero_zero_nat )
% 0.35/0.61 = one_one_rat ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(1)
% 0.35/0.61 thf(fact_4976_gbinomial__0_I1_J,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( gbinomial_nat @ A2 @ zero_zero_nat )
% 0.35/0.61 = one_one_nat ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(1)
% 0.35/0.61 thf(fact_4977_gbinomial__0_I1_J,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( gbinomial_int @ A2 @ zero_zero_nat )
% 0.35/0.61 = one_one_int ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_0(1)
% 0.35/0.61 thf(fact_4978_ceiling__le__zero,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_zero
% 0.35/0.61 thf(fact_4979_ceiling__le__zero,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_zero
% 0.35/0.61 thf(fact_4980_zero__less__ceiling,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.61 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_less_ceiling
% 0.35/0.61 thf(fact_4981_zero__less__ceiling,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_less_ceiling
% 0.35/0.61 thf(fact_4982_ceiling__less__one,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_one
% 0.35/0.61 thf(fact_4983_ceiling__less__one,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_one
% 0.35/0.61 thf(fact_4984_one__le__ceiling,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.61 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_le_ceiling
% 0.35/0.61 thf(fact_4985_one__le__ceiling,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_le_ceiling
% 0.35/0.61 thf(fact_4986_ceiling__le__one,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_one
% 0.35/0.61 thf(fact_4987_ceiling__le__one,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_one
% 0.35/0.61 thf(fact_4988_one__less__ceiling,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.61 = ( ord_less_rat @ one_one_rat @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_ceiling
% 0.35/0.61 thf(fact_4989_one__less__ceiling,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ one_one_real @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_ceiling
% 0.35/0.61 thf(fact_4990_ceiling__less__zero,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_zero
% 0.35/0.61 thf(fact_4991_ceiling__less__zero,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_zero
% 0.35/0.61 thf(fact_4992_zero__le__ceiling,axiom,
% 0.35/0.61 ! [X2: real] :
% 0.35/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_le_ceiling
% 0.35/0.61 thf(fact_4993_zero__le__ceiling,axiom,
% 0.35/0.61 ! [X2: rat] :
% 0.35/0.61 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.61 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % zero_le_ceiling
% 0.35/0.61 thf(fact_4994_ceiling__mono,axiom,
% 0.35/0.61 ! [Y3: real,X2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.35/0.61 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_mono
% 0.35/0.61 thf(fact_4995_ceiling__mono,axiom,
% 0.35/0.61 ! [Y3: rat,X2: rat] :
% 0.35/0.61 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.35/0.61 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_mono
% 0.35/0.61 thf(fact_4996_ceiling__less__cancel,axiom,
% 0.35/0.61 ! [X2: rat,Y3: rat] :
% 0.35/0.61 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 0.35/0.61 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_cancel
% 0.35/0.61 thf(fact_4997_ceiling__less__cancel,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] :
% 0.35/0.61 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 0.35/0.61 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_less_cancel
% 0.35/0.61 thf(fact_4998_gbinomial__of__nat__symmetric,axiom,
% 0.35/0.61 ! [K: nat,N: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.61 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 0.35/0.61 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_of_nat_symmetric
% 0.35/0.61 thf(fact_4999_set__subset__Cons,axiom,
% 0.35/0.61 ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X2 @ Xs ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % set_subset_Cons
% 0.35/0.61 thf(fact_5000_set__subset__Cons,axiom,
% 0.35/0.61 ! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % set_subset_Cons
% 0.35/0.61 thf(fact_5001_set__subset__Cons,axiom,
% 0.35/0.61 ! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % set_subset_Cons
% 0.35/0.61 thf(fact_5002_impossible__Cons,axiom,
% 0.35/0.61 ! [Xs: list_int,Ys2: list_int,X2: int] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
% 0.35/0.61 => ( Xs
% 0.35/0.61 != ( cons_int @ X2 @ Ys2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % impossible_Cons
% 0.35/0.61 thf(fact_5003_impossible__Cons,axiom,
% 0.35/0.61 ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 0.35/0.61 => ( Xs
% 0.35/0.61 != ( cons_VEBT_VEBT @ X2 @ Ys2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % impossible_Cons
% 0.35/0.61 thf(fact_5004_impossible__Cons,axiom,
% 0.35/0.61 ! [Xs: list_o,Ys2: list_o,X2: $o] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) )
% 0.35/0.61 => ( Xs
% 0.35/0.61 != ( cons_o @ X2 @ Ys2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % impossible_Cons
% 0.35/0.61 thf(fact_5005_impossible__Cons,axiom,
% 0.35/0.61 ! [Xs: list_nat,Ys2: list_nat,X2: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
% 0.35/0.61 => ( Xs
% 0.35/0.61 != ( cons_nat @ X2 @ Ys2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % impossible_Cons
% 0.35/0.61 thf(fact_5006_find_Osimps_I2_J,axiom,
% 0.35/0.61 ! [P: int > $o,X2: int,Xs: list_int] :
% 0.35/0.61 ( ( ( P @ X2 )
% 0.35/0.61 => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
% 0.35/0.61 = ( some_int @ X2 ) ) )
% 0.35/0.61 & ( ~ ( P @ X2 )
% 0.35/0.61 => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
% 0.35/0.61 = ( find_int @ P @ Xs ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % find.simps(2)
% 0.35/0.61 thf(fact_5007_find_Osimps_I2_J,axiom,
% 0.35/0.61 ! [P: nat > $o,X2: nat,Xs: list_nat] :
% 0.35/0.61 ( ( ( P @ X2 )
% 0.35/0.61 => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
% 0.35/0.61 = ( some_nat @ X2 ) ) )
% 0.35/0.61 & ( ~ ( P @ X2 )
% 0.35/0.61 => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
% 0.35/0.61 = ( find_nat @ P @ Xs ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % find.simps(2)
% 0.35/0.61 thf(fact_5008_find_Osimps_I2_J,axiom,
% 0.35/0.61 ! [P: product_prod_nat_nat > $o,X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.35/0.61 ( ( ( P @ X2 )
% 0.35/0.61 => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
% 0.35/0.61 = ( some_P7363390416028606310at_nat @ X2 ) ) )
% 0.35/0.61 & ( ~ ( P @ X2 )
% 0.35/0.61 => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
% 0.35/0.61 = ( find_P8199882355184865565at_nat @ P @ Xs ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % find.simps(2)
% 0.35/0.61 thf(fact_5009_find_Osimps_I2_J,axiom,
% 0.35/0.61 ! [P: num > $o,X2: num,Xs: list_num] :
% 0.35/0.61 ( ( ( P @ X2 )
% 0.35/0.61 => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
% 0.35/0.61 = ( some_num @ X2 ) ) )
% 0.35/0.61 & ( ~ ( P @ X2 )
% 0.35/0.61 => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
% 0.35/0.61 = ( find_num @ P @ Xs ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % find.simps(2)
% 0.35/0.61 thf(fact_5010_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.61 ! [K: nat,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A2 )
% 0.35/0.61 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A2 @ K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_ge_n_over_k_pow_k
% 0.35/0.61 thf(fact_5011_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.35/0.61 ! [K: nat,A2: rat] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A2 )
% 0.35/0.61 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A2 @ K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_ge_n_over_k_pow_k
% 0.35/0.61 thf(fact_5012_ceiling__add__le,axiom,
% 0.35/0.61 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_add_le
% 0.35/0.61 thf(fact_5013_ceiling__add__le,axiom,
% 0.35/0.61 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_add_le
% 0.35/0.61 thf(fact_5014_Suc__le__length__iff,axiom,
% 0.35/0.61 ! [N: nat,Xs: list_int] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
% 0.35/0.61 = ( ? [X: int,Ys3: list_int] :
% 0.35/0.61 ( ( Xs
% 0.35/0.61 = ( cons_int @ X @ Ys3 ) )
% 0.35/0.61 & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_le_length_iff
% 0.35/0.61 thf(fact_5015_Suc__le__length__iff,axiom,
% 0.35/0.61 ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.61 = ( ? [X: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.35/0.61 ( ( Xs
% 0.35/0.61 = ( cons_VEBT_VEBT @ X @ Ys3 ) )
% 0.35/0.61 & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_le_length_iff
% 0.35/0.61 thf(fact_5016_Suc__le__length__iff,axiom,
% 0.35/0.61 ! [N: nat,Xs: list_o] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
% 0.35/0.61 = ( ? [X: $o,Ys3: list_o] :
% 0.35/0.61 ( ( Xs
% 0.35/0.61 = ( cons_o @ X @ Ys3 ) )
% 0.35/0.61 & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_le_length_iff
% 0.35/0.61 thf(fact_5017_Suc__le__length__iff,axiom,
% 0.35/0.61 ! [N: nat,Xs: list_nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
% 0.35/0.61 = ( ? [X: nat,Ys3: list_nat] :
% 0.35/0.61 ( ( Xs
% 0.35/0.61 = ( cons_nat @ X @ Ys3 ) )
% 0.35/0.61 & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % Suc_le_length_iff
% 0.35/0.61 thf(fact_5018_gbinomial__trinomial__revision,axiom,
% 0.35/0.61 ! [K: nat,M: nat,A2: complex] :
% 0.35/0.61 ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.61 => ( ( times_times_complex @ ( gbinomial_complex @ A2 @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.35/0.61 = ( times_times_complex @ ( gbinomial_complex @ A2 @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_trinomial_revision
% 0.35/0.61 thf(fact_5019_gbinomial__trinomial__revision,axiom,
% 0.35/0.61 ! [K: nat,M: nat,A2: rat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.61 => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.35/0.61 = ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_trinomial_revision
% 0.35/0.61 thf(fact_5020_gbinomial__trinomial__revision,axiom,
% 0.35/0.61 ! [K: nat,M: nat,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.61 => ( ( times_times_real @ ( gbinomial_real @ A2 @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.35/0.61 = ( times_times_real @ ( gbinomial_real @ A2 @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_trinomial_revision
% 0.35/0.61 thf(fact_5021_list_Osize_I4_J,axiom,
% 0.35/0.61 ! [X21: int,X222: list_int] :
% 0.35/0.61 ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
% 0.35/0.61 = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % list.size(4)
% 0.35/0.61 thf(fact_5022_list_Osize_I4_J,axiom,
% 0.35/0.61 ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] :
% 0.35/0.61 ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X222 ) )
% 0.35/0.61 = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % list.size(4)
% 0.35/0.61 thf(fact_5023_list_Osize_I4_J,axiom,
% 0.35/0.61 ! [X21: $o,X222: list_o] :
% 0.35/0.61 ( ( size_size_list_o @ ( cons_o @ X21 @ X222 ) )
% 0.35/0.61 = ( plus_plus_nat @ ( size_size_list_o @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % list.size(4)
% 0.35/0.61 thf(fact_5024_list_Osize_I4_J,axiom,
% 0.35/0.61 ! [X21: nat,X222: list_nat] :
% 0.35/0.61 ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
% 0.35/0.61 = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % list.size(4)
% 0.35/0.61 thf(fact_5025_nth__Cons_H,axiom,
% 0.35/0.61 ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.35/0.61 ( ( ( N = zero_zero_nat )
% 0.35/0.61 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 ) )
% 0.35/0.61 & ( ( N != zero_zero_nat )
% 0.35/0.61 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons'
% 0.35/0.61 thf(fact_5026_nth__Cons_H,axiom,
% 0.35/0.61 ! [N: nat,X2: int,Xs: list_int] :
% 0.35/0.61 ( ( ( N = zero_zero_nat )
% 0.35/0.61 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 ) )
% 0.35/0.61 & ( ( N != zero_zero_nat )
% 0.35/0.61 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons'
% 0.35/0.61 thf(fact_5027_nth__Cons_H,axiom,
% 0.35/0.61 ! [N: nat,X2: nat,Xs: list_nat] :
% 0.35/0.61 ( ( ( N = zero_zero_nat )
% 0.35/0.61 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 ) )
% 0.35/0.61 & ( ( N != zero_zero_nat )
% 0.35/0.61 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_Cons'
% 0.35/0.61 thf(fact_5028_mult__ceiling__le,axiom,
% 0.35/0.61 ! [A2: real,B3: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 0.35/0.61 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_ceiling_le
% 0.35/0.61 thf(fact_5029_mult__ceiling__le,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat] :
% 0.35/0.61 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.61 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 0.35/0.61 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mult_ceiling_le
% 0.35/0.61 thf(fact_5030_gbinomial__reduce__nat,axiom,
% 0.35/0.61 ! [K: nat,A2: complex] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_complex @ A2 @ K )
% 0.35/0.61 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ K ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_reduce_nat
% 0.35/0.61 thf(fact_5031_gbinomial__reduce__nat,axiom,
% 0.35/0.61 ! [K: nat,A2: real] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_real @ A2 @ K )
% 0.35/0.61 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ K ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_reduce_nat
% 0.35/0.61 thf(fact_5032_gbinomial__reduce__nat,axiom,
% 0.35/0.61 ! [K: nat,A2: rat] :
% 0.35/0.61 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.61 => ( ( gbinomial_rat @ A2 @ K )
% 0.35/0.61 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % gbinomial_reduce_nat
% 0.35/0.61 thf(fact_5033_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: real,Xs: list_real,N: nat] :
% 0.35/0.61 ( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.35/0.61 => ( ( ( nth_real @ ( cons_real @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5034_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: set_nat,Xs: list_set_nat,N: nat] :
% 0.35/0.61 ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.61 => ( ( ( nth_set_nat @ ( cons_set_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5035_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: int,Xs: list_int,N: nat] :
% 0.35/0.61 ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.61 => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5036_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
% 0.35/0.61 ( ~ ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.61 => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5037_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: $o,Xs: list_o,N: nat] :
% 0.35/0.61 ( ~ ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.61 => ( ( ( nth_o @ ( cons_o @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5038_nth__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: nat,Xs: list_nat,N: nat] :
% 0.35/0.61 ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.61 => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = X2 )
% 0.35/0.61 = ( N = zero_zero_nat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_equal_first_eq
% 0.35/0.61 thf(fact_5039_nth__non__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
% 0.35/0.61 ( ( X2 != Y3 )
% 0.35/0.61 => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 0.35/0.61 = Y3 )
% 0.35/0.61 = ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.61 = Y3 )
% 0.35/0.61 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_non_equal_first_eq
% 0.35/0.61 thf(fact_5040_nth__non__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: int,Y3: int,Xs: list_int,N: nat] :
% 0.35/0.61 ( ( X2 != Y3 )
% 0.35/0.61 => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 0.35/0.61 = Y3 )
% 0.35/0.61 = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.61 = Y3 )
% 0.35/0.61 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_non_equal_first_eq
% 0.35/0.61 thf(fact_5041_nth__non__equal__first__eq,axiom,
% 0.35/0.61 ! [X2: nat,Y3: nat,Xs: list_nat,N: nat] :
% 0.35/0.61 ( ( X2 != Y3 )
% 0.35/0.61 => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 0.35/0.61 = Y3 )
% 0.35/0.61 = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 0.35/0.61 = Y3 )
% 0.35/0.61 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % nth_non_equal_first_eq
% 0.35/0.61 thf(fact_5042_upto__aux__rec,axiom,
% 0.35/0.61 ( upto_aux
% 0.35/0.61 = ( ^ [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 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % upto_aux_rec
% 0.35/0.61 thf(fact_5043_ceiling__eq,axiom,
% 0.35/0.61 ! [N: int,X2: real] :
% 0.35/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.61 => ( ( archim7802044766580827645g_real @ X2 )
% 0.35/0.61 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_eq
% 0.35/0.61 thf(fact_5044_ceiling__eq,axiom,
% 0.35/0.61 ! [N: int,X2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X2 )
% 0.35/0.61 => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 0.35/0.61 => ( ( archim2889992004027027881ng_rat @ X2 )
% 0.35/0.61 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_eq
% 0.35/0.61 thf(fact_5045_ceiling__divide__lower,axiom,
% 0.35/0.61 ! [Q3: rat,P6: rat] :
% 0.35/0.61 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.35/0.61 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_divide_lower
% 0.35/0.61 thf(fact_5046_ceiling__divide__lower,axiom,
% 0.35/0.61 ! [Q3: real,P6: real] :
% 0.35/0.61 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.35/0.61 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_divide_lower
% 0.35/0.61 thf(fact_5047__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.35/0.61 ~ ! [TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Info: option4927543243414619207at_nat] :
% 0.35/0.61 ~ ( ( sa
% 0.35/0.61 = ( vEBT_Node @ Info @ deg @ TreeList3 @ Summary3 ) )
% 0.35/0.61 & ( deg
% 0.35/0.61 = ( plus_plus_nat @ na @ m ) )
% 0.35/0.61 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.35/0.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.35/0.61 & ( vEBT_invar_vebt @ Summary3 @ m )
% 0.35/0.61 & ! [X4: vEBT_VEBT] :
% 0.35/0.61 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.35/0.61 => ( vEBT_invar_vebt @ X4 @ na ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.35/0.61 thf(fact_5048_split__pos__lemma,axiom,
% 0.35/0.61 ! [K: int,P: int > int > $o,N: int] :
% 0.35/0.61 ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.61 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.61 = ( ! [I4: int,J3: int] :
% 0.35/0.61 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.61 & ( ord_less_int @ J3 @ K )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.61 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_pos_lemma
% 0.35/0.61 thf(fact_5049_split__neg__lemma,axiom,
% 0.35/0.61 ! [K: int,P: int > int > $o,N: int] :
% 0.35/0.61 ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.61 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.61 = ( ! [I4: int,J3: int] :
% 0.35/0.61 ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.61 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.61 & ( N
% 0.35/0.61 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.61 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % split_neg_lemma
% 0.35/0.61 thf(fact_5050_verit__le__mono__div__int,axiom,
% 0.35/0.61 ! [A3: int,B2: int,N: int] :
% 0.35/0.61 ( ( ord_less_int @ A3 @ B2 )
% 0.35/0.61 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.61 => ( ord_less_eq_int
% 0.35/0.61 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 0.35/0.61 @ ( if_int
% 0.35/0.61 @ ( ( modulo_modulo_int @ B2 @ N )
% 0.35/0.61 = zero_zero_int )
% 0.35/0.61 @ one_one_int
% 0.35/0.61 @ zero_zero_int ) )
% 0.35/0.61 @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % verit_le_mono_div_int
% 0.35/0.61 thf(fact_5051_semiring__norm_I71_J,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(71)
% 0.35/0.61 thf(fact_5052_semiring__norm_I78_J,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(78)
% 0.35/0.61 thf(fact_5053_semiring__norm_I68_J,axiom,
% 0.35/0.61 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(68)
% 0.35/0.61 thf(fact_5054_semiring__norm_I75_J,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ~ ( ord_less_num @ M @ one ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(75)
% 0.35/0.61 thf(fact_5055_case4_I10_J,axiom,
% 0.35/0.61 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.35/0.61
% 0.35/0.61 % case4(10)
% 0.35/0.61 thf(fact_5056_case4_I4_J,axiom,
% 0.35/0.61 ( ( size_s6755466524823107622T_VEBT @ treeList2 )
% 0.35/0.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.35/0.61
% 0.35/0.61 % case4(4)
% 0.35/0.61 thf(fact_5057_pow__sum,axiom,
% 0.35/0.61 ! [A2: nat,B3: nat] :
% 0.35/0.61 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
% 0.35/0.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % pow_sum
% 0.35/0.61 thf(fact_5058_a0,axiom,
% 0.35/0.61 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 0.35/0.61
% 0.35/0.61 % a0
% 0.35/0.61 thf(fact_5059_case4_I7_J,axiom,
% 0.35/0.61 ! [I3: nat] :
% 0.35/0.61 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.35/0.61 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ X8 ) )
% 0.35/0.61 = ( vEBT_V8194947554948674370ptions @ summary2 @ I3 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % case4(7)
% 0.35/0.61 thf(fact_5060_power__minus__is__div,axiom,
% 0.35/0.61 ! [B3: nat,A2: nat] :
% 0.35/0.61 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.61 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A2 @ B3 ) )
% 0.35/0.61 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % power_minus_is_div
% 0.35/0.61 thf(fact_5061_member__bound,axiom,
% 0.35/0.61 ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
% 0.35/0.61 ( ( vEBT_vebt_member @ Tree @ X2 )
% 0.35/0.61 => ( ( vEBT_invar_vebt @ Tree @ N )
% 0.35/0.61 => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % member_bound
% 0.35/0.61 thf(fact_5062_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5063_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5064_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5065_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5066_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5067_numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.61 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_iff
% 0.35/0.61 thf(fact_5068_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5069_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5070_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5071_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5072_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5073_numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 0.35/0.61 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_iff
% 0.35/0.61 thf(fact_5074_semiring__norm_I69_J,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(69)
% 0.35/0.61 thf(fact_5075_semiring__norm_I76_J,axiom,
% 0.35/0.61 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % semiring_norm(76)
% 0.35/0.61 thf(fact_5076_bits__mod__0,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ zero_zero_int @ A2 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_0
% 0.35/0.61 thf(fact_5077_bits__mod__0,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_0
% 0.35/0.61 thf(fact_5078_mod__self,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ A2 @ A2 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_self
% 0.35/0.61 thf(fact_5079_mod__self,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ A2 @ A2 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_self
% 0.35/0.61 thf(fact_5080_mod__by__0,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ A2 @ zero_zero_int )
% 0.35/0.61 = A2 ) ).
% 0.35/0.61
% 0.35/0.61 % mod_by_0
% 0.35/0.61 thf(fact_5081_mod__by__0,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
% 0.35/0.61 = A2 ) ).
% 0.35/0.61
% 0.35/0.61 % mod_by_0
% 0.35/0.61 thf(fact_5082_mod__0,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ zero_zero_int @ A2 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_0
% 0.35/0.61 thf(fact_5083_mod__0,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_0
% 0.35/0.61 thf(fact_5084_misiz,axiom,
% 0.35/0.61 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 0.35/0.61 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.61 => ( ( ( some_nat @ M )
% 0.35/0.61 = ( vEBT_vebt_mint @ T ) )
% 0.35/0.61 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % misiz
% 0.35/0.61 thf(fact_5085_helpypredd,axiom,
% 0.35/0.61 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.35/0.61 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.61 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.35/0.61 = ( some_nat @ Y3 ) )
% 0.35/0.61 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % helpypredd
% 0.35/0.61 thf(fact_5086_helpyd,axiom,
% 0.35/0.61 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.35/0.61 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.61 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.35/0.61 = ( some_nat @ Y3 ) )
% 0.35/0.61 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % helpyd
% 0.35/0.61 thf(fact_5087_delt__out__of__range,axiom,
% 0.35/0.61 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61 ( ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.61 | ( ord_less_nat @ Ma @ X2 ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.61 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % delt_out_of_range
% 0.35/0.61 thf(fact_5088_del__single__cont,axiom,
% 0.35/0.61 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61 ( ( ( X2 = Mi )
% 0.35/0.61 & ( X2 = Ma ) )
% 0.35/0.61 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.61 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % del_single_cont
% 0.35/0.61 thf(fact_5089_set__n__deg__not__0,axiom,
% 0.35/0.61 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 0.35/0.61 ( ! [X5: vEBT_VEBT] :
% 0.35/0.61 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.61 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.35/0.61 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.61 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.61 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % set_n_deg_not_0
% 0.35/0.61 thf(fact_5090_mi__ma__2__deg,axiom,
% 0.35/0.61 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.35/0.61 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.61 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.35/0.61 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mi_ma_2_deg
% 0.35/0.61 thf(fact_5091_pred__max,axiom,
% 0.35/0.61 ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61 => ( ( ord_less_nat @ Ma @ X2 )
% 0.35/0.61 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.61 = ( some_nat @ Ma ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % pred_max
% 0.35/0.61 thf(fact_5092_succ__min,axiom,
% 0.35/0.61 ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.61 => ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.61 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.61 = ( some_nat @ Mi ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % succ_min
% 0.35/0.61 thf(fact_5093_bit__concat__def,axiom,
% 0.35/0.61 ( vEBT_VEBT_bit_concat
% 0.35/0.61 = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % bit_concat_def
% 0.35/0.61 thf(fact_5094_neg__numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.61 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_le_iff
% 0.35/0.61 thf(fact_5095_neg__numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.35/0.61 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_le_iff
% 0.35/0.61 thf(fact_5096_neg__numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.35/0.61 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_le_iff
% 0.35/0.61 thf(fact_5097_neg__numeral__le__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.61 = ( ord_less_eq_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_le_iff
% 0.35/0.61 thf(fact_5098_neg__numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.61 = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_iff
% 0.35/0.61 thf(fact_5099_neg__numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.61 = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_iff
% 0.35/0.61 thf(fact_5100_neg__numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 0.35/0.61 = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_iff
% 0.35/0.61 thf(fact_5101_neg__numeral__less__iff,axiom,
% 0.35/0.61 ! [M: num,N: num] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 0.35/0.61 = ( ord_less_num @ N @ M ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_iff
% 0.35/0.61 thf(fact_5102_power__zero__numeral,axiom,
% 0.35/0.61 ! [K: num] :
% 0.35/0.61 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.35/0.61 = zero_zero_rat ) ).
% 0.35/0.61
% 0.35/0.61 % power_zero_numeral
% 0.35/0.61 thf(fact_5103_power__zero__numeral,axiom,
% 0.35/0.61 ! [K: num] :
% 0.35/0.61 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % power_zero_numeral
% 0.35/0.61 thf(fact_5104_power__zero__numeral,axiom,
% 0.35/0.61 ! [K: num] :
% 0.35/0.61 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % power_zero_numeral
% 0.35/0.61 thf(fact_5105_power__zero__numeral,axiom,
% 0.35/0.61 ! [K: num] :
% 0.35/0.61 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % power_zero_numeral
% 0.35/0.61 thf(fact_5106_power__zero__numeral,axiom,
% 0.35/0.61 ! [K: num] :
% 0.35/0.61 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.35/0.61 = zero_zero_complex ) ).
% 0.35/0.61
% 0.35/0.61 % power_zero_numeral
% 0.35/0.61 thf(fact_5107_mod__mult__self1__is__0,axiom,
% 0.35/0.61 ! [B3: int,A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ ( times_times_int @ B3 @ A2 ) @ B3 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_mult_self1_is_0
% 0.35/0.61 thf(fact_5108_mod__mult__self1__is__0,axiom,
% 0.35/0.61 ! [B3: nat,A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ ( times_times_nat @ B3 @ A2 ) @ B3 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_mult_self1_is_0
% 0.35/0.61 thf(fact_5109_mod__mult__self2__is__0,axiom,
% 0.35/0.61 ! [A2: int,B3: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_mult_self2_is_0
% 0.35/0.61 thf(fact_5110_mod__mult__self2__is__0,axiom,
% 0.35/0.61 ! [A2: nat,B3: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_mult_self2_is_0
% 0.35/0.61 thf(fact_5111_bits__mod__by__1,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ A2 @ one_one_int )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_by_1
% 0.35/0.61 thf(fact_5112_bits__mod__by__1,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ A2 @ one_one_nat )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_by_1
% 0.35/0.61 thf(fact_5113_mod__by__1,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ A2 @ one_one_int )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_by_1
% 0.35/0.61 thf(fact_5114_mod__by__1,axiom,
% 0.35/0.61 ! [A2: nat] :
% 0.35/0.61 ( ( modulo_modulo_nat @ A2 @ one_one_nat )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_by_1
% 0.35/0.61 thf(fact_5115_bits__mod__div__trivial,axiom,
% 0.35/0.61 ! [A2: int,B3: int] :
% 0.35/0.61 ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_div_trivial
% 0.35/0.61 thf(fact_5116_bits__mod__div__trivial,axiom,
% 0.35/0.61 ! [A2: nat,B3: nat] :
% 0.35/0.61 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % bits_mod_div_trivial
% 0.35/0.61 thf(fact_5117_mod__div__trivial,axiom,
% 0.35/0.61 ! [A2: int,B3: int] :
% 0.35/0.61 ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_div_trivial
% 0.35/0.61 thf(fact_5118_mod__div__trivial,axiom,
% 0.35/0.61 ! [A2: nat,B3: nat] :
% 0.35/0.61 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
% 0.35/0.61 = zero_zero_nat ) ).
% 0.35/0.61
% 0.35/0.61 % mod_div_trivial
% 0.35/0.61 thf(fact_5119_of__int__eq__0__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( ( ring_1_of_int_int @ Z )
% 0.35/0.61 = zero_zero_int )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_eq_0_iff
% 0.35/0.61 thf(fact_5120_of__int__eq__0__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( ( ring_1_of_int_real @ Z )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_eq_0_iff
% 0.35/0.61 thf(fact_5121_of__int__eq__0__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( ( ring_1_of_int_rat @ Z )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_eq_0_iff
% 0.35/0.61 thf(fact_5122_of__int__0__eq__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( zero_zero_int
% 0.35/0.61 = ( ring_1_of_int_int @ Z ) )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0_eq_iff
% 0.35/0.61 thf(fact_5123_of__int__0__eq__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( zero_zero_real
% 0.35/0.61 = ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0_eq_iff
% 0.35/0.61 thf(fact_5124_of__int__0__eq__iff,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( zero_zero_rat
% 0.35/0.61 = ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = ( Z = zero_zero_int ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0_eq_iff
% 0.35/0.61 thf(fact_5125_of__int__0,axiom,
% 0.35/0.61 ( ( ring_1_of_int_int @ zero_zero_int )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0
% 0.35/0.61 thf(fact_5126_of__int__0,axiom,
% 0.35/0.61 ( ( ring_1_of_int_real @ zero_zero_int )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0
% 0.35/0.61 thf(fact_5127_of__int__0,axiom,
% 0.35/0.61 ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.35/0.61 = zero_zero_rat ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_0
% 0.35/0.61 thf(fact_5128_of__int__le__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_iff
% 0.35/0.61 thf(fact_5129_of__int__le__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_iff
% 0.35/0.61 thf(fact_5130_of__int__le__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_iff
% 0.35/0.61 thf(fact_5131_of__int__less__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = ( ord_less_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_iff
% 0.35/0.61 thf(fact_5132_of__int__less__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = ( ord_less_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_iff
% 0.35/0.61 thf(fact_5133_of__int__less__iff,axiom,
% 0.35/0.61 ! [W2: int,Z: int] :
% 0.35/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.61 = ( ord_less_int @ W2 @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_iff
% 0.35/0.61 thf(fact_5134_frac__of__int,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = zero_zero_real ) ).
% 0.35/0.61
% 0.35/0.61 % frac_of_int
% 0.35/0.61 thf(fact_5135_frac__of__int,axiom,
% 0.35/0.61 ! [Z: int] :
% 0.35/0.61 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = zero_zero_rat ) ).
% 0.35/0.61
% 0.35/0.61 % frac_of_int
% 0.35/0.61 thf(fact_5136_mintlistlength,axiom,
% 0.35/0.61 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.35/0.61 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.61 => ( ( Mi != Ma )
% 0.35/0.61 => ( ( ord_less_nat @ Mi @ Ma )
% 0.35/0.61 & ? [M4: nat] :
% 0.35/0.61 ( ( ( some_nat @ M4 )
% 0.35/0.61 = ( vEBT_vebt_mint @ Summary ) )
% 0.35/0.61 & ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mintlistlength
% 0.35/0.61 thf(fact_5137_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5138_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5139_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5140_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5141_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5142_numeral__le__one__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 0.35/0.61 = ( ord_less_eq_num @ N @ one ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_le_one_iff
% 0.35/0.61 thf(fact_5143_divide__le__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
% 0.35/0.61 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_le_eq_numeral1(1)
% 0.35/0.61 thf(fact_5144_divide__le__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
% 0.35/0.61 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_le_eq_numeral1(1)
% 0.35/0.61 thf(fact_5145_le__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.61 = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5146_le__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5147_eq__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.61 = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( numeral_numeral_rat @ W2 )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5148_eq__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.61 = ( ( ( ( numeral_numeral_real @ W2 )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( numeral_numeral_real @ W2 )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5149_eq__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: complex,B3: complex,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.61 = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.61 != zero_zero_complex )
% 0.35/0.61 => ( ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5150_divide__eq__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.61 & ( ( ( numeral_numeral_rat @ W2 )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(1)
% 0.35/0.61 thf(fact_5151_divide__eq__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( numeral_numeral_real @ W2 )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.61 & ( ( ( numeral_numeral_real @ W2 )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(1)
% 0.35/0.61 thf(fact_5152_divide__eq__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: complex,W2: num,A2: complex] :
% 0.35/0.61 ( ( ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.61 != zero_zero_complex )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.35/0.61 & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(1)
% 0.35/0.61 thf(fact_5153_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5154_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5155_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5156_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5157_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5158_one__less__numeral__iff,axiom,
% 0.35/0.61 ! [N: num] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 0.35/0.61 = ( ord_less_num @ one @ N ) ) ).
% 0.35/0.61
% 0.35/0.61 % one_less_numeral_iff
% 0.35/0.61 thf(fact_5159_less__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.61 = ( ord_less_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5160_less__divide__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.61 = ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_divide_eq_numeral1(1)
% 0.35/0.61 thf(fact_5161_divide__less__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
% 0.35/0.61 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_less_eq_numeral1(1)
% 0.35/0.61 thf(fact_5162_divide__less__eq__numeral1_I1_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
% 0.35/0.61 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_less_eq_numeral1(1)
% 0.35/0.61 thf(fact_5163_mod__minus1__right,axiom,
% 0.35/0.61 ! [A2: int] :
% 0.35/0.61 ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.61 = zero_zero_int ) ).
% 0.35/0.61
% 0.35/0.61 % mod_minus1_right
% 0.35/0.61 thf(fact_5164_mod__minus1__right,axiom,
% 0.35/0.61 ! [A2: code_integer] :
% 0.35/0.61 ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.61 = zero_z3403309356797280102nteger ) ).
% 0.35/0.61
% 0.35/0.61 % mod_minus1_right
% 0.35/0.61 thf(fact_5165_of__int__numeral__le__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_le_iff
% 0.35/0.61 thf(fact_5166_of__int__numeral__le__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_le_iff
% 0.35/0.61 thf(fact_5167_of__int__numeral__le__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_le_iff
% 0.35/0.61 thf(fact_5168_of__int__numeral__le__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.61 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_le_iff
% 0.35/0.61 thf(fact_5169_of__int__le__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_numeral_iff
% 0.35/0.61 thf(fact_5170_of__int__le__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 0.35/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_numeral_iff
% 0.35/0.61 thf(fact_5171_of__int__le__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_numeral_iff
% 0.35/0.61 thf(fact_5172_of__int__le__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.61 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_le_numeral_iff
% 0.35/0.61 thf(fact_5173_of__int__numeral__less__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_less_iff
% 0.35/0.61 thf(fact_5174_of__int__numeral__less__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_less_iff
% 0.35/0.61 thf(fact_5175_of__int__numeral__less__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_less_iff
% 0.35/0.61 thf(fact_5176_of__int__numeral__less__iff,axiom,
% 0.35/0.61 ! [N: num,Z: int] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 0.35/0.61 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_numeral_less_iff
% 0.35/0.61 thf(fact_5177_of__int__less__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 0.35/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_numeral_iff
% 0.35/0.61 thf(fact_5178_of__int__less__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 0.35/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_numeral_iff
% 0.35/0.61 thf(fact_5179_of__int__less__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_numeral_iff
% 0.35/0.61 thf(fact_5180_of__int__less__numeral__iff,axiom,
% 0.35/0.61 ! [Z: int,N: num] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 0.35/0.61 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % of_int_less_numeral_iff
% 0.35/0.61 thf(fact_5181_ceiling__le__numeral,axiom,
% 0.35/0.61 ! [X2: real,V: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.61 = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_numeral
% 0.35/0.61 thf(fact_5182_ceiling__le__numeral,axiom,
% 0.35/0.61 ! [X2: rat,V: num] :
% 0.35/0.61 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.61 = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % ceiling_le_numeral
% 0.35/0.61 thf(fact_5183_numeral__less__ceiling,axiom,
% 0.35/0.61 ! [V: num,X2: rat] :
% 0.35/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.61 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_ceiling
% 0.35/0.61 thf(fact_5184_numeral__less__ceiling,axiom,
% 0.35/0.61 ! [V: num,X2: real] :
% 0.35/0.61 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.61 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 0.35/0.61
% 0.35/0.61 % numeral_less_ceiling
% 0.35/0.61 thf(fact_5185_mod__neg__neg__trivial,axiom,
% 0.35/0.61 ! [K: int,L: int] :
% 0.35/0.61 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.35/0.61 => ( ( ord_less_int @ L @ K )
% 0.35/0.61 => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.61 = K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mod_neg_neg_trivial
% 0.35/0.61 thf(fact_5186_mod__pos__pos__trivial,axiom,
% 0.35/0.61 ! [K: int,L: int] :
% 0.35/0.61 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.61 => ( ( ord_less_int @ K @ L )
% 0.35/0.61 => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.61 = K ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % mod_pos_pos_trivial
% 0.35/0.61 thf(fact_5187_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % not_neg_one_le_neg_numeral_iff
% 0.35/0.61 thf(fact_5188_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % not_neg_one_le_neg_numeral_iff
% 0.35/0.61 thf(fact_5189_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % not_neg_one_le_neg_numeral_iff
% 0.35/0.61 thf(fact_5190_not__neg__one__le__neg__numeral__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % not_neg_one_le_neg_numeral_iff
% 0.35/0.61 thf(fact_5191_divide__le__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
% 0.35/0.61 = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_le_eq_numeral1(2)
% 0.35/0.61 thf(fact_5192_divide__le__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
% 0.35/0.61 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_le_eq_numeral1(2)
% 0.35/0.61 thf(fact_5193_le__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.61 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5194_le__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.61 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % le_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5195_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.61 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5196_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.61 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5197_eq__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: complex,B3: complex,W2: num] :
% 0.35/0.61 ( ( A2
% 0.35/0.61 = ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.35/0.61 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 != zero_zero_complex )
% 0.35/0.61 => ( ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.61 = B3 ) )
% 0.35/0.61 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % eq_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5198_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 != zero_zero_real )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 0.35/0.61 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.61 = zero_zero_real )
% 0.35/0.61 => ( A2 = zero_zero_real ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(2)
% 0.35/0.61 thf(fact_5199_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 != zero_zero_rat )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 0.35/0.61 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.61 = zero_zero_rat )
% 0.35/0.61 => ( A2 = zero_zero_rat ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(2)
% 0.35/0.61 thf(fact_5200_divide__eq__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: complex,W2: num,A2: complex] :
% 0.35/0.61 ( ( ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.61 = A2 )
% 0.35/0.61 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 != zero_zero_complex )
% 0.35/0.61 => ( B3
% 0.35/0.61 = ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 0.35/0.61 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.61 = zero_zero_complex )
% 0.35/0.61 => ( A2 = zero_zero_complex ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_eq_eq_numeral1(2)
% 0.35/0.61 thf(fact_5201_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_neg_one_iff
% 0.35/0.61 thf(fact_5202_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_neg_one_iff
% 0.35/0.61 thf(fact_5203_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_neg_one_iff
% 0.35/0.61 thf(fact_5204_neg__numeral__less__neg__one__iff,axiom,
% 0.35/0.61 ! [M: num] :
% 0.35/0.61 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.35/0.61 = ( M != one ) ) ).
% 0.35/0.61
% 0.35/0.61 % neg_numeral_less_neg_one_iff
% 0.35/0.61 thf(fact_5205_less__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: real,B3: real,W2: num] :
% 0.35/0.61 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.35/0.61 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5206_less__divide__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [A2: rat,B3: rat,W2: num] :
% 0.35/0.61 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.35/0.61 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.35/0.61
% 0.35/0.61 % less_divide_eq_numeral1(2)
% 0.35/0.61 thf(fact_5207_divide__less__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: real,W2: num,A2: real] :
% 0.35/0.61 ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
% 0.35/0.61 = ( ord_less_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_less_eq_numeral1(2)
% 0.35/0.61 thf(fact_5208_divide__less__eq__numeral1_I2_J,axiom,
% 0.35/0.61 ! [B3: rat,W2: num,A2: rat] :
% 0.35/0.61 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
% 0.35/0.61 = ( ord_less_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% 0.35/0.61
% 0.35/0.61 % divide_less_eq_numeral1(2)
% 0.35/0.61 thf(fact_5209_zero__eq__power2,axiom,
% 0.35/0.61 ! [A2: rat] :
% 0.35/0.61 ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_rat )
% 0.35/0.62 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_eq_power2
% 0.35/0.62 thf(fact_5210_zero__eq__power2,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_int )
% 0.35/0.62 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_eq_power2
% 0.35/0.62 thf(fact_5211_zero__eq__power2,axiom,
% 0.35/0.62 ! [A2: nat] :
% 0.35/0.62 ( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_nat )
% 0.35/0.62 = ( A2 = zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_eq_power2
% 0.35/0.62 thf(fact_5212_zero__eq__power2,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_real )
% 0.35/0.62 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_eq_power2
% 0.35/0.62 thf(fact_5213_zero__eq__power2,axiom,
% 0.35/0.62 ! [A2: complex] :
% 0.35/0.62 ( ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_complex )
% 0.35/0.62 = ( A2 = zero_zero_complex ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_eq_power2
% 0.35/0.62 thf(fact_5214_of__int__0__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_le_iff
% 0.35/0.62 thf(fact_5215_of__int__0__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_le_iff
% 0.35/0.62 thf(fact_5216_of__int__0__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_le_iff
% 0.35/0.62 thf(fact_5217_of__int__le__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_0_iff
% 0.35/0.62 thf(fact_5218_of__int__le__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_0_iff
% 0.35/0.62 thf(fact_5219_of__int__le__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_0_iff
% 0.35/0.62 thf(fact_5220_of__int__less__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 0.35/0.62 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_0_iff
% 0.35/0.62 thf(fact_5221_of__int__less__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 0.35/0.62 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_0_iff
% 0.35/0.62 thf(fact_5222_of__int__less__0__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 0.35/0.62 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_0_iff
% 0.35/0.62 thf(fact_5223_of__int__0__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.62 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_less_iff
% 0.35/0.62 thf(fact_5224_of__int__0__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.62 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_less_iff
% 0.35/0.62 thf(fact_5225_of__int__0__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.62 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_0_less_iff
% 0.35/0.62 thf(fact_5226_of__int__1__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_le_iff
% 0.35/0.62 thf(fact_5227_of__int__1__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_le_iff
% 0.35/0.62 thf(fact_5228_of__int__1__le__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.62 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_le_iff
% 0.35/0.62 thf(fact_5229_of__int__le__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_1_iff
% 0.35/0.62 thf(fact_5230_of__int__le__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_1_iff
% 0.35/0.62 thf(fact_5231_of__int__le__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.35/0.62 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_1_iff
% 0.35/0.62 thf(fact_5232_of__int__1__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.62 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_less_iff
% 0.35/0.62 thf(fact_5233_of__int__1__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.62 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_less_iff
% 0.35/0.62 thf(fact_5234_of__int__1__less__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 0.35/0.62 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_1_less_iff
% 0.35/0.62 thf(fact_5235_of__int__less__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 0.35/0.62 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_1_iff
% 0.35/0.62 thf(fact_5236_of__int__less__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 0.35/0.62 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_1_iff
% 0.35/0.62 thf(fact_5237_of__int__less__1__iff,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 0.35/0.62 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_1_iff
% 0.35/0.62 thf(fact_5238_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5239_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5240_of__int__le__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5241_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5242_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5243_of__int__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5244_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5245_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5246_of__int__less__of__int__power__cancel__iff,axiom,
% 0.35/0.62 ! [B3: int,W2: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_of_int_power_cancel_iff
% 0.35/0.62 thf(fact_5247_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5248_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5249_of__int__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: int,B3: int,W2: nat] :
% 0.35/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
% 0.35/0.62 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5250_one__div__two__eq__zero,axiom,
% 0.35/0.62 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % one_div_two_eq_zero
% 0.35/0.62 thf(fact_5251_one__div__two__eq__zero,axiom,
% 0.35/0.62 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % one_div_two_eq_zero
% 0.35/0.62 thf(fact_5252_one__div__two__eq__zero,axiom,
% 0.35/0.62 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % one_div_two_eq_zero
% 0.35/0.62 thf(fact_5253_bits__1__div__2,axiom,
% 0.35/0.62 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % bits_1_div_2
% 0.35/0.62 thf(fact_5254_bits__1__div__2,axiom,
% 0.35/0.62 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % bits_1_div_2
% 0.35/0.62 thf(fact_5255_bits__1__div__2,axiom,
% 0.35/0.62 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % bits_1_div_2
% 0.35/0.62 thf(fact_5256_power2__eq__iff__nonneg,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_iff_nonneg
% 0.35/0.62 thf(fact_5257_power2__eq__iff__nonneg,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_iff_nonneg
% 0.35/0.62 thf(fact_5258_power2__eq__iff__nonneg,axiom,
% 0.35/0.62 ! [X2: nat,Y3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.35/0.62 => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_iff_nonneg
% 0.35/0.62 thf(fact_5259_power2__eq__iff__nonneg,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.62 => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_iff_nonneg
% 0.35/0.62 thf(fact_5260_power2__less__eq__zero__iff,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.35/0.62 = ( A2 = zero_zero_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_eq_zero_iff
% 0.35/0.62 thf(fact_5261_power2__less__eq__zero__iff,axiom,
% 0.35/0.62 ! [A2: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.35/0.62 = ( A2 = zero_zero_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_eq_zero_iff
% 0.35/0.62 thf(fact_5262_power2__less__eq__zero__iff,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.35/0.62 = ( A2 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_eq_zero_iff
% 0.35/0.62 thf(fact_5263_zero__less__power2,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( A2 != zero_zero_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_power2
% 0.35/0.62 thf(fact_5264_zero__less__power2,axiom,
% 0.35/0.62 ! [A2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( A2 != zero_zero_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_power2
% 0.35/0.62 thf(fact_5265_zero__less__power2,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( A2 != zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_power2
% 0.35/0.62 thf(fact_5266_sum__power2__eq__zero__iff,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_zero_rat )
% 0.35/0.62 = ( ( X2 = zero_zero_rat )
% 0.35/0.62 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_eq_zero_iff
% 0.35/0.62 thf(fact_5267_sum__power2__eq__zero__iff,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_zero_int )
% 0.35/0.62 = ( ( X2 = zero_zero_int )
% 0.35/0.62 & ( Y3 = zero_zero_int ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_eq_zero_iff
% 0.35/0.62 thf(fact_5268_sum__power2__eq__zero__iff,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_zero_real )
% 0.35/0.62 = ( ( X2 = zero_zero_real )
% 0.35/0.62 & ( Y3 = zero_zero_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_eq_zero_iff
% 0.35/0.62 thf(fact_5269_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62 ! [A2: code_integer] :
% 0.35/0.62 ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 != zero_z3403309356797280102nteger )
% 0.35/0.62 = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 = one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_0_eq_1
% 0.35/0.62 thf(fact_5270_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 != zero_zero_int )
% 0.35/0.62 = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 = one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_0_eq_1
% 0.35/0.62 thf(fact_5271_not__mod__2__eq__0__eq__1,axiom,
% 0.35/0.62 ! [A2: nat] :
% 0.35/0.62 ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 != zero_zero_nat )
% 0.35/0.62 = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = one_one_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_0_eq_1
% 0.35/0.62 thf(fact_5272_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62 ! [A2: code_integer] :
% 0.35/0.62 ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 != one_one_Code_integer )
% 0.35/0.62 = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_1_eq_0
% 0.35/0.62 thf(fact_5273_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 != one_one_int )
% 0.35/0.62 = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_1_eq_0
% 0.35/0.62 thf(fact_5274_not__mod__2__eq__1__eq__0,axiom,
% 0.35/0.62 ! [A2: nat] :
% 0.35/0.62 ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 != one_one_nat )
% 0.35/0.62 = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_mod_2_eq_1_eq_0
% 0.35/0.62 thf(fact_5275_ceiling__less__numeral,axiom,
% 0.35/0.62 ! [X2: real,V: num] :
% 0.35/0.62 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.62 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_numeral
% 0.35/0.62 thf(fact_5276_ceiling__less__numeral,axiom,
% 0.35/0.62 ! [X2: rat,V: num] :
% 0.35/0.62 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.62 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_numeral
% 0.35/0.62 thf(fact_5277_numeral__le__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_le_ceiling
% 0.35/0.62 thf(fact_5278_numeral__le__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: real] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.62 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_le_ceiling
% 0.35/0.62 thf(fact_5279_ceiling__le__neg__numeral,axiom,
% 0.35/0.62 ! [X2: real,V: num] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.62 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le_neg_numeral
% 0.35/0.62 thf(fact_5280_ceiling__le__neg__numeral,axiom,
% 0.35/0.62 ! [X2: rat,V: num] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.62 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le_neg_numeral
% 0.35/0.62 thf(fact_5281_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5282_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5283_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5284_numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5285_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5286_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5287_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5288_of__int__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5289_neg__numeral__less__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: real] :
% 0.35/0.62 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.62 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_ceiling
% 0.35/0.62 thf(fact_5290_neg__numeral__less__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: rat] :
% 0.35/0.62 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_ceiling
% 0.35/0.62 thf(fact_5291_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5292_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5293_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5294_numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5295_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5296_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5297_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5298_of__int__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5299_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5300_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5301_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5302_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5303_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5304_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5305_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.35/0.62 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5306_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
% 0.35/0.62 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5307_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.35/0.62 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5308_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.35/0.62 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_less_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5309_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5310_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5311_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5312_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5313_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.35/0.62 ! [I: num,N: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 0.35/0.62 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_power_le_of_nat_cancel_iff
% 0.35/0.62 thf(fact_5314_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5315_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5316_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5317_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5318_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [X2: nat,I: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_le_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5319_ceiling__less__neg__numeral,axiom,
% 0.35/0.62 ! [X2: real,V: num] :
% 0.35/0.62 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.62 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_neg_numeral
% 0.35/0.62 thf(fact_5320_ceiling__less__neg__numeral,axiom,
% 0.35/0.62 ! [X2: rat,V: num] :
% 0.35/0.62 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.62 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_neg_numeral
% 0.35/0.62 thf(fact_5321_neg__numeral__le__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: real] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.62 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_ceiling
% 0.35/0.62 thf(fact_5322_neg__numeral__le__ceiling,axiom,
% 0.35/0.62 ! [V: num,X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_ceiling
% 0.35/0.62 thf(fact_5323_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5324_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5325_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5326_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_le_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5327_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5328_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5329_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5330_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_le_of_int_cancel_iff
% 0.35/0.62 thf(fact_5331_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5332_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5333_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5334_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.35/0.62 ! [A2: int,X2: num,N: nat] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 0.35/0.62 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_less_neg_numeral_power_cancel_iff
% 0.35/0.62 thf(fact_5335_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5336_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5337_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5338_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.35/0.62 ! [X2: num,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 0.35/0.62 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_power_less_of_int_cancel_iff
% 0.35/0.62 thf(fact_5339_sprop1,axiom,
% 0.35/0.62 ( ( sa
% 0.35/0.62 = ( vEBT_Node @ info @ deg @ treeList @ summary ) )
% 0.35/0.62 & ( deg
% 0.35/0.62 = ( plus_plus_nat @ na @ m ) )
% 0.35/0.62 & ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.35/0.62 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.35/0.62 & ( vEBT_invar_vebt @ summary @ m )
% 0.35/0.62 & ! [X4: vEBT_VEBT] :
% 0.35/0.62 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.35/0.62 => ( vEBT_invar_vebt @ X4 @ na ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sprop1
% 0.35/0.62 thf(fact_5340_divmod__digit__0_I2_J,axiom,
% 0.35/0.62 ! [B3: code_integer,A2: code_integer] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.62 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
% 0.35/0.62 = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(2)
% 0.35/0.62 thf(fact_5341_divmod__digit__0_I2_J,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
% 0.35/0.62 = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(2)
% 0.35/0.62 thf(fact_5342_divmod__digit__0_I2_J,axiom,
% 0.35/0.62 ! [B3: nat,A2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
% 0.35/0.62 = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(2)
% 0.35/0.62 thf(fact_5343_pos2,axiom,
% 0.35/0.62 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.35/0.62
% 0.35/0.62 % pos2
% 0.35/0.62 thf(fact_5344_zero__power2,axiom,
% 0.35/0.62 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % zero_power2
% 0.35/0.62 thf(fact_5345_zero__power2,axiom,
% 0.35/0.62 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % zero_power2
% 0.35/0.62 thf(fact_5346_zero__power2,axiom,
% 0.35/0.62 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % zero_power2
% 0.35/0.62 thf(fact_5347_zero__power2,axiom,
% 0.35/0.62 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % zero_power2
% 0.35/0.62 thf(fact_5348_zero__power2,axiom,
% 0.35/0.62 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = zero_zero_complex ) ).
% 0.35/0.62
% 0.35/0.62 % zero_power2
% 0.35/0.62 thf(fact_5349_numeral__2__eq__2,axiom,
% 0.35/0.62 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.35/0.62 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_2_eq_2
% 0.35/0.62 thf(fact_5350_less__exp,axiom,
% 0.35/0.62 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_exp
% 0.35/0.62 thf(fact_5351_self__le__ge2__pow,axiom,
% 0.35/0.62 ! [K: nat,M: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.62 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % self_le_ge2_pow
% 0.35/0.62 thf(fact_5352_power2__nat__le__eq__le,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_nat_le_eq_le
% 0.35/0.62 thf(fact_5353_power2__nat__le__imp__le,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 0.35/0.62 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_nat_le_imp_le
% 0.35/0.62 thf(fact_5354_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62 ! [N: num,Q3: num] :
% 0.35/0.62 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.35/0.62 = zero_z3403309356797280102nteger )
% 0.35/0.62 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(2)
% 0.35/0.62 thf(fact_5355_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62 ! [N: num,Q3: num] :
% 0.35/0.62 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.35/0.62 = zero_zero_int )
% 0.35/0.62 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.35/0.62 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(2)
% 0.35/0.62 thf(fact_5356_cong__exp__iff__simps_I2_J,axiom,
% 0.35/0.62 ! [N: num,Q3: num] :
% 0.35/0.62 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.35/0.62 = zero_zero_nat )
% 0.35/0.62 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.35/0.62 = zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(2)
% 0.35/0.62 thf(fact_5357_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(1)
% 0.35/0.62 thf(fact_5358_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 0.35/0.62 = zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(1)
% 0.35/0.62 thf(fact_5359_cong__exp__iff__simps_I1_J,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 0.35/0.62 = zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % cong_exp_iff_simps(1)
% 0.35/0.62 thf(fact_5360_card__2__iff_H,axiom,
% 0.35/0.62 ! [S: set_complex] :
% 0.35/0.62 ( ( ( finite_card_complex @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: complex] :
% 0.35/0.62 ( ( member_complex @ X @ S )
% 0.35/0.62 & ? [Y: complex] :
% 0.35/0.62 ( ( member_complex @ Y @ S )
% 0.35/0.62 & ( X != Y )
% 0.35/0.62 & ! [Z2: complex] :
% 0.35/0.62 ( ( member_complex @ Z2 @ S )
% 0.35/0.62 => ( ( Z2 = X )
% 0.35/0.62 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff'
% 0.35/0.62 thf(fact_5361_card__2__iff_H,axiom,
% 0.35/0.62 ! [S: set_list_nat] :
% 0.35/0.62 ( ( ( finite_card_list_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: list_nat] :
% 0.35/0.62 ( ( member_list_nat @ X @ S )
% 0.35/0.62 & ? [Y: list_nat] :
% 0.35/0.62 ( ( member_list_nat @ Y @ S )
% 0.35/0.62 & ( X != Y )
% 0.35/0.62 & ! [Z2: list_nat] :
% 0.35/0.62 ( ( member_list_nat @ Z2 @ S )
% 0.35/0.62 => ( ( Z2 = X )
% 0.35/0.62 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff'
% 0.35/0.62 thf(fact_5362_card__2__iff_H,axiom,
% 0.35/0.62 ! [S: set_set_nat] :
% 0.35/0.62 ( ( ( finite_card_set_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: set_nat] :
% 0.35/0.62 ( ( member_set_nat @ X @ S )
% 0.35/0.62 & ? [Y: set_nat] :
% 0.35/0.62 ( ( member_set_nat @ Y @ S )
% 0.35/0.62 & ( X != Y )
% 0.35/0.62 & ! [Z2: set_nat] :
% 0.35/0.62 ( ( member_set_nat @ Z2 @ S )
% 0.35/0.62 => ( ( Z2 = X )
% 0.35/0.62 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff'
% 0.35/0.62 thf(fact_5363_card__2__iff_H,axiom,
% 0.35/0.62 ! [S: set_nat] :
% 0.35/0.62 ( ( ( finite_card_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: nat] :
% 0.35/0.62 ( ( member_nat @ X @ S )
% 0.35/0.62 & ? [Y: nat] :
% 0.35/0.62 ( ( member_nat @ Y @ S )
% 0.35/0.62 & ( X != Y )
% 0.35/0.62 & ! [Z2: nat] :
% 0.35/0.62 ( ( member_nat @ Z2 @ S )
% 0.35/0.62 => ( ( Z2 = X )
% 0.35/0.62 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff'
% 0.35/0.62 thf(fact_5364_card__2__iff_H,axiom,
% 0.35/0.62 ! [S: set_int] :
% 0.35/0.62 ( ( ( finite_card_int @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: int] :
% 0.35/0.62 ( ( member_int @ X @ S )
% 0.35/0.62 & ? [Y: int] :
% 0.35/0.62 ( ( member_int @ Y @ S )
% 0.35/0.62 & ( X != Y )
% 0.35/0.62 & ! [Z2: int] :
% 0.35/0.62 ( ( member_int @ Z2 @ S )
% 0.35/0.62 => ( ( Z2 = X )
% 0.35/0.62 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff'
% 0.35/0.62 thf(fact_5365_le__num__One__iff,axiom,
% 0.35/0.62 ! [X2: num] :
% 0.35/0.62 ( ( ord_less_eq_num @ X2 @ one )
% 0.35/0.62 = ( X2 = one ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_num_One_iff
% 0.35/0.62 thf(fact_5366_bits__stable__imp__add__self,axiom,
% 0.35/0.62 ! [A2: code_integer] :
% 0.35/0.62 ( ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.62 = A2 )
% 0.35/0.62 => ( ( plus_p5714425477246183910nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % bits_stable_imp_add_self
% 0.35/0.62 thf(fact_5367_bits__stable__imp__add__self,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ( ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.62 = A2 )
% 0.35/0.62 => ( ( plus_plus_int @ A2 @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % bits_stable_imp_add_self
% 0.35/0.62 thf(fact_5368_bits__stable__imp__add__self,axiom,
% 0.35/0.62 ! [A2: nat] :
% 0.35/0.62 ( ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = A2 )
% 0.35/0.62 => ( ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % bits_stable_imp_add_self
% 0.35/0.62 thf(fact_5369_divmod__digit__0_I1_J,axiom,
% 0.35/0.62 ! [B3: code_integer,A2: code_integer] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.62 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(1)
% 0.35/0.62 thf(fact_5370_divmod__digit__0_I1_J,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(1)
% 0.35/0.62 thf(fact_5371_divmod__digit__0_I1_J,axiom,
% 0.35/0.62 ! [B3: nat,A2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_0(1)
% 0.35/0.62 thf(fact_5372_sum__squares__bound,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_squares_bound
% 0.35/0.62 thf(fact_5373_sum__squares__bound,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_squares_bound
% 0.35/0.62 thf(fact_5374_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62 ! [M: nat,N: nat,A2: code_integer] :
% 0.35/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_exp_mod_exp_eq
% 0.35/0.62 thf(fact_5375_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62 ! [M: nat,N: nat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62 => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62 = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_exp_mod_exp_eq
% 0.35/0.62 thf(fact_5376_mult__exp__mod__exp__eq,axiom,
% 0.35/0.62 ! [M: nat,N: nat,A2: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.62 => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.62 = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_exp_mod_exp_eq
% 0.35/0.62 thf(fact_5377_half__gt__zero,axiom,
% 0.35/0.62 ! [A2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 0.35/0.62 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % half_gt_zero
% 0.35/0.62 thf(fact_5378_half__gt__zero,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.62 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % half_gt_zero
% 0.35/0.62 thf(fact_5379_half__gt__zero__iff,axiom,
% 0.35/0.62 ! [A2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % half_gt_zero_iff
% 0.35/0.62 thf(fact_5380_half__gt__zero__iff,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % half_gt_zero_iff
% 0.35/0.62 thf(fact_5381_field__less__half__sum,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_rat @ X2 @ Y3 )
% 0.35/0.62 => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % field_less_half_sum
% 0.35/0.62 thf(fact_5382_field__less__half__sum,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.62 => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % field_less_half_sum
% 0.35/0.62 thf(fact_5383_power2__le__imp__le,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_imp_le
% 0.35/0.62 thf(fact_5384_power2__le__imp__le,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_imp_le
% 0.35/0.62 thf(fact_5385_power2__le__imp__le,axiom,
% 0.35/0.62 ! [X2: nat,Y3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.35/0.62 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_imp_le
% 0.35/0.62 thf(fact_5386_power2__le__imp__le,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.62 => ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_imp_le
% 0.35/0.62 thf(fact_5387_power2__eq__imp__eq,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_imp_eq
% 0.35/0.62 thf(fact_5388_power2__eq__imp__eq,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_imp_eq
% 0.35/0.62 thf(fact_5389_power2__eq__imp__eq,axiom,
% 0.35/0.62 ! [X2: nat,Y3: nat] :
% 0.35/0.62 ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.35/0.62 => ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_imp_eq
% 0.35/0.62 thf(fact_5390_power2__eq__imp__eq,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.62 => ( X2 = Y3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_eq_imp_eq
% 0.35/0.62 thf(fact_5391_zero__le__power2,axiom,
% 0.35/0.62 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_power2
% 0.35/0.62 thf(fact_5392_zero__le__power2,axiom,
% 0.35/0.62 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_power2
% 0.35/0.62 thf(fact_5393_zero__le__power2,axiom,
% 0.35/0.62 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_power2
% 0.35/0.62 thf(fact_5394_power2__less__0,axiom,
% 0.35/0.62 ! [A2: real] :
% 0.35/0.62 ~ ( ord_less_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_0
% 0.35/0.62 thf(fact_5395_power2__less__0,axiom,
% 0.35/0.62 ! [A2: rat] :
% 0.35/0.62 ~ ( ord_less_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_0
% 0.35/0.62 thf(fact_5396_power2__less__0,axiom,
% 0.35/0.62 ! [A2: int] :
% 0.35/0.62 ~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_0
% 0.35/0.62 thf(fact_5397_of__nat__less__two__power,axiom,
% 0.35/0.62 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_two_power
% 0.35/0.62 thf(fact_5398_of__nat__less__two__power,axiom,
% 0.35/0.62 ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_two_power
% 0.35/0.62 thf(fact_5399_of__nat__less__two__power,axiom,
% 0.35/0.62 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_two_power
% 0.35/0.62 thf(fact_5400_of__nat__less__two__power,axiom,
% 0.35/0.62 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_two_power
% 0.35/0.62 thf(fact_5401_exp__add__not__zero__imp__right,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_zero_nat )
% 0.35/0.62 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_right
% 0.35/0.62 thf(fact_5402_exp__add__not__zero__imp__right,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_zero_int )
% 0.35/0.62 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_right
% 0.35/0.62 thf(fact_5403_exp__add__not__zero__imp__right,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_z3403309356797280102nteger )
% 0.35/0.62 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_right
% 0.35/0.62 thf(fact_5404_exp__add__not__zero__imp__left,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_zero_nat )
% 0.35/0.62 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.35/0.62 != zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_left
% 0.35/0.62 thf(fact_5405_exp__add__not__zero__imp__left,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_zero_int )
% 0.35/0.62 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.35/0.62 != zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_left
% 0.35/0.62 thf(fact_5406_exp__add__not__zero__imp__left,axiom,
% 0.35/0.62 ! [M: nat,N: nat] :
% 0.35/0.62 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.62 != zero_z3403309356797280102nteger )
% 0.35/0.62 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.35/0.62 != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_add_not_zero_imp_left
% 0.35/0.62 thf(fact_5407_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.35/0.62 ! [N: nat,M: nat] :
% 0.35/0.62 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_zero_nat )
% 0.35/0.62 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.62 != zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_not_zero_imp_exp_diff_not_zero
% 0.35/0.62 thf(fact_5408_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.35/0.62 ! [N: nat,M: nat] :
% 0.35/0.62 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_zero_int )
% 0.35/0.62 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.62 != zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_not_zero_imp_exp_diff_not_zero
% 0.35/0.62 thf(fact_5409_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.35/0.62 ! [N: nat,M: nat] :
% 0.35/0.62 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 0.35/0.62 != zero_z3403309356797280102nteger )
% 0.35/0.62 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.62 != zero_z3403309356797280102nteger ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_not_zero_imp_exp_diff_not_zero
% 0.35/0.62 thf(fact_5410_abs__le__square__iff,axiom,
% 0.35/0.62 ! [X2: code_integer,Y3: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 0.35/0.62 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_le_square_iff
% 0.35/0.62 thf(fact_5411_abs__le__square__iff,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) )
% 0.35/0.62 = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_le_square_iff
% 0.35/0.62 thf(fact_5412_abs__le__square__iff,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y3 ) )
% 0.35/0.62 = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_le_square_iff
% 0.35/0.62 thf(fact_5413_abs__le__square__iff,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y3 ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_le_square_iff
% 0.35/0.62 thf(fact_5414_less__2__cases,axiom,
% 0.35/0.62 ! [N: nat] :
% 0.35/0.62 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 => ( ( N = zero_zero_nat )
% 0.35/0.62 | ( N
% 0.35/0.62 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_2_cases
% 0.35/0.62 thf(fact_5415_less__2__cases__iff,axiom,
% 0.35/0.62 ! [N: nat] :
% 0.35/0.62 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ( N = zero_zero_nat )
% 0.35/0.62 | ( N
% 0.35/0.62 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_2_cases_iff
% 0.35/0.62 thf(fact_5416_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_Pr1261947904930325089at_nat] :
% 0.35/0.62 ( ( ( finite711546835091564841at_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5417_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_complex] :
% 0.35/0.62 ( ( ( finite_card_complex @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: complex,Y: complex] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_complex @ X @ ( insert_complex @ Y @ bot_bot_set_complex ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5418_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_list_nat] :
% 0.35/0.62 ( ( ( finite_card_list_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: list_nat,Y: list_nat] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5419_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_set_nat] :
% 0.35/0.62 ( ( ( finite_card_set_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: set_nat,Y: set_nat] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5420_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_real] :
% 0.35/0.62 ( ( ( finite_card_real @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: real,Y: real] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_real @ X @ ( insert_real @ Y @ bot_bot_set_real ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5421_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_o] :
% 0.35/0.62 ( ( ( finite_card_o @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: $o,Y: $o] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5422_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_nat] :
% 0.35/0.62 ( ( ( finite_card_nat @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: nat,Y: nat] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5423_card__2__iff,axiom,
% 0.35/0.62 ! [S: set_int] :
% 0.35/0.62 ( ( ( finite_card_int @ S )
% 0.35/0.62 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( ? [X: int,Y: int] :
% 0.35/0.62 ( ( S
% 0.35/0.62 = ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 0.35/0.62 & ( X != Y ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % card_2_iff
% 0.35/0.62 thf(fact_5424_nat__induct2,axiom,
% 0.35/0.62 ! [P: nat > $o,N: nat] :
% 0.35/0.62 ( ( P @ zero_zero_nat )
% 0.35/0.62 => ( ( P @ one_one_nat )
% 0.35/0.62 => ( ! [N3: nat] :
% 0.35/0.62 ( ( P @ N3 )
% 0.35/0.62 => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.62 => ( P @ N ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % nat_induct2
% 0.35/0.62 thf(fact_5425_diff__le__diff__pow,axiom,
% 0.35/0.62 ! [K: nat,M: nat,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.62 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % diff_le_diff_pow
% 0.35/0.62 thf(fact_5426_realpow__square__minus__le,axiom,
% 0.35/0.62 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % realpow_square_minus_le
% 0.35/0.62 thf(fact_5427_numeral__1__eq__Suc__0,axiom,
% 0.35/0.62 ( ( numeral_numeral_nat @ one )
% 0.35/0.62 = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % numeral_1_eq_Suc_0
% 0.35/0.62 thf(fact_5428_ex__le__of__int,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_le_of_int
% 0.35/0.62 thf(fact_5429_ex__le__of__int,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_le_of_int
% 0.35/0.62 thf(fact_5430_ex__less__of__int,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_less_of_int
% 0.35/0.62 thf(fact_5431_ex__less__of__int,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_less_of_int
% 0.35/0.62 thf(fact_5432_ex__of__int__less,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).
% 0.35/0.62
% 0.35/0.62 % ex_of_int_less
% 0.35/0.62 thf(fact_5433_ex__of__int__less,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 ) ).
% 0.35/0.62
% 0.35/0.62 % ex_of_int_less
% 0.35/0.62 thf(fact_5434_mod__double__modulus,axiom,
% 0.35/0.62 ! [M: code_integer,X2: code_integer] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.35/0.62 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 0.35/0.62 => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( modulo364778990260209775nteger @ X2 @ M ) )
% 0.35/0.62 | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_double_modulus
% 0.35/0.62 thf(fact_5435_mod__double__modulus,axiom,
% 0.35/0.62 ! [M: nat,X2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 0.35/0.62 => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( modulo_modulo_nat @ X2 @ M ) )
% 0.35/0.62 | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_double_modulus
% 0.35/0.62 thf(fact_5436_mod__double__modulus,axiom,
% 0.35/0.62 ! [M: int,X2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.62 => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( modulo_modulo_int @ X2 @ M ) )
% 0.35/0.62 | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_double_modulus
% 0.35/0.62 thf(fact_5437_divmod__digit__1_I2_J,axiom,
% 0.35/0.62 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.62 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.62 => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(2)
% 0.35/0.62 thf(fact_5438_divmod__digit__1_I2_J,axiom,
% 0.35/0.62 ! [A2: nat,B3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.62 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(2)
% 0.35/0.62 thf(fact_5439_divmod__digit__1_I2_J,axiom,
% 0.35/0.62 ! [A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.62 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( minus_minus_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 0.35/0.62 = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(2)
% 0.35/0.62 thf(fact_5440_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_rat
% 0.35/0.62 != ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5441_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_nat
% 0.35/0.62 != ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5442_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_real
% 0.35/0.62 != ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5443_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_int
% 0.35/0.62 != ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5444_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_z5237406670263579293d_enat
% 0.35/0.62 != ( numera1916890842035813515d_enat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5445_zero__neq__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_z3403309356797280102nteger
% 0.35/0.62 != ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_numeral
% 0.35/0.62 thf(fact_5446_power2__less__imp__less,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_imp_less
% 0.35/0.62 thf(fact_5447_power2__less__imp__less,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_imp_less
% 0.35/0.62 thf(fact_5448_power2__less__imp__less,axiom,
% 0.35/0.62 ! [X2: nat,Y3: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.35/0.62 => ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_imp_less
% 0.35/0.62 thf(fact_5449_power2__less__imp__less,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.62 => ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_less_imp_less
% 0.35/0.62 thf(fact_5450_sum__power2__ge__zero,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_ge_zero
% 0.35/0.62 thf(fact_5451_sum__power2__ge__zero,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_ge_zero
% 0.35/0.62 thf(fact_5452_sum__power2__ge__zero,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_ge_zero
% 0.35/0.62 thf(fact_5453_sum__power2__le__zero__iff,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.35/0.62 = ( ( X2 = zero_zero_real )
% 0.35/0.62 & ( Y3 = zero_zero_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_le_zero_iff
% 0.35/0.62 thf(fact_5454_sum__power2__le__zero__iff,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.35/0.62 = ( ( X2 = zero_zero_rat )
% 0.35/0.62 & ( Y3 = zero_zero_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_le_zero_iff
% 0.35/0.62 thf(fact_5455_sum__power2__le__zero__iff,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.35/0.62 = ( ( X2 = zero_zero_int )
% 0.35/0.62 & ( Y3 = zero_zero_int ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_le_zero_iff
% 0.35/0.62 thf(fact_5456_sum__power2__gt__zero__iff,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.62 = ( ( X2 != zero_zero_real )
% 0.35/0.62 | ( Y3 != zero_zero_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_gt_zero_iff
% 0.35/0.62 thf(fact_5457_sum__power2__gt__zero__iff,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.62 = ( ( X2 != zero_zero_rat )
% 0.35/0.62 | ( Y3 != zero_zero_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_gt_zero_iff
% 0.35/0.62 thf(fact_5458_sum__power2__gt__zero__iff,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.62 = ( ( X2 != zero_zero_int )
% 0.35/0.62 | ( Y3 != zero_zero_int ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % sum_power2_gt_zero_iff
% 0.35/0.62 thf(fact_5459_not__sum__power2__lt__zero,axiom,
% 0.35/0.62 ! [X2: real,Y3: real] :
% 0.35/0.62 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % not_sum_power2_lt_zero
% 0.35/0.62 thf(fact_5460_not__sum__power2__lt__zero,axiom,
% 0.35/0.62 ! [X2: rat,Y3: rat] :
% 0.35/0.62 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % not_sum_power2_lt_zero
% 0.35/0.62 thf(fact_5461_not__sum__power2__lt__zero,axiom,
% 0.35/0.62 ! [X2: int,Y3: int] :
% 0.35/0.62 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % not_sum_power2_lt_zero
% 0.35/0.62 thf(fact_5462_num_Osize_I4_J,axiom,
% 0.35/0.62 ( ( size_size_num @ one )
% 0.35/0.62 = zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % num.size(4)
% 0.35/0.62 thf(fact_5463_square__le__1,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % square_le_1
% 0.35/0.62 thf(fact_5464_square__le__1,axiom,
% 0.35/0.62 ! [X2: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
% 0.35/0.62 => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
% 0.35/0.62 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % square_le_1
% 0.35/0.62 thf(fact_5465_square__le__1,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % square_le_1
% 0.35/0.62 thf(fact_5466_square__le__1,axiom,
% 0.35/0.62 ! [X2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_int @ X2 @ one_one_int )
% 0.35/0.62 => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % square_le_1
% 0.35/0.62 thf(fact_5467_power2__le__iff__abs__le,axiom,
% 0.35/0.62 ! [Y3: code_integer,X2: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 0.35/0.62 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_iff_abs_le
% 0.35/0.62 thf(fact_5468_power2__le__iff__abs__le,axiom,
% 0.35/0.62 ! [Y3: real,X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_iff_abs_le
% 0.35/0.62 thf(fact_5469_power2__le__iff__abs__le,axiom,
% 0.35/0.62 ! [Y3: rat,X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_iff_abs_le
% 0.35/0.62 thf(fact_5470_power2__le__iff__abs__le,axiom,
% 0.35/0.62 ! [Y3: int,X2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.62 => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % power2_le_iff_abs_le
% 0.35/0.62 thf(fact_5471_zero__le__even__power_H,axiom,
% 0.35/0.62 ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_even_power'
% 0.35/0.62 thf(fact_5472_zero__le__even__power_H,axiom,
% 0.35/0.62 ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_even_power'
% 0.35/0.62 thf(fact_5473_zero__le__even__power_H,axiom,
% 0.35/0.62 ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_even_power'
% 0.35/0.62 thf(fact_5474_abs__square__le__1,axiom,
% 0.35/0.62 ! [X2: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.35/0.62 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_le_1
% 0.35/0.62 thf(fact_5475_abs__square__le__1,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.35/0.62 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_le_1
% 0.35/0.62 thf(fact_5476_abs__square__le__1,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.35/0.62 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_le_1
% 0.35/0.62 thf(fact_5477_abs__square__le__1,axiom,
% 0.35/0.62 ! [X2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.35/0.62 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_le_1
% 0.35/0.62 thf(fact_5478_abs__square__less__1,axiom,
% 0.35/0.62 ! [X2: code_integer] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.35/0.62 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_less_1
% 0.35/0.62 thf(fact_5479_abs__square__less__1,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.35/0.62 = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_less_1
% 0.35/0.62 thf(fact_5480_abs__square__less__1,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.35/0.62 = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_less_1
% 0.35/0.62 thf(fact_5481_abs__square__less__1,axiom,
% 0.35/0.62 ! [X2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.35/0.62 = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_square_less_1
% 0.35/0.62 thf(fact_5482_nat__bit__induct,axiom,
% 0.35/0.62 ! [P: nat > $o,N: nat] :
% 0.35/0.62 ( ( P @ zero_zero_nat )
% 0.35/0.62 => ( ! [N3: nat] :
% 0.35/0.62 ( ( P @ N3 )
% 0.35/0.62 => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.62 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.35/0.62 => ( ! [N3: nat] :
% 0.35/0.62 ( ( P @ N3 )
% 0.35/0.62 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.35/0.62 => ( P @ N ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % nat_bit_induct
% 0.35/0.62 thf(fact_5483_div__2__gt__zero,axiom,
% 0.35/0.62 ! [N: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.62 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % div_2_gt_zero
% 0.35/0.62 thf(fact_5484_Suc__n__div__2__gt__zero,axiom,
% 0.35/0.62 ! [N: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.62 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % Suc_n_div_2_gt_zero
% 0.35/0.62 thf(fact_5485_arith__geo__mean,axiom,
% 0.35/0.62 ! [U: real,X2: real,Y3: real] :
% 0.35/0.62 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( times_times_real @ X2 @ Y3 ) )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.62 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % arith_geo_mean
% 0.35/0.62 thf(fact_5486_arith__geo__mean,axiom,
% 0.35/0.62 ! [U: rat,X2: rat,Y3: rat] :
% 0.35/0.62 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.62 = ( times_times_rat @ X2 @ Y3 ) )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.35/0.62 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % arith_geo_mean
% 0.35/0.62 thf(fact_5487_divmod__digit__1_I1_J,axiom,
% 0.35/0.62 ! [A2: code_integer,B3: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 0.35/0.62 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 0.35/0.62 => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_Code_integer )
% 0.35/0.62 = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(1)
% 0.35/0.62 thf(fact_5488_divmod__digit__1_I1_J,axiom,
% 0.35/0.62 ! [A2: nat,B3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.62 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_nat )
% 0.35/0.62 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(1)
% 0.35/0.62 thf(fact_5489_divmod__digit__1_I1_J,axiom,
% 0.35/0.62 ! [A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.62 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 0.35/0.62 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_int )
% 0.35/0.62 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_digit_1(1)
% 0.35/0.62 thf(fact_5490_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.62 ! [A2: real,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.62 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_0_le_power_imp_0_le
% 0.35/0.62 thf(fact_5491_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.62 ! [A2: rat,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_0_le_power_imp_0_le
% 0.35/0.62 thf(fact_5492_odd__0__le__power__imp__0__le,axiom,
% 0.35/0.62 ! [A2: int,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.62 => ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_0_le_power_imp_0_le
% 0.35/0.62 thf(fact_5493_odd__power__less__zero,axiom,
% 0.35/0.62 ! [A2: real,N: nat] :
% 0.35/0.62 ( ( ord_less_real @ A2 @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_power_less_zero
% 0.35/0.62 thf(fact_5494_odd__power__less__zero,axiom,
% 0.35/0.62 ! [A2: rat,N: nat] :
% 0.35/0.62 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_power_less_zero
% 0.35/0.62 thf(fact_5495_odd__power__less__zero,axiom,
% 0.35/0.62 ! [A2: int,N: nat] :
% 0.35/0.62 ( ( ord_less_int @ A2 @ zero_zero_int )
% 0.35/0.62 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % odd_power_less_zero
% 0.35/0.62 thf(fact_5496_ex__power__ivl2,axiom,
% 0.35/0.62 ! [B3: nat,K: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.62 => ? [N3: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
% 0.35/0.62 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_power_ivl2
% 0.35/0.62 thf(fact_5497_ex__power__ivl1,axiom,
% 0.35/0.62 ! [B3: nat,K: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.35/0.62 => ? [N3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
% 0.35/0.62 & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ex_power_ivl1
% 0.35/0.62 thf(fact_5498_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.35/0.62 ! [A2: nat,B3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.62 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.35/0.62 thf(fact_5499_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.35/0.62 ! [A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.62 => ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.35/0.62 thf(fact_5500_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.35/0.62 thf(fact_5501_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.35/0.62 ! [B3: nat,A2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.35/0.62 thf(fact_5502_mod__eq__self__iff__div__eq__0,axiom,
% 0.35/0.62 ! [A2: int,B3: int] :
% 0.35/0.62 ( ( ( modulo_modulo_int @ A2 @ B3 )
% 0.35/0.62 = A2 )
% 0.35/0.62 = ( ( divide_divide_int @ A2 @ B3 )
% 0.35/0.62 = zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_eq_self_iff_div_eq_0
% 0.35/0.62 thf(fact_5503_mod__eq__self__iff__div__eq__0,axiom,
% 0.35/0.62 ! [A2: nat,B3: nat] :
% 0.35/0.62 ( ( ( modulo_modulo_nat @ A2 @ B3 )
% 0.35/0.62 = A2 )
% 0.35/0.62 = ( ( divide_divide_nat @ A2 @ B3 )
% 0.35/0.62 = zero_zero_nat ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_eq_self_iff_div_eq_0
% 0.35/0.62 thf(fact_5504_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5505_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5506_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5507_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5508_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5509_zero__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_le_numeral
% 0.35/0.62 thf(fact_5510_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5511_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5512_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5513_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5514_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5515_not__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_zero
% 0.35/0.62 thf(fact_5516_exp__bound,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % exp_bound
% 0.35/0.62 thf(fact_5517_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5518_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5519_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5520_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5521_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5522_not__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_zero
% 0.35/0.62 thf(fact_5523_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5524_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5525_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5526_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5527_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5528_zero__less__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_less_numeral
% 0.35/0.62 thf(fact_5529_le__of__int__ceiling,axiom,
% 0.35/0.62 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_of_int_ceiling
% 0.35/0.62 thf(fact_5530_le__of__int__ceiling,axiom,
% 0.35/0.62 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_of_int_ceiling
% 0.35/0.62 thf(fact_5531_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5532_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5533_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5534_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5535_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5536_one__le__numeral,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % one_le_numeral
% 0.35/0.62 thf(fact_5537_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5538_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5539_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5540_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5541_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5542_not__numeral__less__one,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_one
% 0.35/0.62 thf(fact_5543_neg__numeral__le__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_numeral
% 0.35/0.62 thf(fact_5544_neg__numeral__le__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_numeral
% 0.35/0.62 thf(fact_5545_neg__numeral__le__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_numeral
% 0.35/0.62 thf(fact_5546_neg__numeral__le__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_numeral
% 0.35/0.62 thf(fact_5547_not__numeral__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_numeral
% 0.35/0.62 thf(fact_5548_not__numeral__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_numeral
% 0.35/0.62 thf(fact_5549_not__numeral__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_numeral
% 0.35/0.62 thf(fact_5550_not__numeral__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_numeral
% 0.35/0.62 thf(fact_5551_zero__neq__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_int
% 0.35/0.62 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_neg_numeral
% 0.35/0.62 thf(fact_5552_zero__neq__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_real
% 0.35/0.62 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_neg_numeral
% 0.35/0.62 thf(fact_5553_zero__neq__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_rat
% 0.35/0.62 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_neg_numeral
% 0.35/0.62 thf(fact_5554_zero__neq__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_z3403309356797280102nteger
% 0.35/0.62 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_neg_numeral
% 0.35/0.62 thf(fact_5555_zero__neq__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ( zero_zero_complex
% 0.35/0.62 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zero_neq_neg_numeral
% 0.35/0.62 thf(fact_5556_zmod__le__nonneg__dividend,axiom,
% 0.35/0.62 ! [M: int,K: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.35/0.62 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % zmod_le_nonneg_dividend
% 0.35/0.62 thf(fact_5557_not__numeral__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_numeral
% 0.35/0.62 thf(fact_5558_not__numeral__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_numeral
% 0.35/0.62 thf(fact_5559_not__numeral__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_numeral
% 0.35/0.62 thf(fact_5560_not__numeral__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_numeral
% 0.35/0.62 thf(fact_5561_neg__numeral__less__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_numeral
% 0.35/0.62 thf(fact_5562_neg__numeral__less__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_numeral
% 0.35/0.62 thf(fact_5563_neg__numeral__less__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_numeral
% 0.35/0.62 thf(fact_5564_neg__numeral__less__numeral,axiom,
% 0.35/0.62 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_numeral
% 0.35/0.62 thf(fact_5565_Euclidean__Division_Opos__mod__bound,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.35/0.62
% 0.35/0.62 % Euclidean_Division.pos_mod_bound
% 0.35/0.62 thf(fact_5566_neg__mod__bound,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_mod_bound
% 0.35/0.62 thf(fact_5567_ln__one__plus__pos__lower__bound,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ln_one_plus_pos_lower_bound
% 0.35/0.62 thf(fact_5568_invar__vebt_Ointros_I2_J,axiom,
% 0.35/0.62 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.35/0.62 ( ! [X5: vEBT_VEBT] :
% 0.35/0.62 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.62 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.35/0.62 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.35/0.62 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.62 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 => ( ( M = N )
% 0.35/0.62 => ( ( Deg
% 0.35/0.62 = ( plus_plus_nat @ N @ M ) )
% 0.35/0.62 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.35/0.62 => ( ! [X5: vEBT_VEBT] :
% 0.35/0.62 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.62 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 0.35/0.62 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % invar_vebt.intros(2)
% 0.35/0.62 thf(fact_5569_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_ln_one_plus_x_minus_x_bound_nonneg
% 0.35/0.62 thf(fact_5570_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.35/0.62 ! [B3: nat,A2: nat] :
% 0.35/0.62 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 0.35/0.62 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.35/0.62 thf(fact_5571_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.35/0.62 thf(fact_5572_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.35/0.62 ! [A2: nat,B3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 0.35/0.62 => ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_nat @ A2 @ B3 )
% 0.35/0.62 = A2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_less
% 0.35/0.62 thf(fact_5573_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.35/0.62 ! [A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.62 => ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ B3 )
% 0.35/0.62 = A2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_less
% 0.35/0.62 thf(fact_5574_invar__vebt_Ointros_I3_J,axiom,
% 0.35/0.62 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.35/0.62 ( ! [X5: vEBT_VEBT] :
% 0.35/0.62 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.62 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.35/0.62 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.35/0.62 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.62 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.62 => ( ( M
% 0.35/0.62 = ( suc @ N ) )
% 0.35/0.62 => ( ( Deg
% 0.35/0.62 = ( plus_plus_nat @ N @ M ) )
% 0.35/0.62 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.35/0.62 => ( ! [X5: vEBT_VEBT] :
% 0.35/0.62 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.62 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 0.35/0.62 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % invar_vebt.intros(3)
% 0.35/0.62 thf(fact_5575_neg__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_zero
% 0.35/0.62 thf(fact_5576_neg__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_zero
% 0.35/0.62 thf(fact_5577_neg__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_zero
% 0.35/0.62 thf(fact_5578_neg__numeral__le__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_zero
% 0.35/0.62 thf(fact_5579_not__zero__le__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_le_neg_numeral
% 0.35/0.62 thf(fact_5580_not__zero__le__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_le_neg_numeral
% 0.35/0.62 thf(fact_5581_not__zero__le__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_le_neg_numeral
% 0.35/0.62 thf(fact_5582_not__zero__le__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_le_neg_numeral
% 0.35/0.62 thf(fact_5583_not__zero__less__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_less_neg_numeral
% 0.35/0.62 thf(fact_5584_not__zero__less__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_less_neg_numeral
% 0.35/0.62 thf(fact_5585_not__zero__less__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_less_neg_numeral
% 0.35/0.62 thf(fact_5586_not__zero__less__neg__numeral,axiom,
% 0.35/0.62 ! [N: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_zero_less_neg_numeral
% 0.35/0.62 thf(fact_5587_neg__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_zero
% 0.35/0.62 thf(fact_5588_neg__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_zero
% 0.35/0.62 thf(fact_5589_neg__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_zero
% 0.35/0.62 thf(fact_5590_neg__numeral__less__zero,axiom,
% 0.35/0.62 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_zero
% 0.35/0.62 thf(fact_5591_ceiling__le,axiom,
% 0.35/0.62 ! [X2: real,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) )
% 0.35/0.62 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le
% 0.35/0.62 thf(fact_5592_ceiling__le,axiom,
% 0.35/0.62 ! [X2: rat,A2: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) )
% 0.35/0.62 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le
% 0.35/0.62 thf(fact_5593_ceiling__le__iff,axiom,
% 0.35/0.62 ! [X2: real,Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 0.35/0.62 = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le_iff
% 0.35/0.62 thf(fact_5594_ceiling__le__iff,axiom,
% 0.35/0.62 ! [X2: rat,Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 0.35/0.62 = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_le_iff
% 0.35/0.62 thf(fact_5595_eq__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ( numeral_numeral_rat @ W2 )
% 0.35/0.62 = ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_rat )
% 0.35/0.62 => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_rat )
% 0.35/0.62 => ( ( numeral_numeral_rat @ W2 )
% 0.35/0.62 = zero_zero_rat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5596_eq__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ( numeral_numeral_real @ W2 )
% 0.35/0.62 = ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_real )
% 0.35/0.62 => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_real )
% 0.35/0.62 => ( ( numeral_numeral_real @ W2 )
% 0.35/0.62 = zero_zero_real ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5597_eq__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: complex,C: complex] :
% 0.35/0.62 ( ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.62 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_complex )
% 0.35/0.62 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_complex )
% 0.35/0.62 => ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.62 = zero_zero_complex ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5598_divide__eq__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ( divide_divide_rat @ B3 @ C )
% 0.35/0.62 = ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = ( ( ( C != zero_zero_rat )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_rat )
% 0.35/0.62 => ( ( numeral_numeral_rat @ W2 )
% 0.35/0.62 = zero_zero_rat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(1)
% 0.35/0.62 thf(fact_5599_divide__eq__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ( divide_divide_real @ B3 @ C )
% 0.35/0.62 = ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = ( ( ( C != zero_zero_real )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_real )
% 0.35/0.62 => ( ( numeral_numeral_real @ W2 )
% 0.35/0.62 = zero_zero_real ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(1)
% 0.35/0.62 thf(fact_5600_divide__eq__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: complex,C: complex,W2: num] :
% 0.35/0.62 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 0.35/0.62 = ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.62 = ( ( ( C != zero_zero_complex )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_complex )
% 0.35/0.62 => ( ( numera6690914467698888265omplex @ W2 )
% 0.35/0.62 = zero_zero_complex ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(1)
% 0.35/0.62 thf(fact_5601_less__ceiling__iff,axiom,
% 0.35/0.62 ! [Z: int,X2: rat] :
% 0.35/0.62 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_ceiling_iff
% 0.35/0.62 thf(fact_5602_less__ceiling__iff,axiom,
% 0.35/0.62 ! [Z: int,X2: real] :
% 0.35/0.62 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.62 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_ceiling_iff
% 0.35/0.62 thf(fact_5603_not__one__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_le_neg_numeral
% 0.35/0.62 thf(fact_5604_not__one__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_le_neg_numeral
% 0.35/0.62 thf(fact_5605_not__one__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_le_neg_numeral
% 0.35/0.62 thf(fact_5606_not__one__le__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_le_neg_numeral
% 0.35/0.62 thf(fact_5607_not__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_one
% 0.35/0.62 thf(fact_5608_not__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_one
% 0.35/0.62 thf(fact_5609_not__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_one
% 0.35/0.62 thf(fact_5610_not__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_le_neg_one
% 0.35/0.62 thf(fact_5611_neg__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_neg_one
% 0.35/0.62 thf(fact_5612_neg__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_neg_one
% 0.35/0.62 thf(fact_5613_neg__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_neg_one
% 0.35/0.62 thf(fact_5614_neg__numeral__le__neg__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_neg_one
% 0.35/0.62 thf(fact_5615_neg__one__le__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_le_numeral
% 0.35/0.62 thf(fact_5616_neg__one__le__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_le_numeral
% 0.35/0.62 thf(fact_5617_neg__one__le__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_le_numeral
% 0.35/0.62 thf(fact_5618_neg__one__le__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_le_numeral
% 0.35/0.62 thf(fact_5619_neg__numeral__le__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_one
% 0.35/0.62 thf(fact_5620_neg__numeral__le__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_one
% 0.35/0.62 thf(fact_5621_neg__numeral__le__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_one
% 0.35/0.62 thf(fact_5622_neg__numeral__le__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_le_one
% 0.35/0.62 thf(fact_5623_neg__numeral__less__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_one
% 0.35/0.62 thf(fact_5624_neg__numeral__less__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_one
% 0.35/0.62 thf(fact_5625_neg__numeral__less__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_one
% 0.35/0.62 thf(fact_5626_neg__numeral__less__one,axiom,
% 0.35/0.62 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.35/0.62
% 0.35/0.62 % neg_numeral_less_one
% 0.35/0.62 thf(fact_5627_neg__one__less__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_less_numeral
% 0.35/0.62 thf(fact_5628_neg__one__less__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_less_numeral
% 0.35/0.62 thf(fact_5629_neg__one__less__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_less_numeral
% 0.35/0.62 thf(fact_5630_neg__one__less__numeral,axiom,
% 0.35/0.62 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_one_less_numeral
% 0.35/0.62 thf(fact_5631_not__numeral__less__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_one
% 0.35/0.62 thf(fact_5632_not__numeral__less__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_one
% 0.35/0.62 thf(fact_5633_not__numeral__less__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_one
% 0.35/0.62 thf(fact_5634_not__numeral__less__neg__one,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_numeral_less_neg_one
% 0.35/0.62 thf(fact_5635_not__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_less_neg_numeral
% 0.35/0.62 thf(fact_5636_not__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_less_neg_numeral
% 0.35/0.62 thf(fact_5637_not__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_less_neg_numeral
% 0.35/0.62 thf(fact_5638_not__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_one_less_neg_numeral
% 0.35/0.62 thf(fact_5639_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_neg_one_less_neg_numeral
% 0.35/0.62 thf(fact_5640_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_neg_one_less_neg_numeral
% 0.35/0.62 thf(fact_5641_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_neg_one_less_neg_numeral
% 0.35/0.62 thf(fact_5642_not__neg__one__less__neg__numeral,axiom,
% 0.35/0.62 ! [M: num] :
% 0.35/0.62 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % not_neg_one_less_neg_numeral
% 0.35/0.62 thf(fact_5643_neg__mod__conj,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ B3 @ zero_zero_int )
% 0.35/0.62 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ zero_zero_int )
% 0.35/0.62 & ( ord_less_int @ B3 @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_mod_conj
% 0.35/0.62 thf(fact_5644_pos__mod__conj,axiom,
% 0.35/0.62 ! [B3: int,A2: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) )
% 0.35/0.62 & ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % pos_mod_conj
% 0.35/0.62 thf(fact_5645_zmod__trivial__iff,axiom,
% 0.35/0.62 ! [I: int,K: int] :
% 0.35/0.62 ( ( ( modulo_modulo_int @ I @ K )
% 0.35/0.62 = I )
% 0.35/0.62 = ( ( K = zero_zero_int )
% 0.35/0.62 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.62 & ( ord_less_int @ I @ K ) )
% 0.35/0.62 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.35/0.62 & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zmod_trivial_iff
% 0.35/0.62 thf(fact_5646_neg__mod__sign,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.62 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.35/0.62
% 0.35/0.62 % neg_mod_sign
% 0.35/0.62 thf(fact_5647_Euclidean__Division_Opos__mod__sign,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % Euclidean_Division.pos_mod_sign
% 0.35/0.62 thf(fact_5648_real__of__int__div4,axiom,
% 0.35/0.62 ! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % real_of_int_div4
% 0.35/0.62 thf(fact_5649_zdiv__mono__strict,axiom,
% 0.35/0.62 ! [A3: int,B2: int,N: int] :
% 0.35/0.62 ( ( ord_less_int @ A3 @ B2 )
% 0.35/0.62 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.62 => ( ( ( modulo_modulo_int @ A3 @ N )
% 0.35/0.62 = zero_zero_int )
% 0.35/0.62 => ( ( ( modulo_modulo_int @ B2 @ N )
% 0.35/0.62 = zero_zero_int )
% 0.35/0.62 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zdiv_mono_strict
% 0.35/0.62 thf(fact_5650_abs__mod__less,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( L != zero_zero_int )
% 0.35/0.62 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % abs_mod_less
% 0.35/0.62 thf(fact_5651_num_Osize_I5_J,axiom,
% 0.35/0.62 ! [X22: num] :
% 0.35/0.62 ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.35/0.62 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % num.size(5)
% 0.35/0.62 thf(fact_5652_of__int__nonneg,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_nonneg
% 0.35/0.62 thf(fact_5653_of__int__nonneg,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_nonneg
% 0.35/0.62 thf(fact_5654_of__int__nonneg,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_nonneg
% 0.35/0.62 thf(fact_5655_of__int__leD,axiom,
% 0.35/0.62 ! [N: int,X2: code_integer] :
% 0.35/0.62 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_leD
% 0.35/0.62 thf(fact_5656_of__int__leD,axiom,
% 0.35/0.62 ! [N: int,X2: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_leD
% 0.35/0.62 thf(fact_5657_of__int__leD,axiom,
% 0.35/0.62 ! [N: int,X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_leD
% 0.35/0.62 thf(fact_5658_of__int__leD,axiom,
% 0.35/0.62 ! [N: int,X2: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_leD
% 0.35/0.62 thf(fact_5659_of__int__pos,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_pos
% 0.35/0.62 thf(fact_5660_of__int__pos,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_pos
% 0.35/0.62 thf(fact_5661_of__int__pos,axiom,
% 0.35/0.62 ! [Z: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.62 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_pos
% 0.35/0.62 thf(fact_5662_of__int__lessD,axiom,
% 0.35/0.62 ! [N: int,X2: code_integer] :
% 0.35/0.62 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_lessD
% 0.35/0.62 thf(fact_5663_of__int__lessD,axiom,
% 0.35/0.62 ! [N: int,X2: real] :
% 0.35/0.62 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_lessD
% 0.35/0.62 thf(fact_5664_of__int__lessD,axiom,
% 0.35/0.62 ! [N: int,X2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_lessD
% 0.35/0.62 thf(fact_5665_of__int__lessD,axiom,
% 0.35/0.62 ! [N: int,X2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 0.35/0.62 => ( ( N = zero_zero_int )
% 0.35/0.62 | ( ord_less_int @ one_one_int @ X2 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_lessD
% 0.35/0.62 thf(fact_5666_floor__exists1,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ? [X5: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X2 )
% 0.35/0.62 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 0.35/0.62 & ! [Y5: int] :
% 0.35/0.62 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X2 )
% 0.35/0.62 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.35/0.62 => ( Y5 = X5 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % floor_exists1
% 0.35/0.62 thf(fact_5667_floor__exists1,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ? [X5: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X2 )
% 0.35/0.62 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 0.35/0.62 & ! [Y5: int] :
% 0.35/0.62 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X2 )
% 0.35/0.62 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 0.35/0.62 => ( Y5 = X5 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % floor_exists1
% 0.35/0.62 thf(fact_5668_floor__exists,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ? [Z4: int] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
% 0.35/0.62 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % floor_exists
% 0.35/0.62 thf(fact_5669_floor__exists,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ? [Z4: int] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
% 0.35/0.62 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % floor_exists
% 0.35/0.62 thf(fact_5670_of__int__ceiling__le__add__one,axiom,
% 0.35/0.62 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_ceiling_le_add_one
% 0.35/0.62 thf(fact_5671_of__int__ceiling__le__add__one,axiom,
% 0.35/0.62 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_ceiling_le_add_one
% 0.35/0.62 thf(fact_5672_of__int__ceiling__diff__one__le,axiom,
% 0.35/0.62 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_ceiling_diff_one_le
% 0.35/0.62 thf(fact_5673_of__int__ceiling__diff__one__le,axiom,
% 0.35/0.62 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.35/0.62
% 0.35/0.62 % of_int_ceiling_diff_one_le
% 0.35/0.62 thf(fact_5674_of__nat__less__of__int__iff,axiom,
% 0.35/0.62 ! [N: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_of_int_iff
% 0.35/0.62 thf(fact_5675_of__nat__less__of__int__iff,axiom,
% 0.35/0.62 ! [N: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_of_int_iff
% 0.35/0.62 thf(fact_5676_of__nat__less__of__int__iff,axiom,
% 0.35/0.62 ! [N: nat,X2: int] :
% 0.35/0.62 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
% 0.35/0.62 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % of_nat_less_of_int_iff
% 0.35/0.62 thf(fact_5677_ceiling__log__nat__eq__if,axiom,
% 0.35/0.62 ! [B3: nat,N: nat,K: nat] :
% 0.35/0.62 ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 0.35/0.62 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.35/0.62 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.62 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.62 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_log_nat_eq_if
% 0.35/0.62 thf(fact_5678_less__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5679_less__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5680_divide__less__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_less_eq_numeral(1)
% 0.35/0.62 thf(fact_5681_divide__less__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_less_eq_numeral(1)
% 0.35/0.62 thf(fact_5682_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_real )
% 0.35/0.62 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_real )
% 0.35/0.62 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = zero_zero_real ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5683_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_rat )
% 0.35/0.62 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_rat )
% 0.35/0.62 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = zero_zero_rat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5684_eq__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: complex,C: complex] :
% 0.35/0.62 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.62 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 0.35/0.62 = ( ( ( C != zero_zero_complex )
% 0.35/0.62 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 0.35/0.62 = B3 ) )
% 0.35/0.62 & ( ( C = zero_zero_complex )
% 0.35/0.62 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.62 = zero_zero_complex ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % eq_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5685_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ( divide_divide_real @ B3 @ C )
% 0.35/0.62 = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.62 = ( ( ( C != zero_zero_real )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_real )
% 0.35/0.62 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = zero_zero_real ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(2)
% 0.35/0.62 thf(fact_5686_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ( divide_divide_rat @ B3 @ C )
% 0.35/0.62 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.62 = ( ( ( C != zero_zero_rat )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_rat )
% 0.35/0.62 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = zero_zero_rat ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(2)
% 0.35/0.62 thf(fact_5687_divide__eq__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: complex,C: complex,W2: num] :
% 0.35/0.62 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 0.35/0.62 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.35/0.62 = ( ( ( C != zero_zero_complex )
% 0.35/0.62 => ( B3
% 0.35/0.62 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ( C = zero_zero_complex )
% 0.35/0.62 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.35/0.62 = zero_zero_complex ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_eq_eq_numeral(2)
% 0.35/0.62 thf(fact_5688_int__le__real__less,axiom,
% 0.35/0.62 ( ord_less_eq_int
% 0.35/0.62 = ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % int_le_real_less
% 0.35/0.62 thf(fact_5689_int__less__real__le,axiom,
% 0.35/0.62 ( ord_less_int
% 0.35/0.62 = ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % int_less_real_le
% 0.35/0.62 thf(fact_5690_mod__pos__neg__trivial,axiom,
% 0.35/0.62 ! [K: int,L: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.62 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.35/0.62 => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.62 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_pos_neg_trivial
% 0.35/0.62 thf(fact_5691_mod__pos__geq,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ L )
% 0.35/0.62 => ( ( ord_less_eq_int @ L @ K )
% 0.35/0.62 => ( ( modulo_modulo_int @ K @ L )
% 0.35/0.62 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mod_pos_geq
% 0.35/0.62 thf(fact_5692_ceiling__log__nat__eq__powr__iff,axiom,
% 0.35/0.62 ! [B3: nat,K: nat,N: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.62 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.62 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.62 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 0.35/0.62 = ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 0.35/0.62 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_log_nat_eq_powr_iff
% 0.35/0.62 thf(fact_5693_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.35/0.62 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.62 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.35/0.62 => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
% 0.35/0.62 = ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.35/0.62 thf(fact_5694_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.35/0.62 ! [C: int,A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.35/0.62 thf(fact_5695_ceiling__correct,axiom,
% 0.35/0.62 ! [X2: real] :
% 0.35/0.62 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
% 0.35/0.62 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_correct
% 0.35/0.62 thf(fact_5696_ceiling__correct,axiom,
% 0.35/0.62 ! [X2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
% 0.35/0.62 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_correct
% 0.35/0.62 thf(fact_5697_ceiling__unique,axiom,
% 0.35/0.62 ! [Z: int,X2: real] :
% 0.35/0.62 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
% 0.35/0.62 => ( ( archim7802044766580827645g_real @ X2 )
% 0.35/0.62 = Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_unique
% 0.35/0.62 thf(fact_5698_ceiling__unique,axiom,
% 0.35/0.62 ! [Z: int,X2: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
% 0.35/0.62 => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
% 0.35/0.62 => ( ( archim2889992004027027881ng_rat @ X2 )
% 0.35/0.62 = Z ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_unique
% 0.35/0.62 thf(fact_5699_ceiling__eq__iff,axiom,
% 0.35/0.62 ! [X2: real,A2: int] :
% 0.35/0.62 ( ( ( archim7802044766580827645g_real @ X2 )
% 0.35/0.62 = A2 )
% 0.35/0.62 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) @ X2 )
% 0.35/0.62 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_eq_iff
% 0.35/0.62 thf(fact_5700_ceiling__eq__iff,axiom,
% 0.35/0.62 ! [X2: rat,A2: int] :
% 0.35/0.62 ( ( ( archim2889992004027027881ng_rat @ X2 )
% 0.35/0.62 = A2 )
% 0.35/0.62 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) @ X2 )
% 0.35/0.62 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_eq_iff
% 0.35/0.62 thf(fact_5701_ceiling__split,axiom,
% 0.35/0.62 ! [P: int > $o,T: real] :
% 0.35/0.62 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.35/0.62 = ( ! [I4: int] :
% 0.35/0.62 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 0.35/0.62 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 0.35/0.62 => ( P @ I4 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_split
% 0.35/0.62 thf(fact_5702_ceiling__split,axiom,
% 0.35/0.62 ! [P: int > $o,T: rat] :
% 0.35/0.62 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.35/0.62 = ( ! [I4: int] :
% 0.35/0.62 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 0.35/0.62 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 0.35/0.62 => ( P @ I4 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_split
% 0.35/0.62 thf(fact_5703_ceiling__less__iff,axiom,
% 0.35/0.62 ! [X2: real,Z: int] :
% 0.35/0.62 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 0.35/0.62 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_iff
% 0.35/0.62 thf(fact_5704_ceiling__less__iff,axiom,
% 0.35/0.62 ! [X2: rat,Z: int] :
% 0.35/0.62 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 0.35/0.62 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_less_iff
% 0.35/0.62 thf(fact_5705_le__ceiling__iff,axiom,
% 0.35/0.62 ! [Z: int,X2: rat] :
% 0.35/0.62 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 0.35/0.62 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_ceiling_iff
% 0.35/0.62 thf(fact_5706_le__ceiling__iff,axiom,
% 0.35/0.62 ! [Z: int,X2: real] :
% 0.35/0.62 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 0.35/0.62 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_ceiling_iff
% 0.35/0.62 thf(fact_5707_divide__le__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_le_eq_numeral(1)
% 0.35/0.62 thf(fact_5708_divide__le__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_le_eq_numeral(1)
% 0.35/0.62 thf(fact_5709_le__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5710_le__divide__eq__numeral_I1_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_divide_eq_numeral(1)
% 0.35/0.62 thf(fact_5711_divide__less__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_less_eq_numeral(2)
% 0.35/0.62 thf(fact_5712_divide__less__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_less_eq_numeral(2)
% 0.35/0.62 thf(fact_5713_less__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5714_less__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % less_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5715_real__of__int__div2,axiom,
% 0.35/0.62 ! [N: int,X2: 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 @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % real_of_int_div2
% 0.35/0.62 thf(fact_5716_split__zmod,axiom,
% 0.35/0.62 ! [P: int > $o,N: int,K: int] :
% 0.35/0.62 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 0.35/0.62 = ( ( ( K = zero_zero_int )
% 0.35/0.62 => ( P @ N ) )
% 0.35/0.62 & ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.62 => ! [I4: int,J3: int] :
% 0.35/0.62 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.35/0.62 & ( ord_less_int @ J3 @ K )
% 0.35/0.62 & ( N
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.62 => ( P @ J3 ) ) )
% 0.35/0.62 & ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.62 => ! [I4: int,J3: int] :
% 0.35/0.62 ( ( ( ord_less_int @ K @ J3 )
% 0.35/0.62 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.35/0.62 & ( N
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 0.35/0.62 => ( P @ J3 ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % split_zmod
% 0.35/0.62 thf(fact_5717_int__mod__neg__eq,axiom,
% 0.35/0.62 ! [A2: int,B3: int,Q3: int,R2: int] :
% 0.35/0.62 ( ( A2
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.62 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.35/0.62 => ( ( ord_less_int @ B3 @ R2 )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ B3 )
% 0.35/0.62 = R2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % int_mod_neg_eq
% 0.35/0.62 thf(fact_5718_int__mod__pos__eq,axiom,
% 0.35/0.62 ! [A2: int,B3: int,Q3: int,R2: int] :
% 0.35/0.62 ( ( A2
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 0.35/0.62 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.35/0.62 => ( ( ord_less_int @ R2 @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ B3 )
% 0.35/0.62 = R2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % int_mod_pos_eq
% 0.35/0.62 thf(fact_5719_real__of__int__div3,axiom,
% 0.35/0.62 ! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).
% 0.35/0.62
% 0.35/0.62 % real_of_int_div3
% 0.35/0.62 thf(fact_5720_minus__mod__int__eq,axiom,
% 0.35/0.62 ! [L: int,K: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.35/0.62 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 0.35/0.62 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % minus_mod_int_eq
% 0.35/0.62 thf(fact_5721_zmod__minus1,axiom,
% 0.35/0.62 ! [B3: int] :
% 0.35/0.62 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.62 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
% 0.35/0.62 = ( minus_minus_int @ B3 @ one_one_int ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zmod_minus1
% 0.35/0.62 thf(fact_5722_zmod__zmult2__eq,axiom,
% 0.35/0.62 ! [C: int,A2: int,B3: int] :
% 0.35/0.62 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.35/0.62 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 0.35/0.62 = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % zmod_zmult2_eq
% 0.35/0.62 thf(fact_5723_ceiling__divide__upper,axiom,
% 0.35/0.62 ! [Q3: real,P6: real] :
% 0.35/0.62 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 0.35/0.62 => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_divide_upper
% 0.35/0.62 thf(fact_5724_ceiling__divide__upper,axiom,
% 0.35/0.62 ! [Q3: rat,P6: rat] :
% 0.35/0.62 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 0.35/0.62 => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % ceiling_divide_upper
% 0.35/0.62 thf(fact_5725_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: real,B3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.62 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5726_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: real,B3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.62 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.62 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5727_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: real,B3: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 0.35/0.62 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 0.35/0.62 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5728_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: rat,B3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.62 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.62 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5729_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: rat,B3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.62 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5730_mult__ceiling__le__Ints,axiom,
% 0.35/0.62 ! [A2: rat,B3: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 0.35/0.62 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 0.35/0.62 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % mult_ceiling_le_Ints
% 0.35/0.62 thf(fact_5731_divide__le__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: real,C: real,W2: num] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_le_eq_numeral(2)
% 0.35/0.62 thf(fact_5732_divide__le__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [B3: rat,C: rat,W2: num] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divide_le_eq_numeral(2)
% 0.35/0.62 thf(fact_5733_le__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: real,C: real] :
% 0.35/0.62 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.62 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.35/0.62 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5734_le__divide__eq__numeral_I2_J,axiom,
% 0.35/0.62 ! [W2: num,B3: rat,C: rat] :
% 0.35/0.62 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
% 0.35/0.62 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.35/0.62 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.35/0.62 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.35/0.62 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % le_divide_eq_numeral(2)
% 0.35/0.62 thf(fact_5735_divmod__step__eq,axiom,
% 0.35/0.62 ! [L: num,R2: nat,Q3: nat] :
% 0.35/0.62 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.35/0.62 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.35/0.62 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 0.35/0.62 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.35/0.62 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 0.35/0.62 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.35/0.62
% 0.35/0.62 % divmod_step_eq
% 0.35/0.62 thf(fact_5736_divmod__step__eq,axiom,
% 0.35/0.62 ! [L: num,R2: int,Q3: int] :
% 0.35/0.62 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.35/0.62 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.35/0.63 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 0.35/0.63 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.35/0.63 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.35/0.63 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % divmod_step_eq
% 0.35/0.63 thf(fact_5737_divmod__step__eq,axiom,
% 0.35/0.63 ! [L: num,R2: code_integer,Q3: code_integer] :
% 0.35/0.63 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.35/0.63 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.35/0.63 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.35/0.63 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.35/0.63 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 0.35/0.63 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % divmod_step_eq
% 0.35/0.63 thf(fact_5738_post__member__pre__member,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y3 )
% 0.35/0.63 => ( ( vEBT_vebt_member @ T @ Y3 )
% 0.35/0.63 | ( X2 = Y3 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % post_member_pre_member
% 0.35/0.63 thf(fact_5739_valid__insert__both__member__options__pres,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 0.35/0.63 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % valid_insert_both_member_options_pres
% 0.35/0.63 thf(fact_5740_valid__insert__both__member__options__add,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % valid_insert_both_member_options_add
% 0.35/0.63 thf(fact_5741_insert__simp__mima,axiom,
% 0.35/0.63 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.63 ( ( ( X2 = Mi )
% 0.35/0.63 | ( X2 = Ma ) )
% 0.35/0.63 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.63 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_simp_mima
% 0.35/0.63 thf(fact_5742_valid__pres__insert,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X2 ) @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % valid_pres_insert
% 0.35/0.63 thf(fact_5743_inrange,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( 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 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % inrange
% 0.35/0.63 thf(fact_5744_mod__less,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_nat @ M @ N )
% 0.35/0.63 => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.63 = M ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_less
% 0.35/0.63 thf(fact_5745_mod__by__Suc__0,axiom,
% 0.35/0.63 ! [M: nat] :
% 0.35/0.63 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.63 = zero_zero_nat ) ).
% 0.35/0.63
% 0.35/0.63 % mod_by_Suc_0
% 0.35/0.63 thf(fact_5746_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.35/0.63 ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 0.35/0.63 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5747_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 0.35/0.63 ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 0.35/0.63 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5748_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: num,H2: num,L3: num,H3: num] :
% 0.35/0.63 ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 0.35/0.63 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5749_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 0.35/0.63 ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 0.35/0.63 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5750_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: int,H2: int,L3: int,H3: int] :
% 0.35/0.63 ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 0.35/0.63 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5751_Icc__eq__Icc,axiom,
% 0.35/0.63 ! [L: real,H2: real,L3: real,H3: real] :
% 0.35/0.63 ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 0.35/0.63 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.35/0.63 = ( ( ( L = L3 )
% 0.35/0.63 & ( H2 = H3 ) )
% 0.35/0.63 | ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.35/0.63 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_Icc
% 0.35/0.63 thf(fact_5752_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: $o,L: $o,U: $o] :
% 0.35/0.63 ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_o @ L @ I )
% 0.35/0.63 & ( ord_less_eq_o @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5753_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: set_nat,L: set_nat,U: set_nat] :
% 0.35/0.63 ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_set_nat @ L @ I )
% 0.35/0.63 & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5754_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: set_int,L: set_int,U: set_int] :
% 0.35/0.63 ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_set_int @ L @ I )
% 0.35/0.63 & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5755_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: rat,L: rat,U: rat] :
% 0.35/0.63 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_rat @ L @ I )
% 0.35/0.63 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5756_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: num,L: num,U: num] :
% 0.35/0.63 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_num @ L @ I )
% 0.35/0.63 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5757_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: nat,L: nat,U: nat] :
% 0.35/0.63 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_nat @ L @ I )
% 0.35/0.63 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5758_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: int,L: int,U: int] :
% 0.35/0.63 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_int @ L @ I )
% 0.35/0.63 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5759_atLeastAtMost__iff,axiom,
% 0.35/0.63 ! [I: real,L: real,U: real] :
% 0.35/0.63 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.35/0.63 = ( ( ord_less_eq_real @ L @ I )
% 0.35/0.63 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_iff
% 0.35/0.63 thf(fact_5760_finite__atLeastAtMost,axiom,
% 0.35/0.63 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_atLeastAtMost
% 0.35/0.63 thf(fact_5761_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: $o,B3: $o] :
% 0.35/0.63 ( ( bot_bot_set_o
% 0.35/0.63 = ( set_or8904488021354931149Most_o @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5762_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: set_int,B3: set_int] :
% 0.35/0.63 ( ( bot_bot_set_set_int
% 0.35/0.63 = ( set_or370866239135849197et_int @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5763_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat] :
% 0.35/0.63 ( ( bot_bot_set_rat
% 0.35/0.63 = ( set_or633870826150836451st_rat @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5764_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: num,B3: num] :
% 0.35/0.63 ( ( bot_bot_set_num
% 0.35/0.63 = ( set_or7049704709247886629st_num @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5765_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat] :
% 0.35/0.63 ( ( bot_bot_set_nat
% 0.35/0.63 = ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5766_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( bot_bot_set_int
% 0.35/0.63 = ( set_or1266510415728281911st_int @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5767_atLeastatMost__empty__iff2,axiom,
% 0.35/0.63 ! [A2: real,B3: real] :
% 0.35/0.63 ( ( bot_bot_set_real
% 0.35/0.63 = ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff2
% 0.35/0.63 thf(fact_5768_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: $o,B3: $o] :
% 0.35/0.63 ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_o )
% 0.35/0.63 = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5769_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: set_int,B3: set_int] :
% 0.35/0.63 ( ( ( set_or370866239135849197et_int @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_set_int )
% 0.35/0.63 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5770_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat] :
% 0.35/0.63 ( ( ( set_or633870826150836451st_rat @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_rat )
% 0.35/0.63 = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5771_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: num,B3: num] :
% 0.35/0.63 ( ( ( set_or7049704709247886629st_num @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_num )
% 0.35/0.63 = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5772_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat] :
% 0.35/0.63 ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_nat )
% 0.35/0.63 = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5773_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_int )
% 0.35/0.63 = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5774_atLeastatMost__empty__iff,axiom,
% 0.35/0.63 ! [A2: real,B3: real] :
% 0.35/0.63 ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_real )
% 0.35/0.63 = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty_iff
% 0.35/0.63 thf(fact_5775_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
% 0.35/0.63 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_set_int @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_set_int @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5776_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.63 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_rat @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5777_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: num,B3: num,C: num,D: num] :
% 0.35/0.63 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_num @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_num @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5778_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5779_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_int @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5780_atLeastatMost__subset__iff,axiom,
% 0.35/0.63 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.63 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.35/0.63 = ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_real @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_real @ B3 @ D ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_subset_iff
% 0.35/0.63 thf(fact_5781_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: $o,A2: $o] :
% 0.35/0.63 ( ( ord_less_o @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_o ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5782_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: rat,A2: rat] :
% 0.35/0.63 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or633870826150836451st_rat @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_rat ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5783_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: num,A2: num] :
% 0.35/0.63 ( ( ord_less_num @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or7049704709247886629st_num @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_num ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5784_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: nat,A2: nat] :
% 0.35/0.63 ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_nat ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5785_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: int,A2: int] :
% 0.35/0.63 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_int ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5786_atLeastatMost__empty,axiom,
% 0.35/0.63 ! [B3: real,A2: real] :
% 0.35/0.63 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.63 => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 0.35/0.63 = bot_bot_set_real ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_empty
% 0.35/0.63 thf(fact_5787_infinite__Icc__iff,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat] :
% 0.35/0.63 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) )
% 0.35/0.63 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Icc_iff
% 0.35/0.63 thf(fact_5788_infinite__Icc__iff,axiom,
% 0.35/0.63 ! [A2: real,B3: real] :
% 0.35/0.63 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) )
% 0.35/0.63 = ( ord_less_real @ A2 @ B3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Icc_iff
% 0.35/0.63 thf(fact_5789_atLeastAtMost__singleton,axiom,
% 0.35/0.63 ! [A2: $o] :
% 0.35/0.63 ( ( set_or8904488021354931149Most_o @ A2 @ A2 )
% 0.35/0.63 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton
% 0.35/0.63 thf(fact_5790_atLeastAtMost__singleton,axiom,
% 0.35/0.63 ! [A2: nat] :
% 0.35/0.63 ( ( set_or1269000886237332187st_nat @ A2 @ A2 )
% 0.35/0.63 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton
% 0.35/0.63 thf(fact_5791_atLeastAtMost__singleton,axiom,
% 0.35/0.63 ! [A2: int] :
% 0.35/0.63 ( ( set_or1266510415728281911st_int @ A2 @ A2 )
% 0.35/0.63 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton
% 0.35/0.63 thf(fact_5792_atLeastAtMost__singleton,axiom,
% 0.35/0.63 ! [A2: real] :
% 0.35/0.63 ( ( set_or1222579329274155063t_real @ A2 @ A2 )
% 0.35/0.63 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton
% 0.35/0.63 thf(fact_5793_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.63 ! [A2: $o,B3: $o,C: $o] :
% 0.35/0.63 ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 0.35/0.63 = ( insert_o @ C @ bot_bot_set_o ) )
% 0.35/0.63 = ( ( A2 = B3 )
% 0.35/0.63 & ( B3 = C ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton_iff
% 0.35/0.63 thf(fact_5794_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.63 ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 0.35/0.63 = ( insert_nat @ C @ bot_bot_set_nat ) )
% 0.35/0.63 = ( ( A2 = B3 )
% 0.35/0.63 & ( B3 = C ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton_iff
% 0.35/0.63 thf(fact_5795_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.63 ! [A2: int,B3: int,C: int] :
% 0.35/0.63 ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 0.35/0.63 = ( insert_int @ C @ bot_bot_set_int ) )
% 0.35/0.63 = ( ( A2 = B3 )
% 0.35/0.63 & ( B3 = C ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton_iff
% 0.35/0.63 thf(fact_5796_atLeastAtMost__singleton__iff,axiom,
% 0.35/0.63 ! [A2: real,B3: real,C: real] :
% 0.35/0.63 ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 0.35/0.63 = ( insert_real @ C @ bot_bot_set_real ) )
% 0.35/0.63 = ( ( A2 = B3 )
% 0.35/0.63 & ( B3 = C ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton_iff
% 0.35/0.63 thf(fact_5797_real__of__nat__less__numeral__iff,axiom,
% 0.35/0.63 ! [N: nat,W2: num] :
% 0.35/0.63 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
% 0.35/0.63 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % real_of_nat_less_numeral_iff
% 0.35/0.63 thf(fact_5798_numeral__less__real__of__nat__iff,axiom,
% 0.35/0.63 ! [W2: num,N: nat] :
% 0.35/0.63 ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
% 0.35/0.63 = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% 0.35/0.63
% 0.35/0.63 % numeral_less_real_of_nat_iff
% 0.35/0.63 thf(fact_5799_numeral__le__real__of__nat__iff,axiom,
% 0.35/0.63 ! [N: num,M: nat] :
% 0.35/0.63 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.35/0.63 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 0.35/0.63
% 0.35/0.63 % numeral_le_real_of_nat_iff
% 0.35/0.63 thf(fact_5800_card__atLeastAtMost,axiom,
% 0.35/0.63 ! [L: nat,U: nat] :
% 0.35/0.63 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.35/0.63 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_atLeastAtMost
% 0.35/0.63 thf(fact_5801_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.35/0.63 ! [N: nat] :
% 0.35/0.63 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63 != ( suc @ zero_zero_nat ) )
% 0.35/0.63 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63 = zero_zero_nat ) ) ).
% 0.35/0.63
% 0.35/0.63 % not_mod2_eq_Suc_0_eq_0
% 0.35/0.63 thf(fact_5802_add__self__mod__2,axiom,
% 0.35/0.63 ! [M: nat] :
% 0.35/0.63 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63 = zero_zero_nat ) ).
% 0.35/0.63
% 0.35/0.63 % add_self_mod_2
% 0.35/0.63 thf(fact_5803_powr__numeral,axiom,
% 0.35/0.63 ! [X2: real,N: num] :
% 0.35/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.63 => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N ) )
% 0.35/0.63 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % powr_numeral
% 0.35/0.63 thf(fact_5804_mod2__gr__0,axiom,
% 0.35/0.63 ! [M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.63 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.63 = one_one_nat ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod2_gr_0
% 0.35/0.63 thf(fact_5805_half__nonnegative__int__iff,axiom,
% 0.35/0.63 ! [K: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.35/0.63 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.63
% 0.35/0.63 % half_nonnegative_int_iff
% 0.35/0.63 thf(fact_5806_half__negative__int__iff,axiom,
% 0.35/0.63 ! [K: int] :
% 0.35/0.63 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.35/0.63 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.63
% 0.35/0.63 % half_negative_int_iff
% 0.35/0.63 thf(fact_5807_mod__less__eq__dividend,axiom,
% 0.35/0.63 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 0.35/0.63
% 0.35/0.63 % mod_less_eq_dividend
% 0.35/0.63 thf(fact_5808_infinite__Icc,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat] :
% 0.35/0.63 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.63 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Icc
% 0.35/0.63 thf(fact_5809_infinite__Icc,axiom,
% 0.35/0.63 ! [A2: real,B3: real] :
% 0.35/0.63 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.63 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Icc
% 0.35/0.63 thf(fact_5810_atLeastAtMost__singleton_H,axiom,
% 0.35/0.63 ! [A2: $o,B3: $o] :
% 0.35/0.63 ( ( A2 = B3 )
% 0.35/0.63 => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 0.35/0.63 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton'
% 0.35/0.63 thf(fact_5811_atLeastAtMost__singleton_H,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat] :
% 0.35/0.63 ( ( A2 = B3 )
% 0.35/0.63 => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 0.35/0.63 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton'
% 0.35/0.63 thf(fact_5812_atLeastAtMost__singleton_H,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( A2 = B3 )
% 0.35/0.63 => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 0.35/0.63 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton'
% 0.35/0.63 thf(fact_5813_atLeastAtMost__singleton_H,axiom,
% 0.35/0.63 ! [A2: real,B3: real] :
% 0.35/0.63 ( ( A2 = B3 )
% 0.35/0.63 => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 0.35/0.63 = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_singleton'
% 0.35/0.63 thf(fact_5814_mod__Suc,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.63 = N )
% 0.35/0.63 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.35/0.63 = zero_zero_nat ) )
% 0.35/0.63 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.63 != N )
% 0.35/0.63 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 0.35/0.63 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_Suc
% 0.35/0.63 thf(fact_5815_all__nat__less,axiom,
% 0.35/0.63 ! [N: nat,P: nat > $o] :
% 0.35/0.63 ( ( ! [M2: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M2 @ N )
% 0.35/0.63 => ( P @ M2 ) ) )
% 0.35/0.63 = ( ! [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.63 => ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % all_nat_less
% 0.35/0.63 thf(fact_5816_ex__nat__less,axiom,
% 0.35/0.63 ! [N: nat,P: nat > $o] :
% 0.35/0.63 ( ( ? [M2: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M2 @ N )
% 0.35/0.63 & ( P @ M2 ) ) )
% 0.35/0.63 = ( ? [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.63 & ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ex_nat_less
% 0.35/0.63 thf(fact_5817_mod__induct,axiom,
% 0.35/0.63 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 0.35/0.63 ( ( P @ N )
% 0.35/0.63 => ( ( ord_less_nat @ N @ P6 )
% 0.35/0.63 => ( ( ord_less_nat @ M @ P6 )
% 0.35/0.63 => ( ! [N3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ N3 @ P6 )
% 0.35/0.63 => ( ( P @ N3 )
% 0.35/0.63 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P6 ) ) ) )
% 0.35/0.63 => ( P @ M ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_induct
% 0.35/0.63 thf(fact_5818_mod__less__divisor,axiom,
% 0.35/0.63 ! [N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_less_divisor
% 0.35/0.63 thf(fact_5819_gcd__nat__induct,axiom,
% 0.35/0.63 ! [P: nat > nat > $o,M: nat,N: nat] :
% 0.35/0.63 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 0.35/0.63 => ( ! [M4: nat,N3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.63 => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
% 0.35/0.63 => ( P @ M4 @ N3 ) ) )
% 0.35/0.63 => ( P @ M @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % gcd_nat_induct
% 0.35/0.63 thf(fact_5820_mod__Suc__le__divisor,axiom,
% 0.35/0.63 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 0.35/0.63
% 0.35/0.63 % mod_Suc_le_divisor
% 0.35/0.63 thf(fact_5821_mod__eq__0D,axiom,
% 0.35/0.63 ! [M: nat,D: nat] :
% 0.35/0.63 ( ( ( modulo_modulo_nat @ M @ D )
% 0.35/0.63 = zero_zero_nat )
% 0.35/0.63 => ? [Q4: nat] :
% 0.35/0.63 ( M
% 0.35/0.63 = ( times_times_nat @ D @ Q4 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_eq_0D
% 0.35/0.63 thf(fact_5822_mod__geq,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ~ ( ord_less_nat @ M @ N )
% 0.35/0.63 => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.63 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_geq
% 0.35/0.63 thf(fact_5823_mod__if,axiom,
% 0.35/0.63 ( modulo_modulo_nat
% 0.35/0.63 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_if
% 0.35/0.63 thf(fact_5824_le__mod__geq,axiom,
% 0.35/0.63 ! [N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63 => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.63 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % le_mod_geq
% 0.35/0.63 thf(fact_5825_vebt__insert_Osimps_I2_J,axiom,
% 0.35/0.63 ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 0.35/0.63 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X2 )
% 0.35/0.63 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % vebt_insert.simps(2)
% 0.35/0.63 thf(fact_5826_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
% 0.35/0.63 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_set_int @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_set_int @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_set_int @ C @ A2 )
% 0.35/0.63 | ( ord_less_set_int @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5827_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 0.35/0.63 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_rat @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_rat @ C @ A2 )
% 0.35/0.63 | ( ord_less_rat @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5828_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: num,B3: num,C: num,D: num] :
% 0.35/0.63 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_num @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_num @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_num @ C @ A2 )
% 0.35/0.63 | ( ord_less_num @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5829_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 0.35/0.63 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_nat @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_nat @ C @ A2 )
% 0.35/0.63 | ( ord_less_nat @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5830_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: int,B3: int,C: int,D: int] :
% 0.35/0.63 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_int @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_int @ C @ A2 )
% 0.35/0.63 | ( ord_less_int @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5831_atLeastatMost__psubset__iff,axiom,
% 0.35/0.63 ! [A2: real,B3: real,C: real,D: real] :
% 0.35/0.63 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.35/0.63 = ( ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.63 | ( ( ord_less_eq_real @ C @ A2 )
% 0.35/0.63 & ( ord_less_eq_real @ B3 @ D )
% 0.35/0.63 & ( ( ord_less_real @ C @ A2 )
% 0.35/0.63 | ( ord_less_real @ B3 @ D ) ) ) )
% 0.35/0.63 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastatMost_psubset_iff
% 0.35/0.63 thf(fact_5832_mod__le__divisor,axiom,
% 0.35/0.63 ! [N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_le_divisor
% 0.35/0.63 thf(fact_5833_div__less__mono,axiom,
% 0.35/0.63 ! [A3: nat,B2: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 0.35/0.63 = zero_zero_nat )
% 0.35/0.63 => ( ( ( modulo_modulo_nat @ B2 @ N )
% 0.35/0.63 = zero_zero_nat )
% 0.35/0.63 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % div_less_mono
% 0.35/0.63 thf(fact_5834_mod__eq__nat1E,axiom,
% 0.35/0.63 ! [M: nat,Q3: nat,N: nat] :
% 0.35/0.63 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.35/0.63 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.35/0.63 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.63 => ~ ! [S3: nat] :
% 0.35/0.63 ( M
% 0.35/0.63 != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_eq_nat1E
% 0.35/0.63 thf(fact_5835_mod__eq__nat2E,axiom,
% 0.35/0.63 ! [M: nat,Q3: nat,N: nat] :
% 0.35/0.63 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.35/0.63 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.35/0.63 => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63 => ~ ! [S3: nat] :
% 0.35/0.63 ( N
% 0.35/0.63 != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % mod_eq_nat2E
% 0.35/0.63 thf(fact_5836_nat__mod__eq__lemma,axiom,
% 0.35/0.63 ! [X2: nat,N: nat,Y3: nat] :
% 0.35/0.63 ( ( ( modulo_modulo_nat @ X2 @ N )
% 0.35/0.63 = ( modulo_modulo_nat @ Y3 @ N ) )
% 0.35/0.63 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.35/0.63 => ? [Q4: nat] :
% 0.35/0.63 ( X2
% 0.35/0.63 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nat_mod_eq_lemma
% 0.35/0.63 thf(fact_5837_atLeast0__atMost__Suc,axiom,
% 0.35/0.63 ! [N: nat] :
% 0.35/0.63 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.63 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeast0_atMost_Suc
% 0.35/0.63 thf(fact_5838_Icc__eq__insert__lb__nat,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 0.35/0.63 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Icc_eq_insert_lb_nat
% 0.35/0.63 thf(fact_5839_atLeastAtMostSuc__conv,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.63 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 0.35/0.63 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMostSuc_conv
% 0.35/0.63 thf(fact_5840_atLeastAtMost__insertL,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.63 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.63 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMost_insertL
% 0.35/0.63 thf(fact_5841_subset__eq__atLeast0__atMost__finite,axiom,
% 0.35/0.63 ! [N6: set_nat,N: nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.63 => ( finite_finite_nat @ N6 ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_eq_atLeast0_atMost_finite
% 0.35/0.63 thf(fact_5842_vebt__insert_Osimps_I3_J,axiom,
% 0.35/0.63 ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 0.35/0.63 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X2 )
% 0.35/0.63 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % vebt_insert.simps(3)
% 0.35/0.63 thf(fact_5843_vebt__insert_Osimps_I1_J,axiom,
% 0.35/0.63 ! [X2: nat,A2: $o,B3: $o] :
% 0.35/0.63 ( ( ( X2 = zero_zero_nat )
% 0.35/0.63 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.35/0.63 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 0.35/0.63 & ( ( X2 != zero_zero_nat )
% 0.35/0.63 => ( ( ( X2 = one_one_nat )
% 0.35/0.63 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.35/0.63 = ( vEBT_Leaf @ A2 @ $true ) ) )
% 0.35/0.63 & ( ( X2 != one_one_nat )
% 0.35/0.63 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.35/0.63 = ( vEBT_Leaf @ A2 @ B3 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % vebt_insert.simps(1)
% 0.35/0.63 thf(fact_5844_split__mod,axiom,
% 0.35/0.63 ! [P: nat > $o,M: nat,N: nat] :
% 0.35/0.63 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.63 = ( ( ( N = zero_zero_nat )
% 0.35/0.63 => ( P @ M ) )
% 0.35/0.63 & ( ( N != zero_zero_nat )
% 0.35/0.63 => ! [I4: nat,J3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ J3 @ N )
% 0.35/0.63 => ( ( M
% 0.35/0.63 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 0.35/0.63 => ( P @ J3 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % split_mod
% 0.35/0.63 thf(fact_5845_Suc__times__mod__eq,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.35/0.63 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 0.35/0.63 = one_one_nat ) ) ).
% 0.35/0.63
% 0.35/0.63 % Suc_times_mod_eq
% 0.35/0.63 thf(fact_5846_two__realpow__ge__one,axiom,
% 0.35/0.63 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.63
% 0.35/0.63 % two_realpow_ge_one
% 0.35/0.63 thf(fact_5847_nth__rotate1,axiom,
% 0.35/0.63 ! [N: nat,Xs: list_int] :
% 0.35/0.63 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.63 => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
% 0.35/0.63 = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nth_rotate1
% 0.35/0.63 thf(fact_5848_nth__rotate1,axiom,
% 0.35/0.63 ! [N: nat,Xs: list_VEBT_VEBT] :
% 0.35/0.63 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.63 => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
% 0.35/0.63 = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nth_rotate1
% 0.35/0.63 thf(fact_5849_nth__rotate1,axiom,
% 0.35/0.63 ! [N: nat,Xs: list_o] :
% 0.35/0.63 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.63 => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
% 0.35/0.63 = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nth_rotate1
% 0.35/0.63 thf(fact_5850_nth__rotate1,axiom,
% 0.35/0.63 ! [N: nat,Xs: list_nat] :
% 0.35/0.63 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.63 => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
% 0.35/0.63 = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nth_rotate1
% 0.35/0.63 thf(fact_5851_ln__2__less__1,axiom,
% 0.35/0.63 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.35/0.63
% 0.35/0.63 % ln_2_less_1
% 0.35/0.63 thf(fact_5852_not__exp__less__eq__0__int,axiom,
% 0.35/0.63 ! [N: nat] :
% 0.35/0.63 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 0.35/0.63
% 0.35/0.63 % not_exp_less_eq_0_int
% 0.35/0.63 thf(fact_5853_verit__le__mono__div,axiom,
% 0.35/0.63 ! [A3: nat,B2: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63 => ( ord_less_eq_nat
% 0.35/0.63 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 0.35/0.63 @ ( if_nat
% 0.35/0.63 @ ( ( modulo_modulo_nat @ B2 @ N )
% 0.35/0.63 = zero_zero_nat )
% 0.35/0.63 @ one_one_nat
% 0.35/0.63 @ zero_zero_nat ) )
% 0.35/0.63 @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % verit_le_mono_div
% 0.35/0.63 thf(fact_5854_exp__half__le2,axiom,
% 0.35/0.63 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.63
% 0.35/0.63 % exp_half_le2
% 0.35/0.63 thf(fact_5855_L2__set__mult__ineq__lemma,axiom,
% 0.35/0.63 ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C ) ) @ ( times_times_real @ B3 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % L2_set_mult_ineq_lemma
% 0.35/0.63 thf(fact_5856_less__log2__of__power,axiom,
% 0.35/0.63 ! [N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.35/0.63 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % less_log2_of_power
% 0.35/0.63 thf(fact_5857_le__log2__of__power,axiom,
% 0.35/0.63 ! [N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 0.35/0.63 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % le_log2_of_power
% 0.35/0.63 thf(fact_5858_powr__neg__numeral,axiom,
% 0.35/0.63 ! [X2: real,N: num] :
% 0.35/0.63 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.63 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 0.35/0.63 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % powr_neg_numeral
% 0.35/0.63 thf(fact_5859_pos__zdiv__mult__2,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.63 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.35/0.63 = ( divide_divide_int @ B3 @ A2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % pos_zdiv_mult_2
% 0.35/0.63 thf(fact_5860_neg__zdiv__mult__2,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.63 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.35/0.63 = ( divide_divide_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % neg_zdiv_mult_2
% 0.35/0.63 thf(fact_5861_pos__zmod__mult__2,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 0.35/0.63 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.35/0.63 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ A2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % pos_zmod_mult_2
% 0.35/0.63 thf(fact_5862_log2__of__power__less,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.63 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % log2_of_power_less
% 0.35/0.63 thf(fact_5863_real__exp__bound__lemma,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.63 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.63 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % real_exp_bound_lemma
% 0.35/0.63 thf(fact_5864_neg__zmod__mult__2,axiom,
% 0.35/0.63 ! [A2: int,B3: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 0.35/0.63 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 0.35/0.63 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % neg_zmod_mult_2
% 0.35/0.63 thf(fact_5865_pos__eucl__rel__int__mult__2,axiom,
% 0.35/0.63 ! [B3: int,A2: int,Q3: int,R2: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.63 => ( ( eucl_rel_int @ A2 @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.35/0.63 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % pos_eucl_rel_int_mult_2
% 0.35/0.63 thf(fact_5866_log2__of__power__le,axiom,
% 0.35/0.63 ! [M: nat,N: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.63 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % log2_of_power_le
% 0.35/0.63 thf(fact_5867_exp__lower__Taylor__quadratic,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.63 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % exp_lower_Taylor_quadratic
% 0.35/0.63 thf(fact_5868_neg__eucl__rel__int__mult__2,axiom,
% 0.35/0.63 ! [B3: int,A2: int,Q3: int,R2: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 0.35/0.63 => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 0.35/0.63 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % neg_eucl_rel_int_mult_2
% 0.35/0.63 thf(fact_5869_arctan__double,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.63 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
% 0.35/0.63 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % arctan_double
% 0.35/0.63 thf(fact_5870_ln__one__minus__pos__lower__bound,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.63 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.63 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ln_one_minus_pos_lower_bound
% 0.35/0.63 thf(fact_5871_abs__ln__one__plus__x__minus__x__bound,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_ln_one_plus_x_minus_x_bound
% 0.35/0.63 thf(fact_5872_ceiling__log2__div2,axiom,
% 0.35/0.63 ! [N: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.63 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.63 = ( 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 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ceiling_log2_div2
% 0.35/0.63 thf(fact_5873_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 0.35/0.63 ! [X2: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.63 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.63 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_ln_one_plus_x_minus_x_bound_nonpos
% 0.35/0.63 thf(fact_5874_VEBT__internal_Oinsert_H_Oelims,axiom,
% 0.35/0.63 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.63 ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
% 0.35/0.63 = Y3 )
% 0.35/0.63 => ( ! [A: $o,B: $o] :
% 0.35/0.63 ( ( X2
% 0.35/0.63 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.63 => ( Y3
% 0.35/0.63 != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) )
% 0.35/0.63 => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.63 ( ( X2
% 0.35/0.63 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.63 => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 0.35/0.63 => ( Y3
% 0.35/0.63 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
% 0.35/0.63 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 0.35/0.63 => ( Y3
% 0.35/0.63 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % VEBT_internal.insert'.elims
% 0.35/0.63 thf(fact_5875_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
% 0.35/0.63 ! [Deg: nat,X2: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.63 ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
% 0.35/0.63 => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) )
% 0.35/0.63 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
% 0.35/0.63 => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % VEBT_internal.insert'.simps(2)
% 0.35/0.63 thf(fact_5876_insert__correct,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_correct
% 0.35/0.63 thf(fact_5877_insert__corr,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_corr
% 0.35/0.63 thf(fact_5878_abs__sqrt__wlog,axiom,
% 0.35/0.63 ! [P: code_integer > code_integer > $o,X2: code_integer] :
% 0.35/0.63 ( ! [X5: code_integer] :
% 0.35/0.63 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 0.35/0.63 => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_sqrt_wlog
% 0.35/0.63 thf(fact_5879_abs__sqrt__wlog,axiom,
% 0.35/0.63 ! [P: real > real > $o,X2: real] :
% 0.35/0.63 ( ! [X5: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.35/0.63 => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_sqrt_wlog
% 0.35/0.63 thf(fact_5880_abs__sqrt__wlog,axiom,
% 0.35/0.63 ! [P: rat > rat > $o,X2: rat] :
% 0.35/0.63 ( ! [X5: rat] :
% 0.35/0.63 ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 0.35/0.63 => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_sqrt_wlog
% 0.35/0.63 thf(fact_5881_abs__sqrt__wlog,axiom,
% 0.35/0.63 ! [P: int > int > $o,X2: int] :
% 0.35/0.63 ( ! [X5: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 0.35/0.63 => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % abs_sqrt_wlog
% 0.35/0.63 thf(fact_5882_pred__list__to__short,axiom,
% 0.35/0.63 ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.35/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.63 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 0.35/0.63 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 = none_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % pred_list_to_short
% 0.35/0.63 thf(fact_5883_succ__list__to__short,axiom,
% 0.35/0.63 ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.63 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.63 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.35/0.63 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 = none_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % succ_list_to_short
% 0.35/0.63 thf(fact_5884_UnCI,axiom,
% 0.35/0.63 ! [C: real,B2: set_real,A3: set_real] :
% 0.35/0.63 ( ( ~ ( member_real @ C @ B2 )
% 0.35/0.63 => ( member_real @ C @ A3 ) )
% 0.35/0.63 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5885_UnCI,axiom,
% 0.35/0.63 ! [C: $o,B2: set_o,A3: set_o] :
% 0.35/0.63 ( ( ~ ( member_o @ C @ B2 )
% 0.35/0.63 => ( member_o @ C @ A3 ) )
% 0.35/0.63 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5886_UnCI,axiom,
% 0.35/0.63 ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 0.35/0.63 ( ( ~ ( member_set_nat @ C @ B2 )
% 0.35/0.63 => ( member_set_nat @ C @ A3 ) )
% 0.35/0.63 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5887_UnCI,axiom,
% 0.35/0.63 ! [C: int,B2: set_int,A3: set_int] :
% 0.35/0.63 ( ( ~ ( member_int @ C @ B2 )
% 0.35/0.63 => ( member_int @ C @ A3 ) )
% 0.35/0.63 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5888_UnCI,axiom,
% 0.35/0.63 ! [C: nat,B2: set_nat,A3: set_nat] :
% 0.35/0.63 ( ( ~ ( member_nat @ C @ B2 )
% 0.35/0.63 => ( member_nat @ C @ A3 ) )
% 0.35/0.63 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5889_UnCI,axiom,
% 0.35/0.63 ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ~ ( member8206827879206165904at_nat @ C @ B2 )
% 0.35/0.63 => ( member8206827879206165904at_nat @ C @ A3 ) )
% 0.35/0.63 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5890_UnCI,axiom,
% 0.35/0.63 ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ~ ( member8757157785044589968at_nat @ C @ B2 )
% 0.35/0.63 => ( member8757157785044589968at_nat @ C @ A3 ) )
% 0.35/0.63 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnCI
% 0.35/0.63 thf(fact_5891_Un__iff,axiom,
% 0.35/0.63 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_real @ C @ A3 )
% 0.35/0.63 | ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5892_Un__iff,axiom,
% 0.35/0.63 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_o @ C @ A3 )
% 0.35/0.63 | ( member_o @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5893_Un__iff,axiom,
% 0.35/0.63 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_set_nat @ C @ A3 )
% 0.35/0.63 | ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5894_Un__iff,axiom,
% 0.35/0.63 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_int @ C @ A3 )
% 0.35/0.63 | ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5895_Un__iff,axiom,
% 0.35/0.63 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_nat @ C @ A3 )
% 0.35/0.63 | ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5896_Un__iff,axiom,
% 0.35/0.63 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member8206827879206165904at_nat @ C @ A3 )
% 0.35/0.63 | ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5897_Un__iff,axiom,
% 0.35/0.63 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member8757157785044589968at_nat @ C @ A3 )
% 0.35/0.63 | ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_iff
% 0.35/0.63 thf(fact_5898_finite__atLeastAtMost__int,axiom,
% 0.35/0.63 ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_atLeastAtMost_int
% 0.35/0.63 thf(fact_5899_high__def,axiom,
% 0.35/0.63 ( vEBT_VEBT_high
% 0.35/0.63 = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % high_def
% 0.35/0.63 thf(fact_5900_high__bound__aux,axiom,
% 0.35/0.63 ! [Ma: nat,N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.35/0.63 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % high_bound_aux
% 0.35/0.63 thf(fact_5901_high__inv,axiom,
% 0.35/0.63 ! [X2: nat,N: nat,Y3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 0.35/0.63 = Y3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % high_inv
% 0.35/0.63 thf(fact_5902_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 0.35/0.63 = bot_bo5327735625951526323at_nat )
% 0.35/0.63 = ( ( A3 = bot_bo5327735625951526323at_nat )
% 0.35/0.63 & ( B2 = bot_bo5327735625951526323at_nat ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5903_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 0.35/0.63 = bot_bo228742789529271731at_nat )
% 0.35/0.63 = ( ( A3 = bot_bo228742789529271731at_nat )
% 0.35/0.63 & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5904_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( ( sup_sup_set_real @ A3 @ B2 )
% 0.35/0.63 = bot_bot_set_real )
% 0.35/0.63 = ( ( A3 = bot_bot_set_real )
% 0.35/0.63 & ( B2 = bot_bot_set_real ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5905_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( ( sup_sup_set_o @ A3 @ B2 )
% 0.35/0.63 = bot_bot_set_o )
% 0.35/0.63 = ( ( A3 = bot_bot_set_o )
% 0.35/0.63 & ( B2 = bot_bot_set_o ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5906_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ( sup_sup_set_nat @ A3 @ B2 )
% 0.35/0.63 = bot_bot_set_nat )
% 0.35/0.63 = ( ( A3 = bot_bot_set_nat )
% 0.35/0.63 & ( B2 = bot_bot_set_nat ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5907_Un__empty,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ( sup_sup_set_int @ A3 @ B2 )
% 0.35/0.63 = bot_bot_set_int )
% 0.35/0.63 = ( ( A3 = bot_bot_set_int )
% 0.35/0.63 & ( B2 = bot_bot_set_int ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty
% 0.35/0.63 thf(fact_5908_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_int,G2: set_int] :
% 0.35/0.63 ( ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite_finite_int @ F2 )
% 0.35/0.63 & ( finite_finite_int @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5909_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_complex,G2: set_complex] :
% 0.35/0.63 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite3207457112153483333omplex @ F2 )
% 0.35/0.63 & ( finite3207457112153483333omplex @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5910_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite6177210948735845034at_nat @ F2 )
% 0.35/0.63 & ( finite6177210948735845034at_nat @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5911_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 0.35/0.63 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite4001608067531595151d_enat @ F2 )
% 0.35/0.63 & ( finite4001608067531595151d_enat @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5912_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_nat,G2: set_nat] :
% 0.35/0.63 ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite_finite_nat @ F2 )
% 0.35/0.63 & ( finite_finite_nat @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5913_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite4392333629123659920at_nat @ F2 )
% 0.35/0.63 & ( finite4392333629123659920at_nat @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5914_finite__Un,axiom,
% 0.35/0.63 ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) )
% 0.35/0.63 = ( ( finite4343798906461161616at_nat @ F2 )
% 0.35/0.63 & ( finite4343798906461161616at_nat @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Un
% 0.35/0.63 thf(fact_5915_Un__subset__iff,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 0.35/0.63 & ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_subset_iff
% 0.35/0.63 thf(fact_5916_Un__subset__iff,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 0.35/0.63 & ( ord_le3000389064537975527at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_subset_iff
% 0.35/0.63 thf(fact_5917_Un__subset__iff,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 0.35/0.63 & ( ord_le1268244103169919719at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_subset_iff
% 0.35/0.63 thf(fact_5918_Un__subset__iff,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ( ord_less_eq_set_int @ A3 @ C2 )
% 0.35/0.63 & ( ord_less_eq_set_int @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_subset_iff
% 0.35/0.63 thf(fact_5919_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5920_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: real,B2: set_real,C2: set_real] :
% 0.35/0.63 ( ( sup_sup_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_real @ A2 @ ( sup_sup_set_real @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5921_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: $o,B2: set_o,C2: set_o] :
% 0.35/0.63 ( ( sup_sup_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_o @ A2 @ ( sup_sup_set_o @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5922_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: int,B2: set_int,C2: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_int @ A2 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5923_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5924_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5925_Un__insert__left,axiom,
% 0.35/0.63 ! [A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_left
% 0.35/0.63 thf(fact_5926_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5927_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_real,A2: real,B2: set_real] :
% 0.35/0.63 ( ( sup_sup_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_real @ A2 @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5928_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_o,A2: $o,B2: set_o] :
% 0.35/0.63 ( ( sup_sup_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_o @ A2 @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5929_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_int,A2: int,B2: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_int @ A2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5930_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5931_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5932_Un__insert__right,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ ( insert9069300056098147895at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_insert_right
% 0.35/0.63 thf(fact_5933_Un__Diff__cancel,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel
% 0.35/0.63 thf(fact_5934_Un__Diff__cancel,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel
% 0.35/0.63 thf(fact_5935_Un__Diff__cancel,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 0.35/0.63 = ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel
% 0.35/0.63 thf(fact_5936_Un__Diff__cancel2,axiom,
% 0.35/0.63 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) @ A3 )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ B2 @ A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel2
% 0.35/0.63 thf(fact_5937_Un__Diff__cancel2,axiom,
% 0.35/0.63 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) @ A3 )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ B2 @ A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel2
% 0.35/0.63 thf(fact_5938_Un__Diff__cancel2,axiom,
% 0.35/0.63 ! [B2: set_nat,A3: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A3 ) @ A3 )
% 0.35/0.63 = ( sup_sup_set_nat @ B2 @ A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff_cancel2
% 0.35/0.63 thf(fact_5939_Compl__Diff__eq,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( uminus4384627049435823934at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Compl_Diff_eq
% 0.35/0.63 thf(fact_5940_Compl__Diff__eq,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Compl_Diff_eq
% 0.35/0.63 thf(fact_5941_Compl__Diff__eq,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Compl_Diff_eq
% 0.35/0.63 thf(fact_5942_case4_I11_J,axiom,
% 0.35/0.63 ( ( mi != ma )
% 0.35/0.63 => ! [I3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.35/0.63 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 0.35/0.63 = I3 )
% 0.35/0.63 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 0.35/0.63 & ! [X4: nat] :
% 0.35/0.63 ( ( ( ( vEBT_VEBT_high @ X4 @ na )
% 0.35/0.63 = I3 )
% 0.35/0.63 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ na ) ) )
% 0.35/0.63 => ( ( ord_less_nat @ mi @ X4 )
% 0.35/0.63 & ( ord_less_eq_nat @ X4 @ ma ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % case4(11)
% 0.35/0.63 thf(fact_5943_UnE,axiom,
% 0.35/0.63 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member_real @ C @ A3 )
% 0.35/0.63 => ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5944_UnE,axiom,
% 0.35/0.63 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member_o @ C @ A3 )
% 0.35/0.63 => ( member_o @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5945_UnE,axiom,
% 0.35/0.63 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member_set_nat @ C @ A3 )
% 0.35/0.63 => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5946_UnE,axiom,
% 0.35/0.63 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member_int @ C @ A3 )
% 0.35/0.63 => ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5947_UnE,axiom,
% 0.35/0.63 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member_nat @ C @ A3 )
% 0.35/0.63 => ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5948_UnE,axiom,
% 0.35/0.63 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member8206827879206165904at_nat @ C @ A3 )
% 0.35/0.63 => ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5949_UnE,axiom,
% 0.35/0.63 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( ~ ( member8757157785044589968at_nat @ C @ A3 )
% 0.35/0.63 => ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnE
% 0.35/0.63 thf(fact_5950_UnI1,axiom,
% 0.35/0.63 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ C @ A3 )
% 0.35/0.63 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5951_UnI1,axiom,
% 0.35/0.63 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ C @ A3 )
% 0.35/0.63 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5952_UnI1,axiom,
% 0.35/0.63 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ A3 )
% 0.35/0.63 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5953_UnI1,axiom,
% 0.35/0.63 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ C @ A3 )
% 0.35/0.63 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5954_UnI1,axiom,
% 0.35/0.63 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ A3 )
% 0.35/0.63 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5955_UnI1,axiom,
% 0.35/0.63 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ C @ A3 )
% 0.35/0.63 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5956_UnI1,axiom,
% 0.35/0.63 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ C @ A3 )
% 0.35/0.63 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI1
% 0.35/0.63 thf(fact_5957_UnI2,axiom,
% 0.35/0.63 ! [C: real,B2: set_real,A3: set_real] :
% 0.35/0.63 ( ( member_real @ C @ B2 )
% 0.35/0.63 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5958_UnI2,axiom,
% 0.35/0.63 ! [C: $o,B2: set_o,A3: set_o] :
% 0.35/0.63 ( ( member_o @ C @ B2 )
% 0.35/0.63 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5959_UnI2,axiom,
% 0.35/0.63 ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ B2 )
% 0.35/0.63 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5960_UnI2,axiom,
% 0.35/0.63 ! [C: int,B2: set_int,A3: set_int] :
% 0.35/0.63 ( ( member_int @ C @ B2 )
% 0.35/0.63 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5961_UnI2,axiom,
% 0.35/0.63 ! [C: nat,B2: set_nat,A3: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ B2 )
% 0.35/0.63 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5962_UnI2,axiom,
% 0.35/0.63 ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ C @ B2 )
% 0.35/0.63 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5963_UnI2,axiom,
% 0.35/0.63 ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ C @ B2 )
% 0.35/0.63 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % UnI2
% 0.35/0.63 thf(fact_5964_bex__Un,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,P: nat > $o] :
% 0.35/0.63 ( ( ? [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 & ( P @ X ) ) )
% 0.35/0.63 = ( ? [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ A3 )
% 0.35/0.63 & ( P @ X ) )
% 0.35/0.63 | ? [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ B2 )
% 0.35/0.63 & ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bex_Un
% 0.35/0.63 thf(fact_5965_bex__Un,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 0.35/0.63 ( ( ? [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 & ( P @ X ) ) )
% 0.35/0.63 = ( ? [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ A3 )
% 0.35/0.63 & ( P @ X ) )
% 0.35/0.63 | ? [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ B2 )
% 0.35/0.63 & ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bex_Un
% 0.35/0.63 thf(fact_5966_bex__Un,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
% 0.35/0.63 ( ( ? [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 & ( P @ X ) ) )
% 0.35/0.63 = ( ? [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ A3 )
% 0.35/0.63 & ( P @ X ) )
% 0.35/0.63 | ? [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.63 & ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bex_Un
% 0.35/0.63 thf(fact_5967_ball__Un,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,P: nat > $o] :
% 0.35/0.63 ( ( ! [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( P @ X ) ) )
% 0.35/0.63 = ( ! [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ A3 )
% 0.35/0.63 => ( P @ X ) )
% 0.35/0.63 & ! [X: nat] :
% 0.35/0.63 ( ( member_nat @ X @ B2 )
% 0.35/0.63 => ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ball_Un
% 0.35/0.63 thf(fact_5968_ball__Un,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 0.35/0.63 ( ( ! [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( P @ X ) ) )
% 0.35/0.63 = ( ! [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ A3 )
% 0.35/0.63 => ( P @ X ) )
% 0.35/0.63 & ! [X: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( member8206827879206165904at_nat @ X @ B2 )
% 0.35/0.63 => ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ball_Un
% 0.35/0.63 thf(fact_5969_ball__Un,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
% 0.35/0.63 ( ( ! [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ( P @ X ) ) )
% 0.35/0.63 = ( ! [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ A3 )
% 0.35/0.63 => ( P @ X ) )
% 0.35/0.63 & ! [X: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( member8757157785044589968at_nat @ X @ B2 )
% 0.35/0.63 => ( P @ X ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ball_Un
% 0.35/0.63 thf(fact_5970_Un__assoc,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_assoc
% 0.35/0.63 thf(fact_5971_Un__assoc,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_assoc
% 0.35/0.63 thf(fact_5972_Un__assoc,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_assoc
% 0.35/0.63 thf(fact_5973_Un__absorb,axiom,
% 0.35/0.63 ! [A3: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ A3 )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb
% 0.35/0.63 thf(fact_5974_Un__absorb,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ A3 )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb
% 0.35/0.63 thf(fact_5975_Un__absorb,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ A3 )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb
% 0.35/0.63 thf(fact_5976_Un__commute,axiom,
% 0.35/0.63 ( sup_sup_set_nat
% 0.35/0.63 = ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_commute
% 0.35/0.63 thf(fact_5977_Un__commute,axiom,
% 0.35/0.63 ( sup_su718114333110466843at_nat
% 0.35/0.63 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B6 @ A6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_commute
% 0.35/0.63 thf(fact_5978_Un__commute,axiom,
% 0.35/0.63 ( sup_su5525570899277871387at_nat
% 0.35/0.63 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_commute
% 0.35/0.63 thf(fact_5979_Un__left__absorb,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_absorb
% 0.35/0.63 thf(fact_5980_Un__left__absorb,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_absorb
% 0.35/0.63 thf(fact_5981_Un__left__absorb,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_absorb
% 0.35/0.63 thf(fact_5982_Un__left__commute,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 0.35/0.63 = ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_commute
% 0.35/0.63 thf(fact_5983_Un__left__commute,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_commute
% 0.35/0.63 thf(fact_5984_Un__left__commute,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_left_commute
% 0.35/0.63 thf(fact_5985_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5987_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_real] :
% 0.35/0.63 ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5988_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_o] :
% 0.35/0.63 ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5989_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5990_boolean__algebra_Odisj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.disj_zero_right
% 0.35/0.63 thf(fact_5991_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A3 @ bot_bo5327735625951526323at_nat )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5992_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A3 @ bot_bo228742789529271731at_nat )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5993_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_real] :
% 0.35/0.63 ( ( sup_sup_set_real @ A3 @ bot_bot_set_real )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5994_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_o] :
% 0.35/0.63 ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5995_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5996_Un__empty__right,axiom,
% 0.35/0.63 ! [A3: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ A3 @ bot_bot_set_int )
% 0.35/0.63 = A3 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_right
% 0.35/0.63 thf(fact_5997_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_5998_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_5999_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_real] :
% 0.35/0.63 ( ( sup_sup_set_real @ bot_bot_set_real @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_6000_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_o] :
% 0.35/0.63 ( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_6001_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_6002_Un__empty__left,axiom,
% 0.35/0.63 ! [B2: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ bot_bot_set_int @ B2 )
% 0.35/0.63 = B2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_empty_left
% 0.35/0.63 thf(fact_6003_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_int,T2: set_int] :
% 0.35/0.63 ( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite_finite_int @ S )
% 0.35/0.63 | ~ ( finite_finite_int @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6004_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_complex,T2: set_complex] :
% 0.35/0.63 ( ( ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite3207457112153483333omplex @ S )
% 0.35/0.63 | ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6005_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite6177210948735845034at_nat @ S )
% 0.35/0.63 | ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6006_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 0.35/0.63 ( ( ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite4001608067531595151d_enat @ S )
% 0.35/0.63 | ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6007_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_nat,T2: set_nat] :
% 0.35/0.63 ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite_finite_nat @ S )
% 0.35/0.63 | ~ ( finite_finite_nat @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6008_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite4392333629123659920at_nat @ S )
% 0.35/0.63 | ~ ( finite4392333629123659920at_nat @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6009_infinite__Un,axiom,
% 0.35/0.63 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) )
% 0.35/0.63 = ( ~ ( finite4343798906461161616at_nat @ S )
% 0.35/0.63 | ~ ( finite4343798906461161616at_nat @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % infinite_Un
% 0.35/0.63 thf(fact_6010_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_int,T2: set_int] :
% 0.35/0.63 ( ~ ( finite_finite_int @ S )
% 0.35/0.63 => ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6011_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_complex,T2: set_complex] :
% 0.35/0.63 ( ~ ( finite3207457112153483333omplex @ S )
% 0.35/0.63 => ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6012_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ~ ( finite6177210948735845034at_nat @ S )
% 0.35/0.63 => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6013_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 0.35/0.63 ( ~ ( finite4001608067531595151d_enat @ S )
% 0.35/0.63 => ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6014_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_nat,T2: set_nat] :
% 0.35/0.63 ( ~ ( finite_finite_nat @ S )
% 0.35/0.63 => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6015_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ~ ( finite4392333629123659920at_nat @ S )
% 0.35/0.63 => ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6016_Un__infinite,axiom,
% 0.35/0.63 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ~ ( finite4343798906461161616at_nat @ S )
% 0.35/0.63 => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_infinite
% 0.35/0.63 thf(fact_6017_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_int,G2: set_int] :
% 0.35/0.63 ( ( finite_finite_int @ F2 )
% 0.35/0.63 => ( ( finite_finite_int @ G2 )
% 0.35/0.63 => ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6018_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_complex,G2: set_complex] :
% 0.35/0.63 ( ( finite3207457112153483333omplex @ F2 )
% 0.35/0.63 => ( ( finite3207457112153483333omplex @ G2 )
% 0.35/0.63 => ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6019_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( finite6177210948735845034at_nat @ F2 )
% 0.35/0.63 => ( ( finite6177210948735845034at_nat @ G2 )
% 0.35/0.63 => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6020_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 0.35/0.63 ( ( finite4001608067531595151d_enat @ F2 )
% 0.35/0.63 => ( ( finite4001608067531595151d_enat @ G2 )
% 0.35/0.63 => ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6021_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_nat,G2: set_nat] :
% 0.35/0.63 ( ( finite_finite_nat @ F2 )
% 0.35/0.63 => ( ( finite_finite_nat @ G2 )
% 0.35/0.63 => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6022_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( finite4392333629123659920at_nat @ F2 )
% 0.35/0.63 => ( ( finite4392333629123659920at_nat @ G2 )
% 0.35/0.63 => ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6023_finite__UnI,axiom,
% 0.35/0.63 ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( finite4343798906461161616at_nat @ F2 )
% 0.35/0.63 => ( ( finite4343798906461161616at_nat @ G2 )
% 0.35/0.63 => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_UnI
% 0.35/0.63 thf(fact_6024_subset__Un__eq,axiom,
% 0.35/0.63 ( ord_less_eq_set_nat
% 0.35/0.63 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.63 ( ( sup_sup_set_nat @ A6 @ B6 )
% 0.35/0.63 = B6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_Un_eq
% 0.35/0.63 thf(fact_6025_subset__Un__eq,axiom,
% 0.35/0.63 ( ord_le3000389064537975527at_nat
% 0.35/0.63 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( sup_su718114333110466843at_nat @ A6 @ B6 )
% 0.35/0.63 = B6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_Un_eq
% 0.35/0.63 thf(fact_6026_subset__Un__eq,axiom,
% 0.35/0.63 ( ord_le1268244103169919719at_nat
% 0.35/0.63 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( sup_su5525570899277871387at_nat @ A6 @ B6 )
% 0.35/0.63 = B6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_Un_eq
% 0.35/0.63 thf(fact_6027_subset__Un__eq,axiom,
% 0.35/0.63 ( ord_less_eq_set_int
% 0.35/0.63 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.63 ( ( sup_sup_set_int @ A6 @ B6 )
% 0.35/0.63 = B6 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_Un_eq
% 0.35/0.63 thf(fact_6028_subset__UnE,axiom,
% 0.35/0.63 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 => ~ ! [A8: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ A8 @ A3 )
% 0.35/0.63 => ! [B10: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ B10 @ B2 )
% 0.35/0.63 => ( C2
% 0.35/0.63 != ( sup_sup_set_nat @ A8 @ B10 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_UnE
% 0.35/0.63 thf(fact_6029_subset__UnE,axiom,
% 0.35/0.63 ! [C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ~ ! [A8: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ A8 @ A3 )
% 0.35/0.63 => ! [B10: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ B10 @ B2 )
% 0.35/0.63 => ( C2
% 0.35/0.63 != ( sup_su718114333110466843at_nat @ A8 @ B10 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_UnE
% 0.35/0.63 thf(fact_6030_subset__UnE,axiom,
% 0.35/0.63 ! [C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 => ~ ! [A8: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ A8 @ A3 )
% 0.35/0.63 => ! [B10: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ B10 @ B2 )
% 0.35/0.63 => ( C2
% 0.35/0.63 != ( sup_su5525570899277871387at_nat @ A8 @ B10 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_UnE
% 0.35/0.63 thf(fact_6031_subset__UnE,axiom,
% 0.35/0.63 ! [C2: set_int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.63 => ~ ! [A8: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ A8 @ A3 )
% 0.35/0.63 => ! [B10: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ B10 @ B2 )
% 0.35/0.63 => ( C2
% 0.35/0.63 != ( sup_sup_set_int @ A8 @ B10 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % subset_UnE
% 0.35/0.63 thf(fact_6032_Un__absorb2,axiom,
% 0.35/0.63 ! [B2: set_nat,A3: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.63 => ( ( sup_sup_set_nat @ A3 @ B2 )
% 0.35/0.63 = A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb2
% 0.35/0.63 thf(fact_6033_Un__absorb2,axiom,
% 0.35/0.63 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ B2 @ A3 )
% 0.35/0.63 => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 0.35/0.63 = A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb2
% 0.35/0.63 thf(fact_6034_Un__absorb2,axiom,
% 0.35/0.63 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ B2 @ A3 )
% 0.35/0.63 => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 0.35/0.63 = A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb2
% 0.35/0.63 thf(fact_6035_Un__absorb2,axiom,
% 0.35/0.63 ! [B2: set_int,A3: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.35/0.63 => ( ( sup_sup_set_int @ A3 @ B2 )
% 0.35/0.63 = A3 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb2
% 0.35/0.63 thf(fact_6036_Un__absorb1,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_sup_set_nat @ A3 @ B2 )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb1
% 0.35/0.63 thf(fact_6037_Un__absorb1,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb1
% 0.35/0.63 thf(fact_6038_Un__absorb1,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb1
% 0.35/0.63 thf(fact_6039_Un__absorb1,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_sup_set_int @ A3 @ B2 )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_absorb1
% 0.35/0.63 thf(fact_6040_Un__upper2,axiom,
% 0.35/0.63 ! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper2
% 0.35/0.63 thf(fact_6041_Un__upper2,axiom,
% 0.35/0.63 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper2
% 0.35/0.63 thf(fact_6042_Un__upper2,axiom,
% 0.35/0.63 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper2
% 0.35/0.63 thf(fact_6043_Un__upper2,axiom,
% 0.35/0.63 ! [B2: set_int,A3: set_int] : ( ord_less_eq_set_int @ B2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper2
% 0.35/0.63 thf(fact_6044_Un__upper1,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper1
% 0.35/0.63 thf(fact_6045_Un__upper1,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper1
% 0.35/0.63 thf(fact_6046_Un__upper1,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper1
% 0.35/0.63 thf(fact_6047_Un__upper1,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_upper1
% 0.35/0.63 thf(fact_6048_Un__least,axiom,
% 0.35/0.63 ! [A3: set_nat,C2: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 0.35/0.63 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_least
% 0.35/0.63 thf(fact_6049_Un__least,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_le3000389064537975527at_nat @ B2 @ C2 )
% 0.35/0.63 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_least
% 0.35/0.63 thf(fact_6050_Un__least,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_le1268244103169919719at_nat @ B2 @ C2 )
% 0.35/0.63 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_least
% 0.35/0.63 thf(fact_6051_Un__least,axiom,
% 0.35/0.63 ! [A3: set_int,C2: set_int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 0.35/0.63 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_least
% 0.35/0.63 thf(fact_6052_Un__mono,axiom,
% 0.35/0.63 ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_less_eq_set_nat @ B2 @ D4 )
% 0.35/0.63 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_mono
% 0.35/0.63 thf(fact_6053_Un__mono,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,D4: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_le3000389064537975527at_nat @ B2 @ D4 )
% 0.35/0.63 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ C2 @ D4 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_mono
% 0.35/0.63 thf(fact_6054_Un__mono,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_le1268244103169919719at_nat @ B2 @ D4 )
% 0.35/0.63 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D4 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_mono
% 0.35/0.63 thf(fact_6055_Un__mono,axiom,
% 0.35/0.63 ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 0.35/0.63 => ( ( ord_less_eq_set_int @ B2 @ D4 )
% 0.35/0.63 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ ( sup_sup_set_int @ C2 @ D4 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_mono
% 0.35/0.63 thf(fact_6056_Un__Diff,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( minus_8321449233255521966at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) @ ( minus_8321449233255521966at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff
% 0.35/0.63 thf(fact_6057_Un__Diff,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff
% 0.35/0.63 thf(fact_6058_Un__Diff,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Diff
% 0.35/0.63 thf(fact_6059_ivl__disj__un__two__touch_I4_J,axiom,
% 0.35/0.63 ! [L: rat,M: rat,U: rat] :
% 0.35/0.63 ( ( ord_less_eq_rat @ L @ M )
% 0.35/0.63 => ( ( ord_less_eq_rat @ M @ U )
% 0.35/0.63 => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
% 0.35/0.63 = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ivl_disj_un_two_touch(4)
% 0.35/0.63 thf(fact_6060_ivl__disj__un__two__touch_I4_J,axiom,
% 0.35/0.63 ! [L: num,M: num,U: num] :
% 0.35/0.63 ( ( ord_less_eq_num @ L @ M )
% 0.35/0.63 => ( ( ord_less_eq_num @ M @ U )
% 0.35/0.63 => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M ) @ ( set_or7049704709247886629st_num @ M @ U ) )
% 0.35/0.63 = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ivl_disj_un_two_touch(4)
% 0.35/0.63 thf(fact_6061_ivl__disj__un__two__touch_I4_J,axiom,
% 0.35/0.63 ! [L: nat,M: nat,U: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ L @ M )
% 0.35/0.63 => ( ( ord_less_eq_nat @ M @ U )
% 0.35/0.63 => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
% 0.35/0.63 = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ivl_disj_un_two_touch(4)
% 0.35/0.63 thf(fact_6062_ivl__disj__un__two__touch_I4_J,axiom,
% 0.35/0.63 ! [L: int,M: int,U: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ L @ M )
% 0.35/0.63 => ( ( ord_less_eq_int @ M @ U )
% 0.35/0.63 => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
% 0.35/0.63 = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ivl_disj_un_two_touch(4)
% 0.35/0.63 thf(fact_6063_ivl__disj__un__two__touch_I4_J,axiom,
% 0.35/0.63 ! [L: real,M: real,U: real] :
% 0.35/0.63 ( ( ord_less_eq_real @ L @ M )
% 0.35/0.63 => ( ( ord_less_eq_real @ M @ U )
% 0.35/0.63 => ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M ) @ ( set_or1222579329274155063t_real @ M @ U ) )
% 0.35/0.63 = ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % ivl_disj_un_two_touch(4)
% 0.35/0.63 thf(fact_6064_insert__is__Un,axiom,
% 0.35/0.63 ( insert8211810215607154385at_nat
% 0.35/0.63 = ( ^ [A4: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6065_insert__is__Un,axiom,
% 0.35/0.63 ( insert5050368324300391991at_nat
% 0.35/0.63 = ( ^ [A4: produc859450856879609959at_nat] : ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6066_insert__is__Un,axiom,
% 0.35/0.63 ( insert9069300056098147895at_nat
% 0.35/0.63 = ( ^ [A4: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6067_insert__is__Un,axiom,
% 0.35/0.63 ( insert_real
% 0.35/0.63 = ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6068_insert__is__Un,axiom,
% 0.35/0.63 ( insert_o
% 0.35/0.63 = ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6069_insert__is__Un,axiom,
% 0.35/0.63 ( insert_nat
% 0.35/0.63 = ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6070_insert__is__Un,axiom,
% 0.35/0.63 ( insert_int
% 0.35/0.63 = ( ^ [A4: int] : ( sup_sup_set_int @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_is_Un
% 0.35/0.63 thf(fact_6071_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 0.35/0.63 ( ( ( sup_su6327502436637775413at_nat @ A3 @ B2 )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo2099793752762293965at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6072_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,X2: produc859450856879609959at_nat] :
% 0.35/0.63 ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo5327735625951526323at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6073_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat] :
% 0.35/0.63 ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo228742789529271731at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo228742789529271731at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6074_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_real,B2: set_real,X2: real] :
% 0.35/0.63 ( ( ( sup_sup_set_real @ A3 @ B2 )
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_real )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.63 & ( B2 = bot_bot_set_real ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6075_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_o,B2: set_o,X2: $o] :
% 0.35/0.63 ( ( ( sup_sup_set_o @ A3 @ B2 )
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_o )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.63 & ( B2 = bot_bot_set_o ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6076_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,X2: nat] :
% 0.35/0.63 ( ( ( sup_sup_set_nat @ A3 @ B2 )
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 & ( B2 = bot_bot_set_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6077_Un__singleton__iff,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int,X2: int] :
% 0.35/0.63 ( ( ( sup_sup_set_int @ A3 @ B2 )
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_int )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.63 & ( B2 = bot_bot_set_int ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Un_singleton_iff
% 0.35/0.63 thf(fact_6078_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 0.35/0.63 = ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo2099793752762293965at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6079_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 0.35/0.63 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo5327735625951526323at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6080_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 0.35/0.63 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bo228742789529271731at_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.63 & ( B2 = bot_bo228742789529271731at_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6081_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( ( insert_real @ X2 @ bot_bot_set_real )
% 0.35/0.63 = ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_real )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.63 & ( B2 = bot_bot_set_real ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6082_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( ( insert_o @ X2 @ bot_bot_set_o )
% 0.35/0.63 = ( sup_sup_set_o @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_o )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.63 & ( B2 = bot_bot_set_o ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6083_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ( insert_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.63 = ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_nat )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 & ( B2 = bot_bot_set_nat ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6084_singleton__Un__iff,axiom,
% 0.35/0.63 ! [X2: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ( insert_int @ X2 @ bot_bot_set_int )
% 0.35/0.63 = ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ( A3 = bot_bot_set_int )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.63 & ( B2 = bot_bot_set_int ) )
% 0.35/0.63 | ( ( A3
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.63 & ( B2
% 0.35/0.63 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % singleton_Un_iff
% 0.35/0.63 thf(fact_6085_Diff__subset__conv,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_subset_conv
% 0.35/0.63 thf(fact_6086_Diff__subset__conv,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_subset_conv
% 0.35/0.63 thf(fact_6087_Diff__subset__conv,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_subset_conv
% 0.35/0.63 thf(fact_6088_Diff__subset__conv,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ C2 )
% 0.35/0.63 = ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_subset_conv
% 0.35/0.63 thf(fact_6089_Diff__partition,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_partition
% 0.35/0.63 thf(fact_6090_Diff__partition,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_partition
% 0.35/0.63 thf(fact_6091_Diff__partition,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_partition
% 0.35/0.63 thf(fact_6092_Diff__partition,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.35/0.63 => ( ( sup_sup_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
% 0.35/0.63 = B2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % Diff_partition
% 0.35/0.63 thf(fact_6093_atLeastAtMostPlus1__int__conv,axiom,
% 0.35/0.63 ! [M: int,N: int] :
% 0.35/0.63 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.35/0.63 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 0.35/0.63 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % atLeastAtMostPlus1_int_conv
% 0.35/0.63 thf(fact_6094_simp__from__to,axiom,
% 0.35/0.63 ( set_or1266510415728281911st_int
% 0.35/0.63 = ( ^ [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 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % simp_from_to
% 0.35/0.63 thf(fact_6095_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_complex,B2: set_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6096_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_list_nat,B2: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6097_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_set_nat,B2: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6098_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_int,B2: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6099_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6100_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6101_card__Un__le,axiom,
% 0.35/0.63 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % card_Un_le
% 0.35/0.63 thf(fact_6102_periodic__finite__ex,axiom,
% 0.35/0.63 ! [D: int,P: int > $o] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D )
% 0.35/0.63 => ( ! [X5: int,K2: int] :
% 0.35/0.63 ( ( P @ X5 )
% 0.35/0.63 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.35/0.63 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.35/0.63 = ( ? [X: int] :
% 0.35/0.63 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.35/0.63 & ( P @ X ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % periodic_finite_ex
% 0.35/0.63 thf(fact_6103_aset_I7_J,axiom,
% 0.35/0.63 ! [D4: int,A3: set_int,T: int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_int @ T @ X4 )
% 0.35/0.63 => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(7)
% 0.35/0.63 thf(fact_6104_aset_I5_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,A3: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ T @ A3 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_int @ X4 @ T )
% 0.35/0.63 => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(5)
% 0.35/0.63 thf(fact_6105_aset_I4_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,A3: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ T @ A3 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( X4 != T )
% 0.35/0.63 => ( ( plus_plus_int @ X4 @ D4 )
% 0.35/0.63 != T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(4)
% 0.35/0.63 thf(fact_6106_aset_I3_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,A3: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( X4 = T )
% 0.35/0.63 => ( ( plus_plus_int @ X4 @ D4 )
% 0.35/0.63 = T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(3)
% 0.35/0.63 thf(fact_6107_bset_I7_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ T @ B2 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_int @ T @ X4 )
% 0.35/0.63 => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(7)
% 0.35/0.63 thf(fact_6108_bset_I5_J,axiom,
% 0.35/0.63 ! [D4: int,B2: set_int,T: int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_int @ X4 @ T )
% 0.35/0.63 => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(5)
% 0.35/0.63 thf(fact_6109_bset_I4_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ T @ B2 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( X4 != T )
% 0.35/0.63 => ( ( minus_minus_int @ X4 @ D4 )
% 0.35/0.63 != T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(4)
% 0.35/0.63 thf(fact_6110_bset_I3_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( X4 = T )
% 0.35/0.63 => ( ( minus_minus_int @ X4 @ D4 )
% 0.35/0.63 = T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(3)
% 0.35/0.63 thf(fact_6111_aset_I8_J,axiom,
% 0.35/0.63 ! [D4: int,A3: set_int,T: int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_eq_int @ T @ X4 )
% 0.35/0.63 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(8)
% 0.35/0.63 thf(fact_6112_aset_I6_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,A3: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ A3 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_eq_int @ X4 @ T )
% 0.35/0.63 => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % aset(6)
% 0.35/0.63 thf(fact_6113_bset_I8_J,axiom,
% 0.35/0.63 ! [D4: int,T: int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_eq_int @ T @ X4 )
% 0.35/0.63 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(8)
% 0.35/0.63 thf(fact_6114_bset_I6_J,axiom,
% 0.35/0.63 ! [D4: int,B2: set_int,T: int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ! [X4: int] :
% 0.35/0.63 ( ! [Xa3: int] :
% 0.35/0.63 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb2: int] :
% 0.35/0.63 ( ( member_int @ Xb2 @ B2 )
% 0.35/0.63 => ( X4
% 0.35/0.63 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.35/0.63 => ( ( ord_less_eq_int @ X4 @ T )
% 0.35/0.63 => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % bset(6)
% 0.35/0.63 thf(fact_6115_cppi,axiom,
% 0.35/0.63 ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ? [Z5: int] :
% 0.35/0.63 ! [X5: int] :
% 0.35/0.63 ( ( ord_less_int @ Z5 @ X5 )
% 0.35/0.63 => ( ( P @ X5 )
% 0.35/0.63 = ( P4 @ X5 ) ) )
% 0.35/0.63 => ( ! [X5: int] :
% 0.35/0.63 ( ! [Xa: int] :
% 0.35/0.63 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb3: int] :
% 0.35/0.63 ( ( member_int @ Xb3 @ A3 )
% 0.35/0.63 => ( X5
% 0.35/0.63 != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
% 0.35/0.63 => ( ( P @ X5 )
% 0.35/0.63 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 0.35/0.63 => ( ! [X5: int,K2: int] :
% 0.35/0.63 ( ( P4 @ X5 )
% 0.35/0.63 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.35/0.63 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.35/0.63 = ( ? [X: int] :
% 0.35/0.63 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 & ( P4 @ X ) )
% 0.35/0.63 | ? [X: int] :
% 0.35/0.63 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 & ? [Y: int] :
% 0.35/0.63 ( ( member_int @ Y @ A3 )
% 0.35/0.63 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % cppi
% 0.35/0.63 thf(fact_6116_cpmi,axiom,
% 0.35/0.63 ! [D4: int,P: int > $o,P4: int > $o,B2: set_int] :
% 0.35/0.63 ( ( ord_less_int @ zero_zero_int @ D4 )
% 0.35/0.63 => ( ? [Z5: int] :
% 0.35/0.63 ! [X5: int] :
% 0.35/0.63 ( ( ord_less_int @ X5 @ Z5 )
% 0.35/0.63 => ( ( P @ X5 )
% 0.35/0.63 = ( P4 @ X5 ) ) )
% 0.35/0.63 => ( ! [X5: int] :
% 0.35/0.63 ( ! [Xa: int] :
% 0.35/0.63 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 => ! [Xb3: int] :
% 0.35/0.63 ( ( member_int @ Xb3 @ B2 )
% 0.35/0.63 => ( X5
% 0.35/0.63 != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
% 0.35/0.63 => ( ( P @ X5 )
% 0.35/0.63 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 0.35/0.63 => ( ! [X5: int,K2: int] :
% 0.35/0.63 ( ( P4 @ X5 )
% 0.35/0.63 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 0.35/0.63 => ( ( ? [X8: int] : ( P @ X8 ) )
% 0.35/0.63 = ( ? [X: int] :
% 0.35/0.63 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 & ( P4 @ X ) )
% 0.35/0.63 | ? [X: int] :
% 0.35/0.63 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 0.35/0.63 & ? [Y: int] :
% 0.35/0.63 ( ( member_int @ Y @ B2 )
% 0.35/0.63 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % cpmi
% 0.35/0.63 thf(fact_6117_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.35/0.63 ! [X2: nat,N: nat,M: nat] :
% 0.35/0.63 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.63 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.63 => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % VEBT_internal.exp_split_high_low(1)
% 0.35/0.63 thf(fact_6118_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
% 0.35/0.63 ! [A2: $o,B3: $o,X2: nat] :
% 0.35/0.63 ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 0.35/0.63 = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % VEBT_internal.insert'.simps(1)
% 0.35/0.63 thf(fact_6119_insert_H__correct,axiom,
% 0.35/0.63 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.63 => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) )
% 0.35/0.63 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert'_correct
% 0.35/0.63 thf(fact_6120_nested__mint,axiom,
% 0.35/0.63 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.63 => ( ( N
% 0.35/0.63 = ( suc @ ( suc @ Va ) ) )
% 0.35/0.63 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 0.35/0.63 => ( ( Ma != Mi )
% 0.35/0.63 => ( 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 ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % nested_mint
% 0.35/0.63 thf(fact_6121_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.35/0.63 ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.63 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.63 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.35/0.63 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % both_member_options_from_chilf_to_complete_tree
% 0.35/0.63 thf(fact_6122_member__inv,axiom,
% 0.35/0.63 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.35/0.63 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.63 & ( ( X2 = Mi )
% 0.35/0.63 | ( X2 = Ma )
% 0.35/0.63 | ( ( ord_less_nat @ X2 @ Ma )
% 0.35/0.63 & ( ord_less_nat @ Mi @ X2 )
% 0.35/0.63 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.63 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % member_inv
% 0.35/0.63 thf(fact_6123_both__member__options__from__complete__tree__to__child,axiom,
% 0.35/0.63 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.35/0.63 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.35/0.63 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.63 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 | ( X2 = Mi )
% 0.35/0.63 | ( X2 = Ma ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % both_member_options_from_complete_tree_to_child
% 0.35/0.63 thf(fact_6124_both__member__options__ding,axiom,
% 0.35/0.63 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.63 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.63 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.63 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.63 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % both_member_options_ding
% 0.35/0.63 thf(fact_6125_bit__split__inv,axiom,
% 0.35/0.63 ! [X2: nat,D: nat] :
% 0.35/0.63 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
% 0.35/0.63 = X2 ) ).
% 0.35/0.63
% 0.35/0.63 % bit_split_inv
% 0.35/0.63 thf(fact_6126_Int__iff,axiom,
% 0.35/0.63 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_real @ C @ A3 )
% 0.35/0.63 & ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6127_Int__iff,axiom,
% 0.35/0.63 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_o @ C @ A3 )
% 0.35/0.63 & ( member_o @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6128_Int__iff,axiom,
% 0.35/0.63 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_set_nat @ C @ A3 )
% 0.35/0.63 & ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6129_Int__iff,axiom,
% 0.35/0.63 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_int @ C @ A3 )
% 0.35/0.63 & ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6130_Int__iff,axiom,
% 0.35/0.63 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member_nat @ C @ A3 )
% 0.35/0.63 & ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6131_Int__iff,axiom,
% 0.35/0.63 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( member8440522571783428010at_nat @ C @ A3 )
% 0.35/0.63 & ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_iff
% 0.35/0.63 thf(fact_6132_IntI,axiom,
% 0.35/0.63 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ C @ A3 )
% 0.35/0.63 => ( ( member_real @ C @ B2 )
% 0.35/0.63 => ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6133_IntI,axiom,
% 0.35/0.63 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ C @ A3 )
% 0.35/0.63 => ( ( member_o @ C @ B2 )
% 0.35/0.63 => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6134_IntI,axiom,
% 0.35/0.63 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ C @ A3 )
% 0.35/0.63 => ( ( member_set_nat @ C @ B2 )
% 0.35/0.63 => ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6135_IntI,axiom,
% 0.35/0.63 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ C @ A3 )
% 0.35/0.63 => ( ( member_int @ C @ B2 )
% 0.35/0.63 => ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6136_IntI,axiom,
% 0.35/0.63 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ C @ A3 )
% 0.35/0.63 => ( ( member_nat @ C @ B2 )
% 0.35/0.63 => ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6137_IntI,axiom,
% 0.35/0.63 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( member8440522571783428010at_nat @ C @ A3 )
% 0.35/0.63 => ( ( member8440522571783428010at_nat @ C @ B2 )
% 0.35/0.63 => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % IntI
% 0.35/0.63 thf(fact_6138_low__def,axiom,
% 0.35/0.63 ( vEBT_VEBT_low
% 0.35/0.63 = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % low_def
% 0.35/0.63 thf(fact_6139_low__inv,axiom,
% 0.35/0.63 ! [X2: nat,N: nat,Y3: nat] :
% 0.35/0.63 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.63 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 0.35/0.63 = X2 ) ) ).
% 0.35/0.63
% 0.35/0.63 % low_inv
% 0.35/0.63 thf(fact_6140_boolean__algebra_Oconj__zero__left,axiom,
% 0.35/0.63 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
% 0.35/0.63 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_left
% 0.35/0.63 thf(fact_6141_boolean__algebra_Oconj__zero__left,axiom,
% 0.35/0.63 ! [X2: set_real] :
% 0.35/0.63 ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
% 0.35/0.63 = bot_bot_set_real ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_left
% 0.35/0.63 thf(fact_6142_boolean__algebra_Oconj__zero__left,axiom,
% 0.35/0.63 ! [X2: set_o] :
% 0.35/0.63 ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
% 0.35/0.63 = bot_bot_set_o ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_left
% 0.35/0.63 thf(fact_6143_boolean__algebra_Oconj__zero__left,axiom,
% 0.35/0.63 ! [X2: set_nat] :
% 0.35/0.63 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
% 0.35/0.63 = bot_bot_set_nat ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_left
% 0.35/0.63 thf(fact_6144_boolean__algebra_Oconj__zero__left,axiom,
% 0.35/0.63 ! [X2: set_int] :
% 0.35/0.63 ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
% 0.35/0.63 = bot_bot_set_int ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_left
% 0.35/0.63 thf(fact_6145_boolean__algebra_Oconj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 0.35/0.63 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_right
% 0.35/0.63 thf(fact_6146_boolean__algebra_Oconj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_real] :
% 0.35/0.63 ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
% 0.35/0.63 = bot_bot_set_real ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_right
% 0.35/0.63 thf(fact_6147_boolean__algebra_Oconj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_o] :
% 0.35/0.63 ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
% 0.35/0.63 = bot_bot_set_o ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_right
% 0.35/0.63 thf(fact_6148_boolean__algebra_Oconj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_nat] :
% 0.35/0.63 ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.63 = bot_bot_set_nat ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_right
% 0.35/0.63 thf(fact_6149_boolean__algebra_Oconj__zero__right,axiom,
% 0.35/0.63 ! [X2: set_int] :
% 0.35/0.63 ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
% 0.35/0.63 = bot_bot_set_int ) ).
% 0.35/0.63
% 0.35/0.63 % boolean_algebra.conj_zero_right
% 0.35/0.63 thf(fact_6150_finite__Int,axiom,
% 0.35/0.63 ! [F2: set_int,G2: set_int] :
% 0.35/0.63 ( ( ( finite_finite_int @ F2 )
% 0.35/0.63 | ( finite_finite_int @ G2 ) )
% 0.35/0.63 => ( finite_finite_int @ ( inf_inf_set_int @ F2 @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Int
% 0.35/0.63 thf(fact_6151_finite__Int,axiom,
% 0.35/0.63 ! [F2: set_complex,G2: set_complex] :
% 0.35/0.63 ( ( ( finite3207457112153483333omplex @ F2 )
% 0.35/0.63 | ( finite3207457112153483333omplex @ G2 ) )
% 0.35/0.63 => ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ F2 @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Int
% 0.35/0.63 thf(fact_6152_finite__Int,axiom,
% 0.35/0.63 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 0.35/0.63 ( ( ( finite4001608067531595151d_enat @ F2 )
% 0.35/0.63 | ( finite4001608067531595151d_enat @ G2 ) )
% 0.35/0.63 => ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ F2 @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Int
% 0.35/0.63 thf(fact_6153_finite__Int,axiom,
% 0.35/0.63 ! [F2: set_nat,G2: set_nat] :
% 0.35/0.63 ( ( ( finite_finite_nat @ F2 )
% 0.35/0.63 | ( finite_finite_nat @ G2 ) )
% 0.35/0.63 => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Int
% 0.35/0.63 thf(fact_6154_finite__Int,axiom,
% 0.35/0.63 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( ( finite6177210948735845034at_nat @ F2 )
% 0.35/0.63 | ( finite6177210948735845034at_nat @ G2 ) )
% 0.35/0.63 => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F2 @ G2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % finite_Int
% 0.35/0.63 thf(fact_6155_Int__subset__iff,axiom,
% 0.35/0.63 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ord_less_eq_set_nat @ C2 @ A3 )
% 0.35/0.63 & ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_subset_iff
% 0.35/0.63 thf(fact_6156_Int__subset__iff,axiom,
% 0.35/0.63 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
% 0.35/0.63 & ( ord_le3146513528884898305at_nat @ C2 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_subset_iff
% 0.35/0.63 thf(fact_6157_Int__subset__iff,axiom,
% 0.35/0.63 ! [C2: set_int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.63 = ( ( ord_less_eq_set_int @ C2 @ A3 )
% 0.35/0.63 & ( ord_less_eq_set_int @ C2 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_subset_iff
% 0.35/0.63 thf(fact_6158_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6159_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6160_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6161_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6162_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6163_Int__insert__right__if1,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if1
% 0.35/0.63 thf(fact_6164_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ~ ( member_real @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6165_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ~ ( member_o @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6166_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ~ ( member_set_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6167_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ~ ( member_int @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6168_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ~ ( member_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6169_Int__insert__right__if0,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.63 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_right_if0
% 0.35/0.63 thf(fact_6170_insert__inter__insert,axiom,
% 0.35/0.63 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.63 ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_inter_insert
% 0.35/0.63 thf(fact_6171_insert__inter__insert,axiom,
% 0.35/0.63 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.63 ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_inter_insert
% 0.35/0.63 thf(fact_6172_insert__inter__insert,axiom,
% 0.35/0.63 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.63 ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_inter_insert
% 0.35/0.63 thf(fact_6173_insert__inter__insert,axiom,
% 0.35/0.63 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_inter_insert
% 0.35/0.63 thf(fact_6174_insert__inter__insert,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % insert_inter_insert
% 0.35/0.63 thf(fact_6175_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: real,C2: set_real,B2: set_real] :
% 0.35/0.63 ( ( member_real @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6176_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: $o,C2: set_o,B2: set_o] :
% 0.35/0.63 ( ( member_o @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6177_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ( member_set_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6178_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: int,C2: set_int,B2: set_int] :
% 0.35/0.63 ( ( member_int @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6179_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 0.35/0.63 ( ( member_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6180_Int__insert__left__if1,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( member8440522571783428010at_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if1
% 0.35/0.63 thf(fact_6181_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: real,C2: set_real,B2: set_real] :
% 0.35/0.63 ( ~ ( member_real @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_inf_set_real @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6182_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: $o,C2: set_o,B2: set_o] :
% 0.35/0.63 ( ~ ( member_o @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_inf_set_o @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6183_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 0.35/0.63 ( ~ ( member_set_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6184_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: int,C2: set_int,B2: set_int] :
% 0.35/0.63 ( ~ ( member_int @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_inf_set_int @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6185_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 0.35/0.63 ( ~ ( member_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6186_Int__insert__left__if0,axiom,
% 0.35/0.63 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
% 0.35/0.63 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.63 = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.63
% 0.35/0.63 % Int_insert_left_if0
% 0.35/0.63 thf(fact_6187_Un__Int__eq_I1_J,axiom,
% 0.35/0.63 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ S )
% 0.35/0.63 = S ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(1)
% 0.35/0.63 thf(fact_6188_Un__Int__eq_I1_J,axiom,
% 0.35/0.63 ! [S: set_nat,T2: set_nat] :
% 0.35/0.63 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
% 0.35/0.63 = S ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(1)
% 0.35/0.63 thf(fact_6189_Un__Int__eq_I1_J,axiom,
% 0.35/0.63 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ S )
% 0.35/0.63 = S ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(1)
% 0.35/0.63 thf(fact_6190_Un__Int__eq_I1_J,axiom,
% 0.35/0.63 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ S )
% 0.35/0.63 = S ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(1)
% 0.35/0.63 thf(fact_6191_Un__Int__eq_I2_J,axiom,
% 0.35/0.63 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ T2 )
% 0.35/0.63 = T2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(2)
% 0.35/0.63 thf(fact_6192_Un__Int__eq_I2_J,axiom,
% 0.35/0.63 ! [S: set_nat,T2: set_nat] :
% 0.35/0.63 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
% 0.35/0.63 = T2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(2)
% 0.35/0.63 thf(fact_6193_Un__Int__eq_I2_J,axiom,
% 0.35/0.63 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.63 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ T2 )
% 0.35/0.63 = T2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(2)
% 0.35/0.63 thf(fact_6194_Un__Int__eq_I2_J,axiom,
% 0.35/0.63 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.63 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ T2 )
% 0.35/0.63 = T2 ) ).
% 0.35/0.63
% 0.35/0.63 % Un_Int_eq(2)
% 0.35/0.63 thf(fact_6195_Un__Int__eq_I3_J,axiom,
% 0.35/0.63 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.63 ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(3)
% 0.35/0.64 thf(fact_6196_Un__Int__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_nat,T2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(3)
% 0.35/0.64 thf(fact_6197_Un__Int__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( inf_in4302113700860409141at_nat @ S @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(3)
% 0.35/0.64 thf(fact_6198_Un__Int__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(3)
% 0.35/0.64 thf(fact_6199_Un__Int__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ T2 @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(4)
% 0.35/0.64 thf(fact_6200_Un__Int__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_nat,S: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(4)
% 0.35/0.64 thf(fact_6201_Un__Int__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( inf_in4302113700860409141at_nat @ T2 @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(4)
% 0.35/0.64 thf(fact_6202_Un__Int__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( inf_in7913087082777306421at_nat @ T2 @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_eq(4)
% 0.35/0.64 thf(fact_6203_Int__Un__eq_I1_J,axiom,
% 0.35/0.64 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ S )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(1)
% 0.35/0.64 thf(fact_6204_Int__Un__eq_I1_J,axiom,
% 0.35/0.64 ! [S: set_nat,T2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(1)
% 0.35/0.64 thf(fact_6205_Int__Un__eq_I1_J,axiom,
% 0.35/0.64 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ S )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(1)
% 0.35/0.64 thf(fact_6206_Int__Un__eq_I1_J,axiom,
% 0.35/0.64 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ S )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(1)
% 0.35/0.64 thf(fact_6207_Int__Un__eq_I2_J,axiom,
% 0.35/0.64 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ T2 )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(2)
% 0.35/0.64 thf(fact_6208_Int__Un__eq_I2_J,axiom,
% 0.35/0.64 ! [S: set_nat,T2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(2)
% 0.35/0.64 thf(fact_6209_Int__Un__eq_I2_J,axiom,
% 0.35/0.64 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ T2 )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(2)
% 0.35/0.64 thf(fact_6210_Int__Un__eq_I2_J,axiom,
% 0.35/0.64 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ T2 )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(2)
% 0.35/0.64 thf(fact_6211_Int__Un__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(3)
% 0.35/0.64 thf(fact_6212_Int__Un__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_nat,T2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(3)
% 0.35/0.64 thf(fact_6213_Int__Un__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ S @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(3)
% 0.35/0.64 thf(fact_6214_Int__Un__eq_I3_J,axiom,
% 0.35/0.64 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
% 0.35/0.64 = S ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(3)
% 0.35/0.64 thf(fact_6215_Int__Un__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ T2 @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(4)
% 0.35/0.64 thf(fact_6216_Int__Un__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_nat,S: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(4)
% 0.35/0.64 thf(fact_6217_Int__Un__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ T2 @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(4)
% 0.35/0.64 thf(fact_6218_Int__Un__eq_I4_J,axiom,
% 0.35/0.64 ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ T2 @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
% 0.35/0.64 = T2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_eq(4)
% 0.35/0.64 thf(fact_6219_summaxma,axiom,
% 0.35/0.64 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.64 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.35/0.64 => ( ( Mi != Ma )
% 0.35/0.64 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 0.35/0.64 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % summaxma
% 0.35/0.64 thf(fact_6220_inf__compl__bot__left1,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left1
% 0.35/0.64 thf(fact_6221_inf__compl__bot__left1,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( inf_inf_set_real @ X2 @ Y3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left1
% 0.35/0.64 thf(fact_6222_inf__compl__bot__left1,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ ( inf_inf_set_o @ X2 @ Y3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left1
% 0.35/0.64 thf(fact_6223_inf__compl__bot__left1,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( inf_inf_set_nat @ X2 @ Y3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left1
% 0.35/0.64 thf(fact_6224_inf__compl__bot__left1,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( inf_inf_set_int @ X2 @ Y3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left1
% 0.35/0.64 thf(fact_6225_inf__compl__bot__left2,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ Y3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left2
% 0.35/0.64 thf(fact_6226_inf__compl__bot__left2,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left2
% 0.35/0.64 thf(fact_6227_inf__compl__bot__left2,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ Y3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left2
% 0.35/0.64 thf(fact_6228_inf__compl__bot__left2,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left2
% 0.35/0.64 thf(fact_6229_inf__compl__bot__left2,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_left2
% 0.35/0.64 thf(fact_6230_inf__compl__bot__right,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ ( uminus6524753893492686040at_nat @ X2 ) ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_right
% 0.35/0.64 thf(fact_6231_inf__compl__bot__right,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ Y3 @ ( uminus612125837232591019t_real @ X2 ) ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_right
% 0.35/0.64 thf(fact_6232_inf__compl__bot__right,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ Y3 @ ( uminus_uminus_set_o @ X2 ) ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_right
% 0.35/0.64 thf(fact_6233_inf__compl__bot__right,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X2 ) ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_right
% 0.35/0.64 thf(fact_6234_inf__compl__bot__right,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_compl_bot_right
% 0.35/0.64 thf(fact_6235_boolean__algebra_Oconj__cancel__left,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ X2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_left
% 0.35/0.64 thf(fact_6236_boolean__algebra_Oconj__cancel__left,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ X2 )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_left
% 0.35/0.64 thf(fact_6237_boolean__algebra_Oconj__cancel__left,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ X2 )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_left
% 0.35/0.64 thf(fact_6238_boolean__algebra_Oconj__cancel__left,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_left
% 0.35/0.64 thf(fact_6239_boolean__algebra_Oconj__cancel__left,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ X2 )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_left
% 0.35/0.64 thf(fact_6240_boolean__algebra_Oconj__cancel__right,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ X2 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_right
% 0.35/0.64 thf(fact_6241_boolean__algebra_Oconj__cancel__right,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ X2 @ ( uminus612125837232591019t_real @ X2 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_right
% 0.35/0.64 thf(fact_6242_boolean__algebra_Oconj__cancel__right,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ X2 @ ( uminus_uminus_set_o @ X2 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_right
% 0.35/0.64 thf(fact_6243_boolean__algebra_Oconj__cancel__right,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_right
% 0.35/0.64 thf(fact_6244_boolean__algebra_Oconj__cancel__right,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ X2 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % boolean_algebra.conj_cancel_right
% 0.35/0.64 thf(fact_6245_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( bot_bot_set_set_nat
% 0.35/0.64 = ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member_set_nat @ B3 @ A3 )
% 0.35/0.64 & ( bot_bot_set_set_nat
% 0.35/0.64 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6246_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( bot_bo2099793752762293965at_nat
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 )
% 0.35/0.64 & ( bot_bo2099793752762293965at_nat
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6247_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_real,B3: real,B2: set_real] :
% 0.35/0.64 ( ( bot_bot_set_real
% 0.35/0.64 = ( inf_inf_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member_real @ B3 @ A3 )
% 0.35/0.64 & ( bot_bot_set_real
% 0.35/0.64 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6248_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_o,B3: $o,B2: set_o] :
% 0.35/0.64 ( ( bot_bot_set_o
% 0.35/0.64 = ( inf_inf_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member_o @ B3 @ A3 )
% 0.35/0.64 & ( bot_bot_set_o
% 0.35/0.64 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6249_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_nat,B3: nat,B2: set_nat] :
% 0.35/0.64 ( ( bot_bot_set_nat
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member_nat @ B3 @ A3 )
% 0.35/0.64 & ( bot_bot_set_nat
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6250_disjoint__insert_I2_J,axiom,
% 0.35/0.64 ! [A3: set_int,B3: int,B2: set_int] :
% 0.35/0.64 ( ( bot_bot_set_int
% 0.35/0.64 = ( inf_inf_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) )
% 0.35/0.64 = ( ~ ( member_int @ B3 @ A3 )
% 0.35/0.64 & ( bot_bot_set_int
% 0.35/0.64 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(2)
% 0.35/0.64 thf(fact_6251_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_set_nat,A2: set_nat,A3: set_set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_set_nat )
% 0.35/0.64 = ( ~ ( member_set_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_set_nat @ B2 @ A3 )
% 0.35/0.64 = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6252_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_in2572325071724192079at_nat @ B2 @ A3 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6253_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_real,A2: real,A3: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ~ ( member_real @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_real @ B2 @ A3 )
% 0.35/0.64 = bot_bot_set_real ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6254_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_o,A2: $o,A3: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ~ ( member_o @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_o @ B2 @ A3 )
% 0.35/0.64 = bot_bot_set_o ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6255_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_nat,A2: nat,A3: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ~ ( member_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_nat @ B2 @ A3 )
% 0.35/0.64 = bot_bot_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6256_disjoint__insert_I1_J,axiom,
% 0.35/0.64 ! [B2: set_int,A2: int,A3: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ B2 @ ( insert_int @ A2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ~ ( member_int @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_int @ B2 @ A3 )
% 0.35/0.64 = bot_bot_set_int ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_insert(1)
% 0.35/0.64 thf(fact_6257_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( bot_bot_set_set_nat
% 0.35/0.64 = ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member_set_nat @ A2 @ B2 )
% 0.35/0.64 & ( bot_bot_set_set_nat
% 0.35/0.64 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6258_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( bot_bo2099793752762293965at_nat
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.35/0.64 & ( bot_bo2099793752762293965at_nat
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6259_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( bot_bot_set_real
% 0.35/0.64 = ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member_real @ A2 @ B2 )
% 0.35/0.64 & ( bot_bot_set_real
% 0.35/0.64 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6260_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( bot_bot_set_o
% 0.35/0.64 = ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member_o @ A2 @ B2 )
% 0.35/0.64 & ( bot_bot_set_o
% 0.35/0.64 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6261_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( bot_bot_set_nat
% 0.35/0.64 = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member_nat @ A2 @ B2 )
% 0.35/0.64 & ( bot_bot_set_nat
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6262_insert__disjoint_I2_J,axiom,
% 0.35/0.64 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( bot_bot_set_int
% 0.35/0.64 = ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 ) )
% 0.35/0.64 = ( ~ ( member_int @ A2 @ B2 )
% 0.35/0.64 & ( bot_bot_set_int
% 0.35/0.64 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(2)
% 0.35/0.64 thf(fact_6263_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bot_set_set_nat )
% 0.35/0.64 = ( ~ ( member_set_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6264_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6265_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ~ ( member_real @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6266_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ~ ( member_o @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6267_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ~ ( member_nat @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6268_insert__disjoint_I1_J,axiom,
% 0.35/0.64 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ~ ( member_int @ A2 @ B2 )
% 0.35/0.64 & ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_disjoint(1)
% 0.35/0.64 thf(fact_6269_Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_disjoint
% 0.35/0.64 thf(fact_6270_Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ A3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_disjoint
% 0.35/0.64 thf(fact_6271_Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_disjoint
% 0.35/0.64 thf(fact_6272_Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_disjoint
% 0.35/0.64 thf(fact_6273_Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_disjoint
% 0.35/0.64 thf(fact_6274_option_Ocollapse,axiom,
% 0.35/0.64 ! [Option: option_nat] :
% 0.35/0.64 ( ( Option != none_nat )
% 0.35/0.64 => ( ( some_nat @ ( the_nat @ Option ) )
% 0.35/0.64 = Option ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.collapse
% 0.35/0.64 thf(fact_6275_option_Ocollapse,axiom,
% 0.35/0.64 ! [Option: option4927543243414619207at_nat] :
% 0.35/0.64 ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.64 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.35/0.64 = Option ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.collapse
% 0.35/0.64 thf(fact_6276_option_Ocollapse,axiom,
% 0.35/0.64 ! [Option: option_num] :
% 0.35/0.64 ( ( Option != none_num )
% 0.35/0.64 => ( ( some_num @ ( the_num @ Option ) )
% 0.35/0.64 = Option ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.collapse
% 0.35/0.64 thf(fact_6277_Compl__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint
% 0.35/0.64 thf(fact_6278_Compl__disjoint,axiom,
% 0.35/0.64 ! [A3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint
% 0.35/0.64 thf(fact_6279_Compl__disjoint,axiom,
% 0.35/0.64 ! [A3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint
% 0.35/0.64 thf(fact_6280_Compl__disjoint,axiom,
% 0.35/0.64 ! [A3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint
% 0.35/0.64 thf(fact_6281_Compl__disjoint,axiom,
% 0.35/0.64 ! [A3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint
% 0.35/0.64 thf(fact_6282_Compl__disjoint2,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ A3 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint2
% 0.35/0.64 thf(fact_6283_Compl__disjoint2,axiom,
% 0.35/0.64 ! [A3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A3 ) @ A3 )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint2
% 0.35/0.64 thf(fact_6284_Compl__disjoint2,axiom,
% 0.35/0.64 ! [A3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint2
% 0.35/0.64 thf(fact_6285_Compl__disjoint2,axiom,
% 0.35/0.64 ! [A3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint2
% 0.35/0.64 thf(fact_6286_Compl__disjoint2,axiom,
% 0.35/0.64 ! [A3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ A3 )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_disjoint2
% 0.35/0.64 thf(fact_6287_Diff__Compl,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Compl
% 0.35/0.64 thf(fact_6288_Diff__Compl,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Compl
% 0.35/0.64 thf(fact_6289_Int__left__commute,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_left_commute
% 0.35/0.64 thf(fact_6290_Int__left__commute,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ B2 @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_left_commute
% 0.35/0.64 thf(fact_6291_Int__left__absorb,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_left_absorb
% 0.35/0.64 thf(fact_6292_Int__left__absorb,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_left_absorb
% 0.35/0.64 thf(fact_6293_Int__commute,axiom,
% 0.35/0.64 ( inf_inf_set_nat
% 0.35/0.64 = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ B6 @ A6 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_commute
% 0.35/0.64 thf(fact_6294_Int__commute,axiom,
% 0.35/0.64 ( inf_in2572325071724192079at_nat
% 0.35/0.64 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A6 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_commute
% 0.35/0.64 thf(fact_6295_Int__absorb,axiom,
% 0.35/0.64 ! [A3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ A3 )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb
% 0.35/0.64 thf(fact_6296_Int__absorb,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb
% 0.35/0.64 thf(fact_6297_Int__assoc,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_assoc
% 0.35/0.64 thf(fact_6298_Int__assoc,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_assoc
% 0.35/0.64 thf(fact_6299_IntD2,axiom,
% 0.35/0.64 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.64 => ( member_real @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6300_IntD2,axiom,
% 0.35/0.64 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.64 => ( member_o @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6301_IntD2,axiom,
% 0.35/0.64 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member_set_nat @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6302_IntD2,axiom,
% 0.35/0.64 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.64 => ( member_int @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6303_IntD2,axiom,
% 0.35/0.64 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member_nat @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6304_IntD2,axiom,
% 0.35/0.64 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD2
% 0.35/0.64 thf(fact_6305_IntD1,axiom,
% 0.35/0.64 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.64 => ( member_real @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6306_IntD1,axiom,
% 0.35/0.64 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.64 => ( member_o @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6307_IntD1,axiom,
% 0.35/0.64 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member_set_nat @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6308_IntD1,axiom,
% 0.35/0.64 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.64 => ( member_int @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6309_IntD1,axiom,
% 0.35/0.64 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member_nat @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6310_IntD1,axiom,
% 0.35/0.64 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( member8440522571783428010at_nat @ C @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntD1
% 0.35/0.64 thf(fact_6311_IntE,axiom,
% 0.35/0.64 ! [C: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member_real @ C @ A3 )
% 0.35/0.64 => ~ ( member_real @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6312_IntE,axiom,
% 0.35/0.64 ! [C: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member_o @ C @ A3 )
% 0.35/0.64 => ~ ( member_o @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6313_IntE,axiom,
% 0.35/0.64 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member_set_nat @ C @ A3 )
% 0.35/0.64 => ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6314_IntE,axiom,
% 0.35/0.64 ! [C: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member_int @ C @ A3 )
% 0.35/0.64 => ~ ( member_int @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6315_IntE,axiom,
% 0.35/0.64 ! [C: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member_nat @ C @ A3 )
% 0.35/0.64 => ~ ( member_nat @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6316_IntE,axiom,
% 0.35/0.64 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
% 0.35/0.64 => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % IntE
% 0.35/0.64 thf(fact_6317_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ! [X5: set_nat] :
% 0.35/0.64 ( ( member_set_nat @ X5 @ A3 )
% 0.35/0.64 => ~ ( member_set_nat @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_set_nat ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6318_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ! [X5: product_prod_nat_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 0.35/0.64 => ~ ( member8440522571783428010at_nat @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6319_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ! [X5: real] :
% 0.35/0.64 ( ( member_real @ X5 @ A3 )
% 0.35/0.64 => ~ ( member_real @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6320_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ! [X5: $o] :
% 0.35/0.64 ( ( member_o @ X5 @ A3 )
% 0.35/0.64 => ~ ( member_o @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6321_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ! [X5: nat] :
% 0.35/0.64 ( ( member_nat @ X5 @ A3 )
% 0.35/0.64 => ~ ( member_nat @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6322_Int__emptyI,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ! [X5: int] :
% 0.35/0.64 ( ( member_int @ X5 @ A3 )
% 0.35/0.64 => ~ ( member_int @ X5 @ B2 ) )
% 0.35/0.64 => ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_emptyI
% 0.35/0.64 thf(fact_6323_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_set_nat )
% 0.35/0.64 = ( ! [X: set_nat] :
% 0.35/0.64 ( ( member_set_nat @ X @ A3 )
% 0.35/0.64 => ~ ( member_set_nat @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6324_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ! [X: product_prod_nat_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ X @ A3 )
% 0.35/0.64 => ~ ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6325_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ! [X: real] :
% 0.35/0.64 ( ( member_real @ X @ A3 )
% 0.35/0.64 => ~ ( member_real @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6326_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ! [X: $o] :
% 0.35/0.64 ( ( member_o @ X @ A3 )
% 0.35/0.64 => ~ ( member_o @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6327_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ! [X: nat] :
% 0.35/0.64 ( ( member_nat @ X @ A3 )
% 0.35/0.64 => ~ ( member_nat @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6328_disjoint__iff,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ! [X: int] :
% 0.35/0.64 ( ( member_int @ X @ A3 )
% 0.35/0.64 => ~ ( member_int @ X @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff
% 0.35/0.64 thf(fact_6329_Int__empty__left,axiom,
% 0.35/0.64 ! [B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_left
% 0.35/0.64 thf(fact_6330_Int__empty__left,axiom,
% 0.35/0.64 ! [B2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_left
% 0.35/0.64 thf(fact_6331_Int__empty__left,axiom,
% 0.35/0.64 ! [B2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_left
% 0.35/0.64 thf(fact_6332_Int__empty__left,axiom,
% 0.35/0.64 ! [B2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_left
% 0.35/0.64 thf(fact_6333_Int__empty__left,axiom,
% 0.35/0.64 ! [B2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ bot_bot_set_int @ B2 )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_left
% 0.35/0.64 thf(fact_6334_Int__empty__right,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_right
% 0.35/0.64 thf(fact_6335_Int__empty__right,axiom,
% 0.35/0.64 ! [A3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ A3 @ bot_bot_set_real )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_right
% 0.35/0.64 thf(fact_6336_Int__empty__right,axiom,
% 0.35/0.64 ! [A3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_right
% 0.35/0.64 thf(fact_6337_Int__empty__right,axiom,
% 0.35/0.64 ! [A3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_right
% 0.35/0.64 thf(fact_6338_Int__empty__right,axiom,
% 0.35/0.64 ! [A3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ A3 @ bot_bot_set_int )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Int_empty_right
% 0.35/0.64 thf(fact_6339_disjoint__iff__not__equal,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ! [X: product_prod_nat_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ X @ A3 )
% 0.35/0.64 => ! [Y: product_prod_nat_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ Y @ B2 )
% 0.35/0.64 => ( X != Y ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff_not_equal
% 0.35/0.64 thf(fact_6340_disjoint__iff__not__equal,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ! [X: real] :
% 0.35/0.64 ( ( member_real @ X @ A3 )
% 0.35/0.64 => ! [Y: real] :
% 0.35/0.64 ( ( member_real @ Y @ B2 )
% 0.35/0.64 => ( X != Y ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff_not_equal
% 0.35/0.64 thf(fact_6341_disjoint__iff__not__equal,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ! [X: $o] :
% 0.35/0.64 ( ( member_o @ X @ A3 )
% 0.35/0.64 => ! [Y: $o] :
% 0.35/0.64 ( ( member_o @ Y @ B2 )
% 0.35/0.64 => ( X = ( ~ Y ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff_not_equal
% 0.35/0.64 thf(fact_6342_disjoint__iff__not__equal,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ! [X: nat] :
% 0.35/0.64 ( ( member_nat @ X @ A3 )
% 0.35/0.64 => ! [Y: nat] :
% 0.35/0.64 ( ( member_nat @ Y @ B2 )
% 0.35/0.64 => ( X != Y ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff_not_equal
% 0.35/0.64 thf(fact_6343_disjoint__iff__not__equal,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ! [X: int] :
% 0.35/0.64 ( ( member_int @ X @ A3 )
% 0.35/0.64 => ! [Y: int] :
% 0.35/0.64 ( ( member_int @ Y @ B2 )
% 0.35/0.64 => ( X != Y ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_iff_not_equal
% 0.35/0.64 thf(fact_6344_Int__mono,axiom,
% 0.35/0.64 ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 0.35/0.64 => ( ( ord_less_eq_set_nat @ B2 @ D4 )
% 0.35/0.64 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D4 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_mono
% 0.35/0.64 thf(fact_6345_Int__mono,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ A3 @ C2 )
% 0.35/0.64 => ( ( ord_le3146513528884898305at_nat @ B2 @ D4 )
% 0.35/0.64 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D4 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_mono
% 0.35/0.64 thf(fact_6346_Int__mono,axiom,
% 0.35/0.64 ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 0.35/0.64 => ( ( ord_less_eq_set_int @ B2 @ D4 )
% 0.35/0.64 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( inf_inf_set_int @ C2 @ D4 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_mono
% 0.35/0.64 thf(fact_6347_Int__lower1,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower1
% 0.35/0.64 thf(fact_6348_Int__lower1,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower1
% 0.35/0.64 thf(fact_6349_Int__lower1,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower1
% 0.35/0.64 thf(fact_6350_Int__lower2,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ B2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower2
% 0.35/0.64 thf(fact_6351_Int__lower2,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ B2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower2
% 0.35/0.64 thf(fact_6352_Int__lower2,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ B2 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_lower2
% 0.35/0.64 thf(fact_6353_Int__absorb1,axiom,
% 0.35/0.64 ! [B2: set_nat,A3: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb1
% 0.35/0.64 thf(fact_6354_Int__absorb1,axiom,
% 0.35/0.64 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb1
% 0.35/0.64 thf(fact_6355_Int__absorb1,axiom,
% 0.35/0.64 ! [B2: set_int,A3: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb1
% 0.35/0.64 thf(fact_6356_Int__absorb2,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb2
% 0.35/0.64 thf(fact_6357_Int__absorb2,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb2
% 0.35/0.64 thf(fact_6358_Int__absorb2,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.35/0.64 => ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_absorb2
% 0.35/0.64 thf(fact_6359_Int__greatest,axiom,
% 0.35/0.64 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ C2 @ A3 )
% 0.35/0.64 => ( ( ord_less_eq_set_nat @ C2 @ B2 )
% 0.35/0.64 => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_greatest
% 0.35/0.64 thf(fact_6360_Int__greatest,axiom,
% 0.35/0.64 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
% 0.35/0.64 => ( ( ord_le3146513528884898305at_nat @ C2 @ B2 )
% 0.35/0.64 => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_greatest
% 0.35/0.64 thf(fact_6361_Int__greatest,axiom,
% 0.35/0.64 ! [C2: set_int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ C2 @ A3 )
% 0.35/0.64 => ( ( ord_less_eq_set_int @ C2 @ B2 )
% 0.35/0.64 => ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_greatest
% 0.35/0.64 thf(fact_6362_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o,P: $o > $o,Q: $o > $o] :
% 0.35/0.64 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: $o] :
% 0.35/0.64 ( ( member_o @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B2 @ ( collect_o @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6363_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real,P: real > $o,Q: real > $o] :
% 0.35/0.64 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: real] :
% 0.35/0.64 ( ( member_real @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6364_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_list_nat,B2: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.64 ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: list_nat] :
% 0.35/0.64 ( ( member_list_nat @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B2 @ ( collect_list_nat @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6365_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.64 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: set_nat] :
% 0.35/0.64 ( ( member_set_nat @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6366_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: nat] :
% 0.35/0.64 ( ( member_nat @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6367_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: product_prod_nat_nat] :
% 0.35/0.64 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B2 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6368_Int__Collect__mono,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int,P: int > $o,Q: int > $o] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 0.35/0.64 => ( ! [X5: int] :
% 0.35/0.64 ( ( member_int @ X5 @ A3 )
% 0.35/0.64 => ( ( P @ X5 )
% 0.35/0.64 => ( Q @ X5 ) ) )
% 0.35/0.64 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B2 @ ( collect_int @ Q ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Collect_mono
% 0.35/0.64 thf(fact_6369_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: real,A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( member_real @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.64 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_real @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6370_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: $o,A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( member_o @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.64 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_o @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6371_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( ( member_set_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_set_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6372_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: int,A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( member_int @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.64 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_int @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6373_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( member_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6374_Int__insert__right,axiom,
% 0.35/0.64 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) )
% 0.35/0.64 & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_right
% 0.35/0.64 thf(fact_6375_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: real,C2: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( member_real @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_real @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6376_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: $o,C2: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( member_o @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_o @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6377_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( ( member_set_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_set_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6378_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: int,C2: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( member_int @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_int @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6379_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( member_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6380_Int__insert__left,axiom,
% 0.35/0.64 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( member8440522571783428010at_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) )
% 0.35/0.64 & ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
% 0.35/0.64 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_insert_left
% 0.35/0.64 thf(fact_6381_Un__Int__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib2
% 0.35/0.64 thf(fact_6382_Un__Int__distrib2,axiom,
% 0.35/0.64 ! [B2: set_nat,C2: set_nat,A3: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A3 ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib2
% 0.35/0.64 thf(fact_6383_Un__Int__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ A3 ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib2
% 0.35/0.64 thf(fact_6384_Un__Int__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ A3 ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib2
% 0.35/0.64 thf(fact_6385_Int__Un__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib2
% 0.35/0.64 thf(fact_6386_Int__Un__distrib2,axiom,
% 0.35/0.64 ! [B2: set_nat,C2: set_nat,A3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib2
% 0.35/0.64 thf(fact_6387_Int__Un__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ A3 ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib2
% 0.35/0.64 thf(fact_6388_Int__Un__distrib2,axiom,
% 0.35/0.64 ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A3 )
% 0.35/0.64 = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ A3 ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib2
% 0.35/0.64 thf(fact_6389_Un__Int__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib
% 0.35/0.64 thf(fact_6390_Un__Int__distrib,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib
% 0.35/0.64 thf(fact_6391_Un__Int__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib
% 0.35/0.64 thf(fact_6392_Un__Int__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_distrib
% 0.35/0.64 thf(fact_6393_Int__Un__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib
% 0.35/0.64 thf(fact_6394_Int__Un__distrib,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib
% 0.35/0.64 thf(fact_6395_Int__Un__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib
% 0.35/0.64 thf(fact_6396_Int__Un__distrib,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Un_distrib
% 0.35/0.64 thf(fact_6397_Un__Int__crazy,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_crazy
% 0.35/0.64 thf(fact_6398_Un__Int__crazy,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A3 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_crazy
% 0.35/0.64 thf(fact_6399_Un__Int__crazy,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_crazy
% 0.35/0.64 thf(fact_6400_Un__Int__crazy,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_crazy
% 0.35/0.64 thf(fact_6401_Diff__Int__distrib2,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int_distrib2
% 0.35/0.64 thf(fact_6402_Diff__Int__distrib2,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int_distrib2
% 0.35/0.64 thf(fact_6403_Diff__Int__distrib,axiom,
% 0.35/0.64 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int_distrib
% 0.35/0.64 thf(fact_6404_Diff__Int__distrib,axiom,
% 0.35/0.64 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int_distrib
% 0.35/0.64 thf(fact_6405_Diff__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Diff_Int
% 0.35/0.64 thf(fact_6406_Diff__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Diff_Int
% 0.35/0.64 thf(fact_6407_Diff__Int2,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int2
% 0.35/0.64 thf(fact_6408_Diff__Int2,axiom,
% 0.35/0.64 ! [A3: set_nat,C2: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ B2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int2
% 0.35/0.64 thf(fact_6409_Int__Diff,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff
% 0.35/0.64 thf(fact_6410_Int__Diff,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff
% 0.35/0.64 thf(fact_6411_option_Osel,axiom,
% 0.35/0.64 ! [X22: nat] :
% 0.35/0.64 ( ( the_nat @ ( some_nat @ X22 ) )
% 0.35/0.64 = X22 ) ).
% 0.35/0.64
% 0.35/0.64 % option.sel
% 0.35/0.64 thf(fact_6412_option_Osel,axiom,
% 0.35/0.64 ! [X22: product_prod_nat_nat] :
% 0.35/0.64 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.35/0.64 = X22 ) ).
% 0.35/0.64
% 0.35/0.64 % option.sel
% 0.35/0.64 thf(fact_6413_option_Osel,axiom,
% 0.35/0.64 ! [X22: num] :
% 0.35/0.64 ( ( the_num @ ( some_num @ X22 ) )
% 0.35/0.64 = X22 ) ).
% 0.35/0.64
% 0.35/0.64 % option.sel
% 0.35/0.64 thf(fact_6414_option_Oexpand,axiom,
% 0.35/0.64 ! [Option: option_nat,Option2: option_nat] :
% 0.35/0.64 ( ( ( Option = none_nat )
% 0.35/0.64 = ( Option2 = none_nat ) )
% 0.35/0.64 => ( ( ( Option != none_nat )
% 0.35/0.64 => ( ( Option2 != none_nat )
% 0.35/0.64 => ( ( the_nat @ Option )
% 0.35/0.64 = ( the_nat @ Option2 ) ) ) )
% 0.35/0.64 => ( Option = Option2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.expand
% 0.35/0.64 thf(fact_6415_option_Oexpand,axiom,
% 0.35/0.64 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.35/0.64 ( ( ( Option = none_P5556105721700978146at_nat )
% 0.35/0.64 = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.35/0.64 => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.64 => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.35/0.64 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.35/0.64 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.35/0.64 => ( Option = Option2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.expand
% 0.35/0.64 thf(fact_6416_option_Oexpand,axiom,
% 0.35/0.64 ! [Option: option_num,Option2: option_num] :
% 0.35/0.64 ( ( ( Option = none_num )
% 0.35/0.64 = ( Option2 = none_num ) )
% 0.35/0.64 => ( ( ( Option != none_num )
% 0.35/0.64 => ( ( Option2 != none_num )
% 0.35/0.64 => ( ( the_num @ Option )
% 0.35/0.64 = ( the_num @ Option2 ) ) ) )
% 0.35/0.64 => ( Option = Option2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.expand
% 0.35/0.64 thf(fact_6417_inf__cancel__left1,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ A2 ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ B3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left1
% 0.35/0.64 thf(fact_6418_inf__cancel__left1,axiom,
% 0.35/0.64 ! [X2: set_real,A2: set_real,B3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( inf_inf_set_real @ X2 @ A2 ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ B3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left1
% 0.35/0.64 thf(fact_6419_inf__cancel__left1,axiom,
% 0.35/0.64 ! [X2: set_o,A2: set_o,B3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( inf_inf_set_o @ X2 @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ B3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left1
% 0.35/0.64 thf(fact_6420_inf__cancel__left1,axiom,
% 0.35/0.64 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left1
% 0.35/0.64 thf(fact_6421_inf__cancel__left1,axiom,
% 0.35/0.64 ! [X2: set_int,A2: set_int,B3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( inf_inf_set_int @ X2 @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ B3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left1
% 0.35/0.64 thf(fact_6422_inf__cancel__left2,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ A2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ B3 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left2
% 0.35/0.64 thf(fact_6423_inf__cancel__left2,axiom,
% 0.35/0.64 ! [X2: set_real,A2: set_real,B3: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ A2 ) @ ( inf_inf_set_real @ X2 @ B3 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left2
% 0.35/0.64 thf(fact_6424_inf__cancel__left2,axiom,
% 0.35/0.64 ! [X2: set_o,A2: set_o,B3: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ A2 ) @ ( inf_inf_set_o @ X2 @ B3 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left2
% 0.35/0.64 thf(fact_6425_inf__cancel__left2,axiom,
% 0.35/0.64 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A2 ) @ ( inf_inf_set_nat @ X2 @ B3 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left2
% 0.35/0.64 thf(fact_6426_inf__cancel__left2,axiom,
% 0.35/0.64 ! [X2: set_int,A2: set_int,B3: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ A2 ) @ ( inf_inf_set_int @ X2 @ B3 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_cancel_left2
% 0.35/0.64 thf(fact_6427_Diff__triv,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 => ( ( minus_1356011639430497352at_nat @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_triv
% 0.35/0.64 thf(fact_6428_Diff__triv,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 => ( ( minus_minus_set_real @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_triv
% 0.35/0.64 thf(fact_6429_Diff__triv,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 => ( ( minus_minus_set_o @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_triv
% 0.35/0.64 thf(fact_6430_Diff__triv,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 => ( ( minus_minus_set_int @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_triv
% 0.35/0.64 thf(fact_6431_Diff__triv,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 => ( ( minus_minus_set_nat @ A3 @ B2 )
% 0.35/0.64 = A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_triv
% 0.35/0.64 thf(fact_6432_Int__Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_disjoint
% 0.35/0.64 thf(fact_6433_Int__Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ ( inf_inf_set_real @ A3 @ B2 ) @ ( minus_minus_set_real @ A3 @ B2 ) )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_disjoint
% 0.35/0.64 thf(fact_6434_Int__Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B2 ) @ ( minus_minus_set_o @ A3 @ B2 ) )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_disjoint
% 0.35/0.64 thf(fact_6435_Int__Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_disjoint
% 0.35/0.64 thf(fact_6436_Int__Diff__disjoint,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_disjoint
% 0.35/0.64 thf(fact_6437_Un__Int__assoc__eq,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ C2 @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_assoc_eq
% 0.35/0.64 thf(fact_6438_Un__Int__assoc__eq,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
% 0.35/0.64 = ( ord_less_eq_set_nat @ C2 @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_assoc_eq
% 0.35/0.64 thf(fact_6439_Un__Int__assoc__eq,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) )
% 0.35/0.64 = ( ord_le3000389064537975527at_nat @ C2 @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_assoc_eq
% 0.35/0.64 thf(fact_6440_Un__Int__assoc__eq,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) )
% 0.35/0.64 = ( ord_le1268244103169919719at_nat @ C2 @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_assoc_eq
% 0.35/0.64 thf(fact_6441_Un__Int__assoc__eq,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int,C2: set_int] :
% 0.35/0.64 ( ( ( sup_sup_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ C2 )
% 0.35/0.64 = ( inf_inf_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) )
% 0.35/0.64 = ( ord_less_eq_set_int @ C2 @ A3 ) ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Int_assoc_eq
% 0.35/0.64 thf(fact_6442_Un__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Diff_Int
% 0.35/0.64 thf(fact_6443_Un__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Diff_Int
% 0.35/0.64 thf(fact_6444_Un__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Diff_Int
% 0.35/0.64 thf(fact_6445_Un__Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Un_Diff_Int
% 0.35/0.64 thf(fact_6446_Int__Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_Un
% 0.35/0.64 thf(fact_6447_Int__Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_Un
% 0.35/0.64 thf(fact_6448_Int__Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_Un
% 0.35/0.64 thf(fact_6449_Int__Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = A3 ) ).
% 0.35/0.64
% 0.35/0.64 % Int_Diff_Un
% 0.35/0.64 thf(fact_6450_Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int
% 0.35/0.64 thf(fact_6451_Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( minus_8321449233255521966at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int
% 0.35/0.64 thf(fact_6452_Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( minus_3314409938677909166at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int
% 0.35/0.64 thf(fact_6453_Diff__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Int
% 0.35/0.64 thf(fact_6454_Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Un
% 0.35/0.64 thf(fact_6455_Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( minus_8321449233255521966at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Un
% 0.35/0.64 thf(fact_6456_Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( minus_3314409938677909166at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Un
% 0.35/0.64 thf(fact_6457_Diff__Un,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 0.35/0.64 ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_Un
% 0.35/0.64 thf(fact_6458_Compl__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Int
% 0.35/0.64 thf(fact_6459_Compl__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Int
% 0.35/0.64 thf(fact_6460_Compl__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( uminus4384627049435823934at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Int
% 0.35/0.64 thf(fact_6461_Compl__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Int
% 0.35/0.64 thf(fact_6462_Compl__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Un
% 0.35/0.64 thf(fact_6463_Compl__Un,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Un
% 0.35/0.64 thf(fact_6464_Compl__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( uminus4384627049435823934at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_in4302113700860409141at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Un
% 0.35/0.64 thf(fact_6465_Compl__Un,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Compl_Un
% 0.35/0.64 thf(fact_6466_Diff__eq,axiom,
% 0.35/0.64 ( minus_1356011639430497352at_nat
% 0.35/0.64 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ A6 @ ( uminus6524753893492686040at_nat @ B6 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_eq
% 0.35/0.64 thf(fact_6467_Diff__eq,axiom,
% 0.35/0.64 ( minus_minus_set_nat
% 0.35/0.64 = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ A6 @ ( uminus5710092332889474511et_nat @ B6 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % Diff_eq
% 0.35/0.64 thf(fact_6468_option_Oexhaust__sel,axiom,
% 0.35/0.64 ! [Option: option_nat] :
% 0.35/0.64 ( ( Option != none_nat )
% 0.35/0.64 => ( Option
% 0.35/0.64 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.exhaust_sel
% 0.35/0.64 thf(fact_6469_option_Oexhaust__sel,axiom,
% 0.35/0.64 ! [Option: option4927543243414619207at_nat] :
% 0.35/0.64 ( ( Option != none_P5556105721700978146at_nat )
% 0.35/0.64 => ( Option
% 0.35/0.64 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.exhaust_sel
% 0.35/0.64 thf(fact_6470_option_Oexhaust__sel,axiom,
% 0.35/0.64 ! [Option: option_num] :
% 0.35/0.64 ( ( Option != none_num )
% 0.35/0.64 => ( Option
% 0.35/0.64 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % option.exhaust_sel
% 0.35/0.64 thf(fact_6471_inf__shunt,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf_shunt
% 0.35/0.64 thf(fact_6472_inf__shunt,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf_shunt
% 0.35/0.64 thf(fact_6473_inf__shunt,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ord_less_eq_set_o @ X2 @ ( uminus_uminus_set_o @ Y3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf_shunt
% 0.35/0.64 thf(fact_6474_inf__shunt,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf_shunt
% 0.35/0.64 thf(fact_6475_inf__shunt,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf_shunt
% 0.35/0.64 thf(fact_6476_shunt1,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt1
% 0.35/0.64 thf(fact_6477_shunt1,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt1
% 0.35/0.64 thf(fact_6478_shunt1,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt1
% 0.35/0.64 thf(fact_6479_shunt1,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt1
% 0.35/0.64 thf(fact_6480_shunt1,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt1
% 0.35/0.64 thf(fact_6481_shunt2,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) @ Z )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt2
% 0.35/0.64 thf(fact_6482_shunt2,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) @ Z )
% 0.35/0.64 = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt2
% 0.35/0.64 thf(fact_6483_shunt2,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ ( uminus4384627049435823934at_nat @ Y3 ) ) @ Z )
% 0.35/0.64 = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt2
% 0.35/0.64 thf(fact_6484_shunt2,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ ( uminus935396558254630718at_nat @ Y3 ) ) @ Z )
% 0.35/0.64 = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt2
% 0.35/0.64 thf(fact_6485_shunt2,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) @ Z )
% 0.35/0.64 = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % shunt2
% 0.35/0.64 thf(fact_6486_sup__neg__inf,axiom,
% 0.35/0.64 ! [P6: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ P6 @ ( sup_su6327502436637775413at_nat @ Q3 @ R2 ) )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P6 @ ( uminus6524753893492686040at_nat @ Q3 ) ) @ R2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_neg_inf
% 0.35/0.64 thf(fact_6487_sup__neg__inf,axiom,
% 0.35/0.64 ! [P6: set_nat,Q3: set_nat,R2: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ P6 @ ( sup_sup_set_nat @ Q3 @ R2 ) )
% 0.35/0.64 = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P6 @ ( uminus5710092332889474511et_nat @ Q3 ) ) @ R2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_neg_inf
% 0.35/0.64 thf(fact_6488_sup__neg__inf,axiom,
% 0.35/0.64 ! [P6: set_Pr8693737435421807431at_nat,Q3: set_Pr8693737435421807431at_nat,R2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ord_le3000389064537975527at_nat @ P6 @ ( sup_su718114333110466843at_nat @ Q3 @ R2 ) )
% 0.35/0.64 = ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ P6 @ ( uminus4384627049435823934at_nat @ Q3 ) ) @ R2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_neg_inf
% 0.35/0.64 thf(fact_6489_sup__neg__inf,axiom,
% 0.35/0.64 ! [P6: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ord_le1268244103169919719at_nat @ P6 @ ( sup_su5525570899277871387at_nat @ Q3 @ R2 ) )
% 0.35/0.64 = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P6 @ ( uminus935396558254630718at_nat @ Q3 ) ) @ R2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_neg_inf
% 0.35/0.64 thf(fact_6490_sup__neg__inf,axiom,
% 0.35/0.64 ! [P6: set_int,Q3: set_int,R2: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ P6 @ ( sup_sup_set_int @ Q3 @ R2 ) )
% 0.35/0.64 = ( ord_less_eq_set_int @ ( inf_inf_set_int @ P6 @ ( uminus1532241313380277803et_int @ Q3 ) ) @ R2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_neg_inf
% 0.35/0.64 thf(fact_6491_disjoint__eq__subset__Compl,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 = ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_eq_subset_Compl
% 0.35/0.64 thf(fact_6492_disjoint__eq__subset__Compl,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_eq_subset_Compl
% 0.35/0.64 thf(fact_6493_disjoint__eq__subset__Compl,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_eq_subset_Compl
% 0.35/0.64 thf(fact_6494_disjoint__eq__subset__Compl,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_eq_subset_Compl
% 0.35/0.64 thf(fact_6495_disjoint__eq__subset__Compl,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ B2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % disjoint_eq_subset_Compl
% 0.35/0.64 thf(fact_6496_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_list_nat,B2: set_list_nat] :
% 0.35/0.64 ( ( finite8100373058378681591st_nat @ A3 )
% 0.35/0.64 => ( ( finite8100373058378681591st_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6497_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( finite1152437895449049373et_nat @ A3 )
% 0.35/0.64 => ( ( finite1152437895449049373et_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6498_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( finite_finite_int @ A3 )
% 0.35/0.64 => ( ( finite_finite_int @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6499_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_complex,B2: set_complex] :
% 0.35/0.64 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.64 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6500_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.35/0.64 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.64 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6501_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.64 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6502_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( finite_finite_nat @ A3 )
% 0.35/0.64 => ( ( finite_finite_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6503_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( finite4392333629123659920at_nat @ A3 )
% 0.35/0.64 => ( ( finite4392333629123659920at_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( finite1207074278014112911at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6504_card__Un__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( finite4343798906461161616at_nat @ A3 )
% 0.35/0.64 => ( ( finite4343798906461161616at_nat @ B2 )
% 0.35/0.64 => ( ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( finite3771342082235030671at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_Int
% 0.35/0.64 thf(fact_6505_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_list_nat,B2: set_list_nat] :
% 0.35/0.64 ( ( finite8100373058378681591st_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6506_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6507_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( finite_finite_int @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6508_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_complex,B2: set_complex] :
% 0.35/0.64 ( ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6509_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.35/0.64 ( ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6510_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6511_card__Diff__subset__Int,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 0.35/0.64 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Diff_subset_Int
% 0.35/0.64 thf(fact_6512_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_complex,B2: set_complex] :
% 0.35/0.64 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.64 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_complex )
% 0.35/0.64 => ( ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6513_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.35/0.64 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.64 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.64 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 0.35/0.64 = bot_bo7653980558646680370d_enat )
% 0.35/0.64 => ( ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6514_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_real,B2: set_real] :
% 0.35/0.64 ( ( finite_finite_real @ A3 )
% 0.35/0.64 => ( ( finite_finite_real @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 => ( ( finite_card_real @ ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6515_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_o,B2: set_o] :
% 0.35/0.64 ( ( finite_finite_o @ A3 )
% 0.35/0.64 => ( ( finite_finite_o @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_o @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 => ( ( finite_card_o @ ( sup_sup_set_o @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6516_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.64 ( ( finite_finite_nat @ A3 )
% 0.35/0.64 => ( ( finite_finite_nat @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 => ( ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6517_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_int,B2: set_int] :
% 0.35/0.64 ( ( finite_finite_int @ A3 )
% 0.35/0.64 => ( ( finite_finite_int @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_int @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 => ( ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6518_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_list_nat,B2: set_list_nat] :
% 0.35/0.64 ( ( finite8100373058378681591st_nat @ A3 )
% 0.35/0.64 => ( ( finite8100373058378681591st_nat @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_list_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_list_nat )
% 0.35/0.64 => ( ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6519_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_set_nat,B2: set_set_nat] :
% 0.35/0.64 ( ( finite1152437895449049373et_nat @ A3 )
% 0.35/0.64 => ( ( finite1152437895449049373et_nat @ B2 )
% 0.35/0.64 => ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bot_set_set_nat )
% 0.35/0.64 => ( ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6520_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.64 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.64 => ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat )
% 0.35/0.64 => ( ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6521_card__Un__disjoint,axiom,
% 0.35/0.64 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( finite4392333629123659920at_nat @ A3 )
% 0.35/0.64 => ( ( finite4392333629123659920at_nat @ B2 )
% 0.35/0.64 => ( ( ( inf_in4302113700860409141at_nat @ A3 @ B2 )
% 0.35/0.64 = bot_bo5327735625951526323at_nat )
% 0.35/0.64 => ( ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.64 = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % card_Un_disjoint
% 0.35/0.64 thf(fact_6522_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.35/0.64 ! [X2: nat,N: nat,M: nat] :
% 0.35/0.64 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.64 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.64 => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % VEBT_internal.exp_split_high_low(2)
% 0.35/0.64 thf(fact_6523_invar__vebt_Ointros_I4_J,axiom,
% 0.35/0.64 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.35/0.64 ( ! [X5: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( M = N )
% 0.35/0.64 => ( ( Deg
% 0.35/0.64 = ( plus_plus_nat @ N @ M ) )
% 0.35/0.64 => ( ! [I2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.35/0.64 => ( ( ( Mi = Ma )
% 0.35/0.64 => ! [X5: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.64 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.35/0.64 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.35/0.64 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.64 => ( ( ( Mi != Ma )
% 0.35/0.64 => ! [I2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.35/0.64 = I2 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.35/0.64 & ! [X5: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.35/0.64 = I2 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi @ X5 )
% 0.35/0.64 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % invar_vebt.intros(4)
% 0.35/0.64 thf(fact_6524_invar__vebt_Ointros_I5_J,axiom,
% 0.35/0.64 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.35/0.64 ( ! [X5: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X5 @ N ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( M
% 0.35/0.64 = ( suc @ N ) )
% 0.35/0.64 => ( ( Deg
% 0.35/0.64 = ( plus_plus_nat @ N @ M ) )
% 0.35/0.64 => ( ! [I2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.35/0.64 => ( ( ( Mi = Ma )
% 0.35/0.64 => ! [X5: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.64 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 0.35/0.64 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.35/0.64 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.64 => ( ( ( Mi != Ma )
% 0.35/0.64 => ! [I2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 0.35/0.64 = I2 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 0.35/0.64 & ! [X5: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 0.35/0.64 = I2 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi @ X5 )
% 0.35/0.64 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % invar_vebt.intros(5)
% 0.35/0.64 thf(fact_6525_invar__vebt_Osimps,axiom,
% 0.35/0.64 ( vEBT_invar_vebt
% 0.35/0.64 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 0.35/0.64 ( ( ? [A4: $o,B4: $o] :
% 0.35/0.64 ( A12
% 0.35/0.64 = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.35/0.64 & ( A23
% 0.35/0.64 = ( suc @ zero_zero_nat ) ) )
% 0.35/0.64 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
% 0.35/0.64 ( ( A12
% 0.35/0.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.35/0.64 & ( vEBT_invar_vebt @ Summary4 @ N2 )
% 0.35/0.64 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.64 & ( A23
% 0.35/0.64 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.35/0.64 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.64 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
% 0.35/0.64 ( ( A12
% 0.35/0.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.35/0.64 & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
% 0.35/0.64 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.64 & ( A23
% 0.35/0.64 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.35/0.64 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.64 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.35/0.64 ( ( A12
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.35/0.64 & ( vEBT_invar_vebt @ Summary4 @ N2 )
% 0.35/0.64 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.64 & ( A23
% 0.35/0.64 = ( plus_plus_nat @ N2 @ N2 ) )
% 0.35/0.64 & ! [I4: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
% 0.35/0.64 & ( ( Mi3 = Ma3 )
% 0.35/0.64 => ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.64 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.64 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.35/0.64 & ( ( Mi3 != Ma3 )
% 0.35/0.64 => ! [I4: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.35/0.64 = I4 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.35/0.64 & ! [X: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.35/0.64 = I4 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.64 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 0.35/0.64 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.35/0.64 ( ( A12
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
% 0.35/0.64 & ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X @ N2 ) )
% 0.35/0.64 & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
% 0.35/0.64 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.64 & ( A23
% 0.35/0.64 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 0.35/0.64 & ! [I4: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
% 0.35/0.64 & ( ( Mi3 = Ma3 )
% 0.35/0.64 => ! [X: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 0.35/0.64 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.64 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.64 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 0.35/0.64 & ( ( Mi3 != Ma3 )
% 0.35/0.64 => ! [I4: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 0.35/0.64 = I4 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 0.35/0.64 & ! [X: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 0.35/0.64 = I4 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.64 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % invar_vebt.simps
% 0.35/0.64 thf(fact_6526_invar__vebt_Ocases,axiom,
% 0.35/0.64 ! [A13: vEBT_VEBT,A24: nat] :
% 0.35/0.64 ( ( vEBT_invar_vebt @ A13 @ A24 )
% 0.35/0.64 => ( ( ? [A: $o,B: $o] :
% 0.35/0.64 ( A13
% 0.35/0.64 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.64 => ( A24
% 0.35/0.64 != ( suc @ zero_zero_nat ) ) )
% 0.35/0.64 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.35/0.64 ( ( A13
% 0.35/0.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.64 => ( ( A24 = Deg2 )
% 0.35/0.64 => ( ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( M4 = N3 )
% 0.35/0.64 => ( ( Deg2
% 0.35/0.64 = ( plus_plus_nat @ N3 @ M4 ) )
% 0.35/0.64 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.35/0.64 => ~ ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.35/0.64 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 0.35/0.64 ( ( A13
% 0.35/0.64 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.64 => ( ( A24 = Deg2 )
% 0.35/0.64 => ( ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( M4
% 0.35/0.64 = ( suc @ N3 ) )
% 0.35/0.64 => ( ( Deg2
% 0.35/0.64 = ( plus_plus_nat @ N3 @ M4 ) )
% 0.35/0.64 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.35/0.64 => ~ ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.35/0.64 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.35/0.64 ( ( A13
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.64 => ( ( A24 = Deg2 )
% 0.35/0.64 => ( ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( M4 = N3 )
% 0.35/0.64 => ( ( Deg2
% 0.35/0.64 = ( plus_plus_nat @ N3 @ M4 ) )
% 0.35/0.64 => ( ! [I3: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.35/0.64 => ( ( ( Mi2 = Ma2 )
% 0.35/0.64 => ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 0.35/0.64 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.64 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.64 => ~ ( ( Mi2 != Ma2 )
% 0.35/0.64 => ! [I3: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.35/0.64 = I3 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.35/0.64 & ! [X4: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 0.35/0.64 = I3 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi2 @ X4 )
% 0.35/0.64 & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.64 => ~ ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.35/0.64 ( ( A13
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.64 => ( ( A24 = Deg2 )
% 0.35/0.64 => ( ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 0.35/0.64 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 0.35/0.64 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.64 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( M4
% 0.35/0.64 = ( suc @ N3 ) )
% 0.35/0.64 => ( ( Deg2
% 0.35/0.64 = ( plus_plus_nat @ N3 @ M4 ) )
% 0.35/0.64 => ( ! [I3: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 0.35/0.64 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 0.35/0.64 => ( ( ( Mi2 = Ma2 )
% 0.35/0.64 => ! [X4: vEBT_VEBT] :
% 0.35/0.64 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.64 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 0.35/0.64 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.35/0.64 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.64 => ~ ( ( Mi2 != Ma2 )
% 0.35/0.64 => ! [I3: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 0.35/0.64 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.35/0.64 = I3 )
% 0.35/0.64 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.35/0.64 & ! [X4: nat] :
% 0.35/0.64 ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 0.35/0.64 = I3 )
% 0.35/0.64 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi2 @ X4 )
% 0.35/0.64 & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % invar_vebt.cases
% 0.35/0.64 thf(fact_6527_in__children__def,axiom,
% 0.35/0.64 ( vEBT_V5917875025757280293ildren
% 0.35/0.64 = ( ^ [N2: nat,TreeList4: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % in_children_def
% 0.35/0.64 thf(fact_6528_del__x__mi__lets__in__not__minNull,axiom,
% 0.35/0.64 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.64 ( ( ( X2 = Mi )
% 0.35/0.64 & ( ord_less_nat @ X2 @ Ma ) )
% 0.35/0.64 => ( ( Mi != Ma )
% 0.35/0.64 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.64 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.64 = H2 )
% 0.35/0.64 => ( ( Xn
% 0.35/0.64 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.64 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.64 = L )
% 0.35/0.64 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.64 => ( ( Newnode
% 0.35/0.64 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.64 => ( ( Newlist
% 0.35/0.64 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.64 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.64 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % del_x_mi_lets_in_not_minNull
% 0.35/0.64 thf(fact_6529_del__x__not__mi__newnode__not__nil,axiom,
% 0.35/0.64 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.64 ( ( ( ord_less_nat @ Mi @ X2 )
% 0.35/0.64 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 0.35/0.64 => ( ( Mi != Ma )
% 0.35/0.64 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.64 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.64 = H2 )
% 0.35/0.64 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.64 = L )
% 0.35/0.64 => ( ( Newnode
% 0.35/0.64 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.64 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.64 => ( ( Newlist
% 0.35/0.64 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.64 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.64 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % del_x_not_mi_newnode_not_nil
% 0.35/0.64 thf(fact_6530_set__bit__0,axiom,
% 0.35/0.64 ! [A2: code_integer] :
% 0.35/0.64 ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A2 )
% 0.35/0.64 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_0
% 0.35/0.64 thf(fact_6531_set__bit__0,axiom,
% 0.35/0.64 ! [A2: nat] :
% 0.35/0.64 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
% 0.35/0.64 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_0
% 0.35/0.64 thf(fact_6532_set__bit__0,axiom,
% 0.35/0.64 ! [A2: int] :
% 0.35/0.64 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
% 0.35/0.64 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_0
% 0.35/0.64 thf(fact_6533_enat__ord__number_I1_J,axiom,
% 0.35/0.64 ! [M: num,N: num] :
% 0.35/0.64 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.64 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % enat_ord_number(1)
% 0.35/0.64 thf(fact_6534_set__bit__nonnegative__int__iff,axiom,
% 0.35/0.64 ! [N: nat,K: int] :
% 0.35/0.64 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 0.35/0.64 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_nonnegative_int_iff
% 0.35/0.64 thf(fact_6535_set__bit__negative__int__iff,axiom,
% 0.35/0.64 ! [N: nat,K: int] :
% 0.35/0.64 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.64 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_negative_int_iff
% 0.35/0.64 thf(fact_6536_list__update__beyond,axiom,
% 0.35/0.64 ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 0.35/0.64 => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
% 0.35/0.64 = Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_beyond
% 0.35/0.64 thf(fact_6537_list__update__beyond,axiom,
% 0.35/0.64 ! [Xs: list_o,I: nat,X2: $o] :
% 0.35/0.64 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 0.35/0.64 => ( ( list_update_o @ Xs @ I @ X2 )
% 0.35/0.64 = Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_beyond
% 0.35/0.64 thf(fact_6538_list__update__beyond,axiom,
% 0.35/0.64 ! [Xs: list_nat,I: nat,X2: nat] :
% 0.35/0.64 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 0.35/0.64 => ( ( list_update_nat @ Xs @ I @ X2 )
% 0.35/0.64 = Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_beyond
% 0.35/0.64 thf(fact_6539_enat__ord__number_I2_J,axiom,
% 0.35/0.64 ! [M: num,N: num] :
% 0.35/0.64 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 0.35/0.64 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % enat_ord_number(2)
% 0.35/0.64 thf(fact_6540_nth__list__update__eq,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_int,X2: int] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ I )
% 0.35/0.64 = X2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update_eq
% 0.35/0.64 thf(fact_6541_nth__list__update__eq,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ I )
% 0.35/0.64 = X2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update_eq
% 0.35/0.64 thf(fact_6542_nth__list__update__eq,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_o,X2: $o] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ I )
% 0.35/0.64 = X2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update_eq
% 0.35/0.64 thf(fact_6543_nth__list__update__eq,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_nat,X2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I )
% 0.35/0.64 = X2 ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update_eq
% 0.35/0.64 thf(fact_6544_set__swap,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_int,J: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 0.35/0.64 = ( set_int2 @ Xs ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_swap
% 0.35/0.64 thf(fact_6545_set__swap,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 0.35/0.64 = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_swap
% 0.35/0.64 thf(fact_6546_set__swap,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_o,J: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 0.35/0.64 = ( set_o2 @ Xs ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_swap
% 0.35/0.64 thf(fact_6547_set__swap,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_nat,J: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 0.35/0.64 = ( set_nat2 @ Xs ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_swap
% 0.35/0.64 thf(fact_6548_add__diff__assoc__enat,axiom,
% 0.35/0.64 ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
% 0.35/0.64 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 0.35/0.64 => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 0.35/0.64 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % add_diff_assoc_enat
% 0.35/0.64 thf(fact_6549_ile0__eq,axiom,
% 0.35/0.64 ! [N: extended_enat] :
% 0.35/0.64 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 0.35/0.64 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.35/0.64
% 0.35/0.64 % ile0_eq
% 0.35/0.64 thf(fact_6550_i0__lb,axiom,
% 0.35/0.64 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 0.35/0.64
% 0.35/0.64 % i0_lb
% 0.35/0.64 thf(fact_6551_set__bit__greater__eq,axiom,
% 0.35/0.64 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_bit_greater_eq
% 0.35/0.64 thf(fact_6552_list__update__code_I2_J,axiom,
% 0.35/0.64 ! [X2: int,Xs: list_int,Y3: int] :
% 0.35/0.64 ( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 0.35/0.64 = ( cons_int @ Y3 @ Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_code(2)
% 0.35/0.64 thf(fact_6553_list__update__code_I2_J,axiom,
% 0.35/0.64 ! [X2: nat,Xs: list_nat,Y3: nat] :
% 0.35/0.64 ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 0.35/0.64 = ( cons_nat @ Y3 @ Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_code(2)
% 0.35/0.64 thf(fact_6554_list__update__code_I2_J,axiom,
% 0.35/0.64 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 0.35/0.64 ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 0.35/0.64 = ( cons_VEBT_VEBT @ Y3 @ Xs ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_code(2)
% 0.35/0.64 thf(fact_6555_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_real,A3: set_real,X2: real,I: nat] :
% 0.35/0.64 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_real @ X2 @ A3 )
% 0.35/0.64 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6556_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_o,A3: set_o,X2: $o,I: nat] :
% 0.35/0.64 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_o @ X2 @ A3 )
% 0.35/0.64 => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6557_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_set_nat,A3: set_set_nat,X2: set_nat,I: nat] :
% 0.35/0.64 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.64 => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6558_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_nat,A3: set_nat,X2: nat,I: nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_nat @ X2 @ A3 )
% 0.35/0.64 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6559_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
% 0.35/0.64 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_VEBT_VEBT @ X2 @ A3 )
% 0.35/0.64 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6560_set__update__subsetI,axiom,
% 0.35/0.64 ! [Xs: list_int,A3: set_int,X2: int,I: nat] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
% 0.35/0.64 => ( ( member_int @ X2 @ A3 )
% 0.35/0.64 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subsetI
% 0.35/0.64 thf(fact_6561_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_P6011104703257516679at_nat,I: nat,X2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X2 ) ) @ ( insert8211810215607154385at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6562_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_real,I: nat,X2: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ ( insert_real @ X2 @ ( set_real2 @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6563_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_o,I: nat,X2: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ ( insert_o @ X2 @ ( set_o2 @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6564_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_nat,I: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ ( insert_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6565_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6566_set__update__subset__insert,axiom,
% 0.35/0.64 ! [Xs: list_int,I: nat,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ ( insert_int @ X2 @ ( set_int2 @ Xs ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_subset_insert
% 0.35/0.64 thf(fact_6567_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_real,X2: real] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 0.35/0.64 => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6568_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_set_nat,X2: set_nat] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.35/0.64 => ( member_set_nat @ X2 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6569_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_int,X2: int] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6570_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6571_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_o,X2: $o] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6572_set__update__memI,axiom,
% 0.35/0.64 ! [N: nat,Xs: list_nat,X2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % set_update_memI
% 0.35/0.64 thf(fact_6573_nth__list__update,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_int,J: nat,X2: int] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( ( ( I = J )
% 0.35/0.64 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = X2 ) )
% 0.35/0.64 & ( ( I != J )
% 0.35/0.64 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update
% 0.35/0.64 thf(fact_6574_nth__list__update,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( ( ( I = J )
% 0.35/0.64 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = X2 ) )
% 0.35/0.64 & ( ( I != J )
% 0.35/0.64 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update
% 0.35/0.64 thf(fact_6575_nth__list__update,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_o,X2: $o,J: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = ( ( ( I = J )
% 0.35/0.64 => X2 )
% 0.35/0.64 & ( ( I != J )
% 0.35/0.64 => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update
% 0.35/0.64 thf(fact_6576_nth__list__update,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_nat,J: nat,X2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( ( ( I = J )
% 0.35/0.64 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = X2 ) )
% 0.35/0.64 & ( ( I != J )
% 0.35/0.64 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
% 0.35/0.64 = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % nth_list_update
% 0.35/0.64 thf(fact_6577_list__update__same__conv,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_int,X2: int] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.35/0.64 => ( ( ( list_update_int @ Xs @ I @ X2 )
% 0.35/0.64 = Xs )
% 0.35/0.64 = ( ( nth_int @ Xs @ I )
% 0.35/0.64 = X2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_same_conv
% 0.35/0.64 thf(fact_6578_list__update__same__conv,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.35/0.64 => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
% 0.35/0.64 = Xs )
% 0.35/0.64 = ( ( nth_VEBT_VEBT @ Xs @ I )
% 0.35/0.64 = X2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_same_conv
% 0.35/0.64 thf(fact_6579_list__update__same__conv,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_o,X2: $o] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.35/0.64 => ( ( ( list_update_o @ Xs @ I @ X2 )
% 0.35/0.64 = Xs )
% 0.35/0.64 = ( ( nth_o @ Xs @ I )
% 0.35/0.64 = X2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_same_conv
% 0.35/0.64 thf(fact_6580_list__update__same__conv,axiom,
% 0.35/0.64 ! [I: nat,Xs: list_nat,X2: nat] :
% 0.35/0.64 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.35/0.64 => ( ( ( list_update_nat @ Xs @ I @ X2 )
% 0.35/0.64 = Xs )
% 0.35/0.64 = ( ( nth_nat @ Xs @ I )
% 0.35/0.64 = X2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % list_update_same_conv
% 0.35/0.64 thf(fact_6581_insert__simp__norm,axiom,
% 0.35/0.64 ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.64 => ( ( ord_less_nat @ Mi @ X2 )
% 0.35/0.64 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.64 => ( ( X2 != Ma )
% 0.35/0.64 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_simp_norm
% 0.35/0.64 thf(fact_6582_insert__simp__excp,axiom,
% 0.35/0.64 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.64 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.64 => ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.64 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.64 => ( ( X2 != Ma )
% 0.35/0.64 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.64 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( 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 ) ) ) ) ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % insert_simp_excp
% 0.35/0.64 thf(fact_6583_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6584_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6585_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( sup_sup_set_real @ bot_bot_set_real @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6586_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6587_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6588_sup__bot__left,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( sup_sup_set_int @ bot_bot_set_int @ X2 )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_left
% 0.35/0.64 thf(fact_6589_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6590_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6591_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6592_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6593_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6594_sup__bot__right,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
% 0.35/0.64 = X2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot_right
% 0.35/0.64 thf(fact_6595_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( bot_bo5327735625951526323at_nat
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bo5327735625951526323at_nat )
% 0.35/0.64 & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6596_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( bot_bo228742789529271731at_nat
% 0.35/0.64 = ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bo228742789529271731at_nat )
% 0.35/0.64 & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6597_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( bot_bot_set_real
% 0.35/0.64 = ( sup_sup_set_real @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bot_set_real )
% 0.35/0.64 & ( Y3 = bot_bot_set_real ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6598_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( bot_bot_set_o
% 0.35/0.64 = ( sup_sup_set_o @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bot_set_o )
% 0.35/0.64 & ( Y3 = bot_bot_set_o ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6599_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( bot_bot_set_nat
% 0.35/0.64 = ( sup_sup_set_nat @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bot_set_nat )
% 0.35/0.64 & ( Y3 = bot_bot_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6600_bot__eq__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( bot_bot_set_int
% 0.35/0.64 = ( sup_sup_set_int @ X2 @ Y3 ) )
% 0.35/0.64 = ( ( X2 = bot_bot_set_int )
% 0.35/0.64 & ( Y3 = bot_bot_set_int ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % bot_eq_sup_iff
% 0.35/0.64 thf(fact_6601_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 0.35/0.64 = bot_bo5327735625951526323at_nat )
% 0.35/0.64 = ( ( X2 = bot_bo5327735625951526323at_nat )
% 0.35/0.64 & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6602_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 0.35/0.64 = bot_bo228742789529271731at_nat )
% 0.35/0.64 = ( ( X2 = bot_bo228742789529271731at_nat )
% 0.35/0.64 & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6603_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_real,Y3: set_real] :
% 0.35/0.64 ( ( ( sup_sup_set_real @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_real )
% 0.35/0.64 = ( ( X2 = bot_bot_set_real )
% 0.35/0.64 & ( Y3 = bot_bot_set_real ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6604_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_o,Y3: set_o] :
% 0.35/0.64 ( ( ( sup_sup_set_o @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_o )
% 0.35/0.64 = ( ( X2 = bot_bot_set_o )
% 0.35/0.64 & ( Y3 = bot_bot_set_o ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6605_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.64 ( ( ( sup_sup_set_nat @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_nat )
% 0.35/0.64 = ( ( X2 = bot_bot_set_nat )
% 0.35/0.64 & ( Y3 = bot_bot_set_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6606_sup__eq__bot__iff,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int] :
% 0.35/0.64 ( ( ( sup_sup_set_int @ X2 @ Y3 )
% 0.35/0.64 = bot_bot_set_int )
% 0.35/0.64 = ( ( X2 = bot_bot_set_int )
% 0.35/0.64 & ( Y3 = bot_bot_set_int ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_eq_bot_iff
% 0.35/0.64 thf(fact_6607_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.35/0.64 & ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6608_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 0.35/0.64 & ( ord_le3146513528884898305at_nat @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6609_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.35/0.64 & ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6610_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.64 ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.64 & ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6611_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.64 ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.35/0.64 & ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6612_le__inf__iff,axiom,
% 0.35/0.64 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.64 ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) )
% 0.35/0.64 = ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.64 & ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_inf_iff
% 0.35/0.64 thf(fact_6613_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.64 & ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6614_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.35/0.64 & ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6615_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.64 & ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6616_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.64 ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.64 & ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6617_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.64 ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.64 & ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6618_inf_Obounded__iff,axiom,
% 0.35/0.64 ! [A2: int,B3: int,C: int] :
% 0.35/0.64 ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 0.35/0.64 = ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.64 & ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % inf.bounded_iff
% 0.35/0.64 thf(fact_6619_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_less_eq_set_nat @ X2 @ Z )
% 0.35/0.64 & ( ord_less_eq_set_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6620_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_le3000389064537975527at_nat @ X2 @ Z )
% 0.35/0.64 & ( ord_le3000389064537975527at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6621_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_le1268244103169919719at_nat @ X2 @ Z )
% 0.35/0.64 & ( ord_le1268244103169919719at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6622_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_less_eq_set_int @ X2 @ Z )
% 0.35/0.64 & ( ord_less_eq_set_int @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6623_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.64 ( ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_less_eq_rat @ X2 @ Z )
% 0.35/0.64 & ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6624_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.64 ( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_less_eq_nat @ X2 @ Z )
% 0.35/0.64 & ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6625_le__sup__iff,axiom,
% 0.35/0.64 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.64 ( ( ord_less_eq_int @ ( sup_sup_int @ X2 @ Y3 ) @ Z )
% 0.35/0.64 = ( ( ord_less_eq_int @ X2 @ Z )
% 0.35/0.64 & ( ord_less_eq_int @ Y3 @ Z ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % le_sup_iff
% 0.35/0.64 thf(fact_6626_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 0.35/0.64 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.64 & ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6627_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 0.35/0.64 & ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6628_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 0.35/0.64 & ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6629_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: set_int,C: set_int,A2: set_int] :
% 0.35/0.64 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.64 & ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6630_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.64 ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.64 & ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6631_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.64 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.64 & ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6632_sup_Obounded__iff,axiom,
% 0.35/0.64 ! [B3: int,C: int,A2: int] :
% 0.35/0.64 ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 0.35/0.64 = ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.64 & ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup.bounded_iff
% 0.35/0.64 thf(fact_6633_inf__bot__left,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_left
% 0.35/0.64 thf(fact_6634_inf__bot__left,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_left
% 0.35/0.64 thf(fact_6635_inf__bot__left,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_left
% 0.35/0.64 thf(fact_6636_inf__bot__left,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_left
% 0.35/0.64 thf(fact_6637_inf__bot__left,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_left
% 0.35/0.64 thf(fact_6638_inf__bot__right,axiom,
% 0.35/0.64 ! [X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.64 ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 0.35/0.64 = bot_bo2099793752762293965at_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_right
% 0.35/0.64 thf(fact_6639_inf__bot__right,axiom,
% 0.35/0.64 ! [X2: set_real] :
% 0.35/0.64 ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
% 0.35/0.64 = bot_bot_set_real ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_right
% 0.35/0.64 thf(fact_6640_inf__bot__right,axiom,
% 0.35/0.64 ! [X2: set_o] :
% 0.35/0.64 ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
% 0.35/0.64 = bot_bot_set_o ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_right
% 0.35/0.64 thf(fact_6641_inf__bot__right,axiom,
% 0.35/0.64 ! [X2: set_nat] :
% 0.35/0.64 ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.64 = bot_bot_set_nat ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_right
% 0.35/0.64 thf(fact_6642_inf__bot__right,axiom,
% 0.35/0.64 ! [X2: set_int] :
% 0.35/0.64 ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
% 0.35/0.64 = bot_bot_set_int ) ).
% 0.35/0.64
% 0.35/0.64 % inf_bot_right
% 0.35/0.64 thf(fact_6643_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( sup_su718114333110466843at_nat @ A2 @ bot_bo5327735625951526323at_nat )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6644_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6645_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_real] :
% 0.35/0.64 ( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6646_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_o] :
% 0.35/0.64 ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6647_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_nat] :
% 0.35/0.64 ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6648_sup__bot_Oright__neutral,axiom,
% 0.35/0.64 ! [A2: set_int] :
% 0.35/0.64 ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
% 0.35/0.64 = A2 ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.right_neutral
% 0.35/0.64 thf(fact_6649_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.64 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.64 ( ( bot_bo5327735625951526323at_nat
% 0.35/0.64 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
% 0.35/0.64 = ( ( A2 = bot_bo5327735625951526323at_nat )
% 0.35/0.64 & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 0.35/0.64
% 0.35/0.64 % sup_bot.neutr_eq_iff
% 0.35/0.64 thf(fact_6650_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.64 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.64 ( ( bot_bo228742789529271731at_nat
% 0.35/0.65 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = bot_bo228742789529271731at_nat )
% 0.35/0.65 & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.neutr_eq_iff
% 0.35/0.65 thf(fact_6651_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.65 ! [A2: set_real,B3: set_real] :
% 0.35/0.65 ( ( bot_bot_set_real
% 0.35/0.65 = ( sup_sup_set_real @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = bot_bot_set_real )
% 0.35/0.65 & ( B3 = bot_bot_set_real ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.neutr_eq_iff
% 0.35/0.65 thf(fact_6652_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.65 ! [A2: set_o,B3: set_o] :
% 0.35/0.65 ( ( bot_bot_set_o
% 0.35/0.65 = ( sup_sup_set_o @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = bot_bot_set_o )
% 0.35/0.65 & ( B3 = bot_bot_set_o ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.neutr_eq_iff
% 0.35/0.65 thf(fact_6653_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( bot_bot_set_nat
% 0.35/0.65 = ( sup_sup_set_nat @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = bot_bot_set_nat )
% 0.35/0.65 & ( B3 = bot_bot_set_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.neutr_eq_iff
% 0.35/0.65 thf(fact_6654_sup__bot_Oneutr__eq__iff,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( bot_bot_set_int
% 0.35/0.65 = ( sup_sup_set_int @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = bot_bot_set_int )
% 0.35/0.65 & ( B3 = bot_bot_set_int ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.neutr_eq_iff
% 0.35/0.65 thf(fact_6655_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6656_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6657_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_real] :
% 0.35/0.65 ( ( sup_sup_set_real @ bot_bot_set_real @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6658_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_o] :
% 0.35/0.65 ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6659_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_nat] :
% 0.35/0.65 ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6660_sup__bot_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: set_int] :
% 0.35/0.65 ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.left_neutral
% 0.35/0.65 thf(fact_6661_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 0.35/0.65 = bot_bo5327735625951526323at_nat )
% 0.35/0.65 = ( ( A2 = bot_bo5327735625951526323at_nat )
% 0.35/0.65 & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6662_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 0.35/0.65 = bot_bo228742789529271731at_nat )
% 0.35/0.65 = ( ( A2 = bot_bo228742789529271731at_nat )
% 0.35/0.65 & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6663_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_real,B3: set_real] :
% 0.35/0.65 ( ( ( sup_sup_set_real @ A2 @ B3 )
% 0.35/0.65 = bot_bot_set_real )
% 0.35/0.65 = ( ( A2 = bot_bot_set_real )
% 0.35/0.65 & ( B3 = bot_bot_set_real ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6664_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_o,B3: set_o] :
% 0.35/0.65 ( ( ( sup_sup_set_o @ A2 @ B3 )
% 0.35/0.65 = bot_bot_set_o )
% 0.35/0.65 = ( ( A2 = bot_bot_set_o )
% 0.35/0.65 & ( B3 = bot_bot_set_o ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6665_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ( sup_sup_set_nat @ A2 @ B3 )
% 0.35/0.65 = bot_bot_set_nat )
% 0.35/0.65 = ( ( A2 = bot_bot_set_nat )
% 0.35/0.65 & ( B3 = bot_bot_set_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6666_sup__bot_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ( sup_sup_set_int @ A2 @ B3 )
% 0.35/0.65 = bot_bot_set_int )
% 0.35/0.65 = ( ( A2 = bot_bot_set_int )
% 0.35/0.65 & ( B3 = bot_bot_set_int ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_bot.eq_neutr_iff
% 0.35/0.65 thf(fact_6667_max_Obounded__iff,axiom,
% 0.35/0.65 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 0.35/0.65 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 & ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.bounded_iff
% 0.35/0.65 thf(fact_6668_max_Obounded__iff,axiom,
% 0.35/0.65 ! [B3: num,C: num,A2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 0.35/0.65 = ( ( ord_less_eq_num @ B3 @ A2 )
% 0.35/0.65 & ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.bounded_iff
% 0.35/0.65 thf(fact_6669_max_Obounded__iff,axiom,
% 0.35/0.65 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 & ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.bounded_iff
% 0.35/0.65 thf(fact_6670_max_Obounded__iff,axiom,
% 0.35/0.65 ! [B3: int,C: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 0.35/0.65 = ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 & ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.bounded_iff
% 0.35/0.65 thf(fact_6671_max_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_rat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb2
% 0.35/0.65 thf(fact_6672_max_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: num,B3: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_num @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb2
% 0.35/0.65 thf(fact_6673_max_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb2
% 0.35/0.65 thf(fact_6674_max_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb2
% 0.35/0.65 thf(fact_6675_max_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_rat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb1
% 0.35/0.65 thf(fact_6676_max_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: num,A2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_num @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb1
% 0.35/0.65 thf(fact_6677_max_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb1
% 0.35/0.65 thf(fact_6678_max_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb1
% 0.35/0.65 thf(fact_6679_max__less__iff__conj,axiom,
% 0.35/0.65 ! [X2: real,Y3: real,Z: real] :
% 0.35/0.65 ( ( ord_less_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ( ord_less_real @ X2 @ Z )
% 0.35/0.65 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_less_iff_conj
% 0.35/0.65 thf(fact_6680_max__less__iff__conj,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.65 ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ( ord_less_rat @ X2 @ Z )
% 0.35/0.65 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_less_iff_conj
% 0.35/0.65 thf(fact_6681_max__less__iff__conj,axiom,
% 0.35/0.65 ! [X2: num,Y3: num,Z: num] :
% 0.35/0.65 ( ( ord_less_num @ ( ord_max_num @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ( ord_less_num @ X2 @ Z )
% 0.35/0.65 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_less_iff_conj
% 0.35/0.65 thf(fact_6682_max__less__iff__conj,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.65 ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ( ord_less_nat @ X2 @ Z )
% 0.35/0.65 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_less_iff_conj
% 0.35/0.65 thf(fact_6683_max__less__iff__conj,axiom,
% 0.35/0.65 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.65 ( ( ord_less_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ( ord_less_int @ X2 @ Z )
% 0.35/0.65 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_less_iff_conj
% 0.35/0.65 thf(fact_6684_max_Oabsorb4,axiom,
% 0.35/0.65 ! [A2: real,B3: real] :
% 0.35/0.65 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_real @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb4
% 0.35/0.65 thf(fact_6685_max_Oabsorb4,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_rat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb4
% 0.35/0.65 thf(fact_6686_max_Oabsorb4,axiom,
% 0.35/0.65 ! [A2: num,B3: num] :
% 0.35/0.65 ( ( ord_less_num @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_num @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb4
% 0.35/0.65 thf(fact_6687_max_Oabsorb4,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb4
% 0.35/0.65 thf(fact_6688_max_Oabsorb4,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_max_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb4
% 0.35/0.65 thf(fact_6689_max_Oabsorb3,axiom,
% 0.35/0.65 ! [B3: real,A2: real] :
% 0.35/0.65 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_real @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb3
% 0.35/0.65 thf(fact_6690_max_Oabsorb3,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_rat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb3
% 0.35/0.65 thf(fact_6691_max_Oabsorb3,axiom,
% 0.35/0.65 ! [B3: num,A2: num] :
% 0.35/0.65 ( ( ord_less_num @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_num @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb3
% 0.35/0.65 thf(fact_6692_max_Oabsorb3,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb3
% 0.35/0.65 thf(fact_6693_max_Oabsorb3,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_max_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb3
% 0.35/0.65 thf(fact_6694_max__bot,axiom,
% 0.35/0.65 ! [X2: set_real] :
% 0.35/0.65 ( ( ord_max_set_real @ bot_bot_set_real @ X2 )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot
% 0.35/0.65 thf(fact_6695_max__bot,axiom,
% 0.35/0.65 ! [X2: set_o] :
% 0.35/0.65 ( ( ord_max_set_o @ bot_bot_set_o @ X2 )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot
% 0.35/0.65 thf(fact_6696_max__bot,axiom,
% 0.35/0.65 ! [X2: set_nat] :
% 0.35/0.65 ( ( ord_max_set_nat @ bot_bot_set_nat @ X2 )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot
% 0.35/0.65 thf(fact_6697_max__bot,axiom,
% 0.35/0.65 ! [X2: set_int] :
% 0.35/0.65 ( ( ord_max_set_int @ bot_bot_set_int @ X2 )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot
% 0.35/0.65 thf(fact_6698_max__bot,axiom,
% 0.35/0.65 ! [X2: nat] :
% 0.35/0.65 ( ( ord_max_nat @ bot_bot_nat @ X2 )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot
% 0.35/0.65 thf(fact_6699_max__bot2,axiom,
% 0.35/0.65 ! [X2: set_real] :
% 0.35/0.65 ( ( ord_max_set_real @ X2 @ bot_bot_set_real )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot2
% 0.35/0.65 thf(fact_6700_max__bot2,axiom,
% 0.35/0.65 ! [X2: set_o] :
% 0.35/0.65 ( ( ord_max_set_o @ X2 @ bot_bot_set_o )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot2
% 0.35/0.65 thf(fact_6701_max__bot2,axiom,
% 0.35/0.65 ! [X2: set_nat] :
% 0.35/0.65 ( ( ord_max_set_nat @ X2 @ bot_bot_set_nat )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot2
% 0.35/0.65 thf(fact_6702_max__bot2,axiom,
% 0.35/0.65 ! [X2: set_int] :
% 0.35/0.65 ( ( ord_max_set_int @ X2 @ bot_bot_set_int )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot2
% 0.35/0.65 thf(fact_6703_max__bot2,axiom,
% 0.35/0.65 ! [X2: nat] :
% 0.35/0.65 ( ( ord_max_nat @ X2 @ bot_bot_nat )
% 0.35/0.65 = X2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_bot2
% 0.35/0.65 thf(fact_6704_max__Suc__Suc,axiom,
% 0.35/0.65 ! [M: nat,N: nat] :
% 0.35/0.65 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.35/0.65 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_Suc_Suc
% 0.35/0.65 thf(fact_6705_max__nat_Oeq__neutr__iff,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ( ord_max_nat @ A2 @ B3 )
% 0.35/0.65 = zero_zero_nat )
% 0.35/0.65 = ( ( A2 = zero_zero_nat )
% 0.35/0.65 & ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_nat.eq_neutr_iff
% 0.35/0.65 thf(fact_6706_max__nat_Oleft__neutral,axiom,
% 0.35/0.65 ! [A2: nat] :
% 0.35/0.65 ( ( ord_max_nat @ zero_zero_nat @ A2 )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_nat.left_neutral
% 0.35/0.65 thf(fact_6707_max__nat_Oneutr__eq__iff,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( zero_zero_nat
% 0.35/0.65 = ( ord_max_nat @ A2 @ B3 ) )
% 0.35/0.65 = ( ( A2 = zero_zero_nat )
% 0.35/0.65 & ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_nat.neutr_eq_iff
% 0.35/0.65 thf(fact_6708_max__nat_Oright__neutral,axiom,
% 0.35/0.65 ! [A2: nat] :
% 0.35/0.65 ( ( ord_max_nat @ A2 @ zero_zero_nat )
% 0.35/0.65 = A2 ) ).
% 0.35/0.65
% 0.35/0.65 % max_nat.right_neutral
% 0.35/0.65 thf(fact_6709_max__0L,axiom,
% 0.35/0.65 ! [N: nat] :
% 0.35/0.65 ( ( ord_max_nat @ zero_zero_nat @ N )
% 0.35/0.65 = N ) ).
% 0.35/0.65
% 0.35/0.65 % max_0L
% 0.35/0.65 thf(fact_6710_max__0R,axiom,
% 0.35/0.65 ! [N: nat] :
% 0.35/0.65 ( ( ord_max_nat @ N @ zero_zero_nat )
% 0.35/0.65 = N ) ).
% 0.35/0.65
% 0.35/0.65 % max_0R
% 0.35/0.65 thf(fact_6711_i0__less,axiom,
% 0.35/0.65 ! [N: extended_enat] :
% 0.35/0.65 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.35/0.65 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 0.35/0.65
% 0.35/0.65 % i0_less
% 0.35/0.65 thf(fact_6712_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 = ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6713_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.65 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.65 = ( numera1916890842035813515d_enat @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.65 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.35/0.65 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6714_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 = ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6715_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 = ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6716_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.65 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.65 = ( numeral_numeral_nat @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.65 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 0.35/0.65 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6717_max__number__of_I1_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 = ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(1)
% 0.35/0.65 thf(fact_6718_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
% 0.35/0.65 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6719_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
% 0.35/0.65 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6720_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
% 0.35/0.65 = ( numeral_numeral_real @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6721_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
% 0.35/0.65 = ( numeral_numeral_int @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6722_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
% 0.35/0.65 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6723_max__0__1_I4_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ zero_z3403309356797280102nteger )
% 0.35/0.65 = ( numera6620942414471956472nteger @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(4)
% 0.35/0.65 thf(fact_6724_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
% 0.35/0.65 = ( numeral_numeral_rat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6725_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
% 0.35/0.65 = ( numeral_numeral_nat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6726_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
% 0.35/0.65 = ( numeral_numeral_real @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6727_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
% 0.35/0.65 = ( numeral_numeral_int @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6728_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 0.35/0.65 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6729_max__0__1_I3_J,axiom,
% 0.35/0.65 ! [X2: num] :
% 0.35/0.65 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X2 ) )
% 0.35/0.65 = ( numera6620942414471956472nteger @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(3)
% 0.35/0.65 thf(fact_6730_max__0__1_I2_J,axiom,
% 0.35/0.65 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 0.35/0.65 = one_one_real ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(2)
% 0.35/0.65 thf(fact_6731_max__0__1_I2_J,axiom,
% 0.35/0.65 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 0.35/0.65 = one_one_rat ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(2)
% 0.35/0.65 thf(fact_6732_max__0__1_I2_J,axiom,
% 0.35/0.65 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 0.35/0.65 = one_one_int ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(2)
% 0.35/0.65 thf(fact_6733_max__0__1_I2_J,axiom,
% 0.35/0.65 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 0.35/0.65 = one_one_nat ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(2)
% 0.35/0.65 thf(fact_6734_max__0__1_I1_J,axiom,
% 0.35/0.65 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 0.35/0.65 = one_one_real ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(1)
% 0.35/0.65 thf(fact_6735_max__0__1_I1_J,axiom,
% 0.35/0.65 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 0.35/0.65 = one_one_rat ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(1)
% 0.35/0.65 thf(fact_6736_max__0__1_I1_J,axiom,
% 0.35/0.65 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 0.35/0.65 = one_one_int ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(1)
% 0.35/0.65 thf(fact_6737_max__0__1_I1_J,axiom,
% 0.35/0.65 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 0.35/0.65 = one_one_nat ) ).
% 0.35/0.65
% 0.35/0.65 % max_0_1(1)
% 0.35/0.65 thf(fact_6738_max__number__of_I2_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 = ( numeral_numeral_real @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(2)
% 0.35/0.65 thf(fact_6739_max__number__of_I2_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(2)
% 0.35/0.65 thf(fact_6740_max__number__of_I2_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(2)
% 0.35/0.65 thf(fact_6741_max__number__of_I2_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 = ( numeral_numeral_int @ U ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(2)
% 0.35/0.65 thf(fact_6742_max__number__of_I3_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 = ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 0.35/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(3)
% 0.35/0.65 thf(fact_6743_max__number__of_I3_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 = ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 0.35/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(3)
% 0.35/0.65 thf(fact_6744_max__number__of_I3_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 = ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 0.35/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(3)
% 0.35/0.65 thf(fact_6745_max__number__of_I3_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 = ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 0.35/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(3)
% 0.35/0.65 thf(fact_6746_max__number__of_I4_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(4)
% 0.35/0.65 thf(fact_6747_max__number__of_I4_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 0.35/0.65 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(4)
% 0.35/0.65 thf(fact_6748_max__number__of_I4_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(4)
% 0.35/0.65 thf(fact_6749_max__number__of_I4_J,axiom,
% 0.35/0.65 ! [U: num,V: num] :
% 0.35/0.65 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 0.35/0.65 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.35/0.65 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_number_of(4)
% 0.35/0.65 thf(fact_6750_max_OcoboundedI2,axiom,
% 0.35/0.65 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ C @ B3 )
% 0.35/0.65 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI2
% 0.35/0.65 thf(fact_6751_max_OcoboundedI2,axiom,
% 0.35/0.65 ! [C: num,B3: num,A2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ C @ B3 )
% 0.35/0.65 => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI2
% 0.35/0.65 thf(fact_6752_max_OcoboundedI2,axiom,
% 0.35/0.65 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ C @ B3 )
% 0.35/0.65 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI2
% 0.35/0.65 thf(fact_6753_max_OcoboundedI2,axiom,
% 0.35/0.65 ! [C: int,B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ C @ B3 )
% 0.35/0.65 => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI2
% 0.35/0.65 thf(fact_6754_max_OcoboundedI1,axiom,
% 0.35/0.65 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI1
% 0.35/0.65 thf(fact_6755_max_OcoboundedI1,axiom,
% 0.35/0.65 ! [C: num,A2: num,B3: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI1
% 0.35/0.65 thf(fact_6756_max_OcoboundedI1,axiom,
% 0.35/0.65 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI1
% 0.35/0.65 thf(fact_6757_max_OcoboundedI1,axiom,
% 0.35/0.65 ! [C: int,A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.coboundedI1
% 0.35/0.65 thf(fact_6758_max_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [A4: rat,B4: rat] :
% 0.35/0.65 ( ( ord_max_rat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff2
% 0.35/0.65 thf(fact_6759_max_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_num
% 0.35/0.65 = ( ^ [A4: num,B4: num] :
% 0.35/0.65 ( ( ord_max_num @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff2
% 0.35/0.65 thf(fact_6760_max_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.65 ( ( ord_max_nat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff2
% 0.35/0.65 thf(fact_6761_max_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [A4: int,B4: int] :
% 0.35/0.65 ( ( ord_max_int @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff2
% 0.35/0.65 thf(fact_6762_max_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.65 ( ( ord_max_rat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff1
% 0.35/0.65 thf(fact_6763_max_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_num
% 0.35/0.65 = ( ^ [B4: num,A4: num] :
% 0.35/0.65 ( ( ord_max_num @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff1
% 0.35/0.65 thf(fact_6764_max_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.65 ( ( ord_max_nat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff1
% 0.35/0.65 thf(fact_6765_max_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [B4: int,A4: int] :
% 0.35/0.65 ( ( ord_max_int @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.absorb_iff1
% 0.35/0.65 thf(fact_6766_le__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_eq_rat @ Z @ X2 )
% 0.35/0.65 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_max_iff_disj
% 0.35/0.65 thf(fact_6767_le__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: num,X2: num,Y3: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_eq_num @ Z @ X2 )
% 0.35/0.65 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_max_iff_disj
% 0.35/0.65 thf(fact_6768_le__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: nat,X2: nat,Y3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_eq_nat @ Z @ X2 )
% 0.35/0.65 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_max_iff_disj
% 0.35/0.65 thf(fact_6769_le__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: int,X2: int,Y3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_eq_int @ Z @ X2 )
% 0.35/0.65 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_max_iff_disj
% 0.35/0.65 thf(fact_6770_max_Ocobounded2,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( ord_max_rat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded2
% 0.35/0.65 thf(fact_6771_max_Ocobounded2,axiom,
% 0.35/0.65 ! [B3: num,A2: num] : ( ord_less_eq_num @ B3 @ ( ord_max_num @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded2
% 0.35/0.65 thf(fact_6772_max_Ocobounded2,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded2
% 0.35/0.65 thf(fact_6773_max_Ocobounded2,axiom,
% 0.35/0.65 ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( ord_max_int @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded2
% 0.35/0.65 thf(fact_6774_max_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( ord_max_rat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded1
% 0.35/0.65 thf(fact_6775_max_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: num,B3: num] : ( ord_less_eq_num @ A2 @ ( ord_max_num @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded1
% 0.35/0.65 thf(fact_6776_max_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded1
% 0.35/0.65 thf(fact_6777_max_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( ord_max_int @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.cobounded1
% 0.35/0.65 thf(fact_6778_max_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.order_iff
% 0.35/0.65 thf(fact_6779_max_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_num
% 0.35/0.65 = ( ^ [B4: num,A4: num] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( ord_max_num @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.order_iff
% 0.35/0.65 thf(fact_6780_max_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.order_iff
% 0.35/0.65 thf(fact_6781_max_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [B4: int,A4: int] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( ord_max_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.order_iff
% 0.35/0.65 thf(fact_6782_max_OboundedI,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedI
% 0.35/0.65 thf(fact_6783_max_OboundedI,axiom,
% 0.35/0.65 ! [B3: num,A2: num,C: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_num @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedI
% 0.35/0.65 thf(fact_6784_max_OboundedI,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedI
% 0.35/0.65 thf(fact_6785_max_OboundedI,axiom,
% 0.35/0.65 ! [B3: int,A2: int,C: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedI
% 0.35/0.65 thf(fact_6786_max_OboundedE,axiom,
% 0.35/0.65 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedE
% 0.35/0.65 thf(fact_6787_max_OboundedE,axiom,
% 0.35/0.65 ! [B3: num,C: num,A2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_num @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedE
% 0.35/0.65 thf(fact_6788_max_OboundedE,axiom,
% 0.35/0.65 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedE
% 0.35/0.65 thf(fact_6789_max_OboundedE,axiom,
% 0.35/0.65 ! [B3: int,C: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.boundedE
% 0.35/0.65 thf(fact_6790_max_OorderI,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( ord_max_rat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderI
% 0.35/0.65 thf(fact_6791_max_OorderI,axiom,
% 0.35/0.65 ! [A2: num,B3: num] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( ord_max_num @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderI
% 0.35/0.65 thf(fact_6792_max_OorderI,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( ord_max_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderI
% 0.35/0.65 thf(fact_6793_max_OorderI,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( ord_max_int @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderI
% 0.35/0.65 thf(fact_6794_max_OorderE,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderE
% 0.35/0.65 thf(fact_6795_max_OorderE,axiom,
% 0.35/0.65 ! [B3: num,A2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( ord_max_num @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderE
% 0.35/0.65 thf(fact_6796_max_OorderE,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderE
% 0.35/0.65 thf(fact_6797_max_OorderE,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( ord_max_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.orderE
% 0.35/0.65 thf(fact_6798_max_Omono,axiom,
% 0.35/0.65 ! [C: rat,A2: rat,D: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.mono
% 0.35/0.65 thf(fact_6799_max_Omono,axiom,
% 0.35/0.65 ! [C: num,A2: num,D: num,B3: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_num @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.mono
% 0.35/0.65 thf(fact_6800_max_Omono,axiom,
% 0.35/0.65 ! [C: nat,A2: nat,D: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.mono
% 0.35/0.65 thf(fact_6801_max_Omono,axiom,
% 0.35/0.65 ! [C: int,A2: int,D: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.mono
% 0.35/0.65 thf(fact_6802_not__iless0,axiom,
% 0.35/0.65 ! [N: extended_enat] :
% 0.35/0.65 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 0.35/0.65
% 0.35/0.65 % not_iless0
% 0.35/0.65 thf(fact_6803_enat__less__induct,axiom,
% 0.35/0.65 ! [P: extended_enat > $o,N: extended_enat] :
% 0.35/0.65 ( ! [N3: extended_enat] :
% 0.35/0.65 ( ! [M3: extended_enat] :
% 0.35/0.65 ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
% 0.35/0.65 => ( P @ M3 ) )
% 0.35/0.65 => ( P @ N3 ) )
% 0.35/0.65 => ( P @ N ) ) ).
% 0.35/0.65
% 0.35/0.65 % enat_less_induct
% 0.35/0.65 thf(fact_6804_enat__0__less__mult__iff,axiom,
% 0.35/0.65 ! [M: extended_enat,N: extended_enat] :
% 0.35/0.65 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 0.35/0.65 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.35/0.65 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % enat_0_less_mult_iff
% 0.35/0.65 thf(fact_6805_bot__enat__def,axiom,
% 0.35/0.65 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.35/0.65
% 0.35/0.65 % bot_enat_def
% 0.35/0.65 thf(fact_6806_of__nat__max,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % of_nat_max
% 0.35/0.65 thf(fact_6807_of__nat__max,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % of_nat_max
% 0.35/0.65 thf(fact_6808_of__nat__max,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % of_nat_max
% 0.35/0.65 thf(fact_6809_less__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: real,X2: real,Y3: real] :
% 0.35/0.65 ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_real @ Z @ X2 )
% 0.35/0.65 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % less_max_iff_disj
% 0.35/0.65 thf(fact_6810_less__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: rat,X2: rat,Y3: rat] :
% 0.35/0.65 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_rat @ Z @ X2 )
% 0.35/0.65 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % less_max_iff_disj
% 0.35/0.65 thf(fact_6811_less__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: num,X2: num,Y3: num] :
% 0.35/0.65 ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_num @ Z @ X2 )
% 0.35/0.65 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % less_max_iff_disj
% 0.35/0.65 thf(fact_6812_less__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: nat,X2: nat,Y3: nat] :
% 0.35/0.65 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_nat @ Z @ X2 )
% 0.35/0.65 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % less_max_iff_disj
% 0.35/0.65 thf(fact_6813_less__max__iff__disj,axiom,
% 0.35/0.65 ! [Z: int,X2: int,Y3: int] :
% 0.35/0.65 ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 0.35/0.65 = ( ( ord_less_int @ Z @ X2 )
% 0.35/0.65 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % less_max_iff_disj
% 0.35/0.65 thf(fact_6814_max_Ostrict__boundedE,axiom,
% 0.35/0.65 ! [B3: real,C: real,A2: real] :
% 0.35/0.65 ( ( ord_less_real @ ( ord_max_real @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_real @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_boundedE
% 0.35/0.65 thf(fact_6815_max_Ostrict__boundedE,axiom,
% 0.35/0.65 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_boundedE
% 0.35/0.65 thf(fact_6816_max_Ostrict__boundedE,axiom,
% 0.35/0.65 ! [B3: num,C: num,A2: num] :
% 0.35/0.65 ( ( ord_less_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_num @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_num @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_boundedE
% 0.35/0.65 thf(fact_6817_max_Ostrict__boundedE,axiom,
% 0.35/0.65 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_boundedE
% 0.35/0.65 thf(fact_6818_max_Ostrict__boundedE,axiom,
% 0.35/0.65 ! [B3: int,C: int,A2: int] :
% 0.35/0.65 ( ( ord_less_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_int @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_boundedE
% 0.35/0.65 thf(fact_6819_max_Ostrict__order__iff,axiom,
% 0.35/0.65 ( ord_less_real
% 0.35/0.65 = ( ^ [B4: real,A4: real] :
% 0.35/0.65 ( ( A4
% 0.35/0.65 = ( ord_max_real @ A4 @ B4 ) )
% 0.35/0.65 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_order_iff
% 0.35/0.65 thf(fact_6820_max_Ostrict__order__iff,axiom,
% 0.35/0.65 ( ord_less_rat
% 0.35/0.65 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.65 ( ( A4
% 0.35/0.65 = ( ord_max_rat @ A4 @ B4 ) )
% 0.35/0.65 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_order_iff
% 0.35/0.65 thf(fact_6821_max_Ostrict__order__iff,axiom,
% 0.35/0.65 ( ord_less_num
% 0.35/0.65 = ( ^ [B4: num,A4: num] :
% 0.35/0.65 ( ( A4
% 0.35/0.65 = ( ord_max_num @ A4 @ B4 ) )
% 0.35/0.65 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_order_iff
% 0.35/0.65 thf(fact_6822_max_Ostrict__order__iff,axiom,
% 0.35/0.65 ( ord_less_nat
% 0.35/0.65 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.65 ( ( A4
% 0.35/0.65 = ( ord_max_nat @ A4 @ B4 ) )
% 0.35/0.65 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_order_iff
% 0.35/0.65 thf(fact_6823_max_Ostrict__order__iff,axiom,
% 0.35/0.65 ( ord_less_int
% 0.35/0.65 = ( ^ [B4: int,A4: int] :
% 0.35/0.65 ( ( A4
% 0.35/0.65 = ( ord_max_int @ A4 @ B4 ) )
% 0.35/0.65 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_order_iff
% 0.35/0.65 thf(fact_6824_max_Ostrict__coboundedI1,axiom,
% 0.35/0.65 ! [C: real,A2: real,B3: real] :
% 0.35/0.65 ( ( ord_less_real @ C @ A2 )
% 0.35/0.65 => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI1
% 0.35/0.65 thf(fact_6825_max_Ostrict__coboundedI1,axiom,
% 0.35/0.65 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_rat @ C @ A2 )
% 0.35/0.65 => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI1
% 0.35/0.65 thf(fact_6826_max_Ostrict__coboundedI1,axiom,
% 0.35/0.65 ! [C: num,A2: num,B3: num] :
% 0.35/0.65 ( ( ord_less_num @ C @ A2 )
% 0.35/0.65 => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI1
% 0.35/0.65 thf(fact_6827_max_Ostrict__coboundedI1,axiom,
% 0.35/0.65 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_nat @ C @ A2 )
% 0.35/0.65 => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI1
% 0.35/0.65 thf(fact_6828_max_Ostrict__coboundedI1,axiom,
% 0.35/0.65 ! [C: int,A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_int @ C @ A2 )
% 0.35/0.65 => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI1
% 0.35/0.65 thf(fact_6829_max_Ostrict__coboundedI2,axiom,
% 0.35/0.65 ! [C: real,B3: real,A2: real] :
% 0.35/0.65 ( ( ord_less_real @ C @ B3 )
% 0.35/0.65 => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI2
% 0.35/0.65 thf(fact_6830_max_Ostrict__coboundedI2,axiom,
% 0.35/0.65 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_rat @ C @ B3 )
% 0.35/0.65 => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI2
% 0.35/0.65 thf(fact_6831_max_Ostrict__coboundedI2,axiom,
% 0.35/0.65 ! [C: num,B3: num,A2: num] :
% 0.35/0.65 ( ( ord_less_num @ C @ B3 )
% 0.35/0.65 => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI2
% 0.35/0.65 thf(fact_6832_max_Ostrict__coboundedI2,axiom,
% 0.35/0.65 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_nat @ C @ B3 )
% 0.35/0.65 => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI2
% 0.35/0.65 thf(fact_6833_max_Ostrict__coboundedI2,axiom,
% 0.35/0.65 ! [C: int,B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_int @ C @ B3 )
% 0.35/0.65 => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max.strict_coboundedI2
% 0.35/0.65 thf(fact_6834_sup__nat__def,axiom,
% 0.35/0.65 sup_sup_nat = ord_max_nat ).
% 0.35/0.65
% 0.35/0.65 % sup_nat_def
% 0.35/0.65 thf(fact_6835_max__def,axiom,
% 0.35/0.65 ( ord_max_set_int
% 0.35/0.65 = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_def
% 0.35/0.65 thf(fact_6836_max__def,axiom,
% 0.35/0.65 ( ord_max_rat
% 0.35/0.65 = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_def
% 0.35/0.65 thf(fact_6837_max__def,axiom,
% 0.35/0.65 ( ord_max_num
% 0.35/0.65 = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_def
% 0.35/0.65 thf(fact_6838_max__def,axiom,
% 0.35/0.65 ( ord_max_nat
% 0.35/0.65 = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_def
% 0.35/0.65 thf(fact_6839_max__def,axiom,
% 0.35/0.65 ( ord_max_int
% 0.35/0.65 = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_def
% 0.35/0.65 thf(fact_6840_max__absorb1,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_max_set_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb1
% 0.35/0.65 thf(fact_6841_max__absorb1,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_max_rat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb1
% 0.35/0.65 thf(fact_6842_max__absorb1,axiom,
% 0.35/0.65 ! [Y3: num,X2: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_max_num @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb1
% 0.35/0.65 thf(fact_6843_max__absorb1,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_max_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb1
% 0.35/0.65 thf(fact_6844_max__absorb1,axiom,
% 0.35/0.65 ! [Y3: int,X2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_max_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb1
% 0.35/0.65 thf(fact_6845_max__absorb2,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_max_set_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb2
% 0.35/0.65 thf(fact_6846_max__absorb2,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_max_rat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb2
% 0.35/0.65 thf(fact_6847_max__absorb2,axiom,
% 0.35/0.65 ! [X2: num,Y3: num] :
% 0.35/0.65 ( ( ord_less_eq_num @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_max_num @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb2
% 0.35/0.65 thf(fact_6848_max__absorb2,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_max_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb2
% 0.35/0.65 thf(fact_6849_max__absorb2,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_max_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_absorb2
% 0.35/0.65 thf(fact_6850_max__add__distrib__right,axiom,
% 0.35/0.65 ! [X2: real,Y3: real,Z: real] :
% 0.35/0.65 ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y3 @ Z ) )
% 0.35/0.65 = ( ord_max_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_right
% 0.35/0.65 thf(fact_6851_max__add__distrib__right,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.65 ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y3 @ Z ) )
% 0.35/0.65 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_right
% 0.35/0.65 thf(fact_6852_max__add__distrib__right,axiom,
% 0.35/0.65 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.65 ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y3 @ Z ) )
% 0.35/0.65 = ( ord_max_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_right
% 0.35/0.65 thf(fact_6853_max__add__distrib__right,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.65 ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y3 @ Z ) )
% 0.35/0.65 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_right
% 0.35/0.65 thf(fact_6854_max__add__distrib__left,axiom,
% 0.35/0.65 ! [X2: real,Y3: real,Z: real] :
% 0.35/0.65 ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_left
% 0.35/0.65 thf(fact_6855_max__add__distrib__left,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.65 ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_left
% 0.35/0.65 thf(fact_6856_max__add__distrib__left,axiom,
% 0.35/0.65 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.65 ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_left
% 0.35/0.65 thf(fact_6857_max__add__distrib__left,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.65 ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_add_distrib_left
% 0.35/0.65 thf(fact_6858_max__diff__distrib__left,axiom,
% 0.35/0.65 ! [X2: real,Y3: real,Z: real] :
% 0.35/0.65 ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_diff_distrib_left
% 0.35/0.65 thf(fact_6859_max__diff__distrib__left,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.65 ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_diff_distrib_left
% 0.35/0.65 thf(fact_6860_max__diff__distrib__left,axiom,
% 0.35/0.65 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.65 ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 0.35/0.65 = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % max_diff_distrib_left
% 0.35/0.65 thf(fact_6861_nat__add__max__right,axiom,
% 0.35/0.65 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.65 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.35/0.65 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % nat_add_max_right
% 0.35/0.65 thf(fact_6862_nat__add__max__left,axiom,
% 0.35/0.65 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.65 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.35/0.65 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % nat_add_max_left
% 0.35/0.65 thf(fact_6863_nat__mult__max__left,axiom,
% 0.35/0.65 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.65 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 0.35/0.65 = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % nat_mult_max_left
% 0.35/0.65 thf(fact_6864_nat__mult__max__right,axiom,
% 0.35/0.65 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.65 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 0.35/0.65 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % nat_mult_max_right
% 0.35/0.65 thf(fact_6865_nat__minus__add__max,axiom,
% 0.35/0.65 ! [N: nat,M: nat] :
% 0.35/0.65 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 0.35/0.65 = ( ord_max_nat @ N @ M ) ) ).
% 0.35/0.65
% 0.35/0.65 % nat_minus_add_max
% 0.35/0.65 thf(fact_6866_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6867_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6868_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6869_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6870_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6871_inf__sup__ord_I2_J,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(2)
% 0.35/0.65 thf(fact_6872_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6873_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6874_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6875_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6876_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6877_inf__sup__ord_I1_J,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(1)
% 0.35/0.65 thf(fact_6878_inf__le1,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6879_inf__le1,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6880_inf__le1,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6881_inf__le1,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6882_inf__le1,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6883_inf__le1,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le1
% 0.35/0.65 thf(fact_6884_inf__le2,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6885_inf__le2,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6886_inf__le2,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6887_inf__le2,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6888_inf__le2,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6889_inf__le2,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf_le2
% 0.35/0.65 thf(fact_6890_le__infE,axiom,
% 0.35/0.65 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_nat @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6891_le__infE,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_le3146513528884898305at_nat @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6892_le__infE,axiom,
% 0.35/0.65 ! [X2: set_int,A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_int @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_int @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6893_le__infE,axiom,
% 0.35/0.65 ! [X2: rat,A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_rat @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_rat @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6894_le__infE,axiom,
% 0.35/0.65 ! [X2: nat,A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_nat @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_nat @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6895_le__infE,axiom,
% 0.35/0.65 ! [X2: int,A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_int @ X2 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_int @ X2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infE
% 0.35/0.65 thf(fact_6896_le__infI,axiom,
% 0.35/0.65 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6897_le__infI,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_le3146513528884898305at_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6898_le__infI,axiom,
% 0.35/0.65 ! [X2: set_int,A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6899_le__infI,axiom,
% 0.35/0.65 ! [X2: rat,A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6900_le__infI,axiom,
% 0.35/0.65 ! [X2: nat,A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6901_le__infI,axiom,
% 0.35/0.65 ! [X2: int,A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI
% 0.35/0.65 thf(fact_6902_inf__mono,axiom,
% 0.35/0.65 ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6903_inf__mono,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_le3146513528884898305at_nat @ B3 @ D )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6904_inf__mono,axiom,
% 0.35/0.65 ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ ( inf_inf_set_int @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6905_inf__mono,axiom,
% 0.35/0.65 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_rat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ ( inf_inf_rat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6906_inf__mono,axiom,
% 0.35/0.65 ! [A2: nat,C: nat,B3: nat,D: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_nat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6907_inf__mono,axiom,
% 0.35/0.65 ! [A2: int,C: int,B3: int,D: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_int @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_mono
% 0.35/0.65 thf(fact_6908_le__infI1,axiom,
% 0.35/0.65 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6909_le__infI1,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ X2 )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6910_le__infI1,axiom,
% 0.35/0.65 ! [A2: set_int,X2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6911_le__infI1,axiom,
% 0.35/0.65 ! [A2: rat,X2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ X2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6912_le__infI1,axiom,
% 0.35/0.65 ! [A2: nat,X2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ X2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6913_le__infI1,axiom,
% 0.35/0.65 ! [A2: int,X2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ X2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI1
% 0.35/0.65 thf(fact_6914_le__infI2,axiom,
% 0.35/0.65 ! [B3: set_nat,X2: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6915_le__infI2,axiom,
% 0.35/0.65 ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6916_le__infI2,axiom,
% 0.35/0.65 ! [B3: set_int,X2: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6917_le__infI2,axiom,
% 0.35/0.65 ! [B3: rat,X2: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6918_le__infI2,axiom,
% 0.35/0.65 ! [B3: nat,X2: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6919_le__infI2,axiom,
% 0.35/0.65 ! [B3: int,X2: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_infI2
% 0.35/0.65 thf(fact_6920_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6921_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6922_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_inf_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6923_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_inf_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6924_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_inf_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6925_inf_OorderE,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( inf_inf_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderE
% 0.35/0.65 thf(fact_6926_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_inf_set_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6927_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6928_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_inf_set_int @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6929_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_inf_rat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6930_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_inf_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6931_inf_OorderI,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( inf_inf_int @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.orderI
% 0.35/0.65 thf(fact_6932_inf__unique,axiom,
% 0.35/0.65 ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
% 0.35/0.65 ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ X5 @ Z4 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6933_inf__unique,axiom,
% 0.35/0.65 ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_le3146513528884898305at_nat @ X5 @ Z4 )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6934_inf__unique,axiom,
% 0.35/0.65 ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
% 0.35/0.65 ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ X5 @ Z4 )
% 0.35/0.65 => ( ord_less_eq_set_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6935_inf__unique,axiom,
% 0.35/0.65 ! [F: rat > rat > rat,X2: rat,Y3: rat] :
% 0.35/0.65 ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: rat,Y4: rat,Z4: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ X5 @ Z4 )
% 0.35/0.65 => ( ord_less_eq_rat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_inf_rat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6936_inf__unique,axiom,
% 0.35/0.65 ! [F: nat > nat > nat,X2: nat,Y3: nat] :
% 0.35/0.65 ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ X5 @ Z4 )
% 0.35/0.65 => ( ord_less_eq_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_inf_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6937_inf__unique,axiom,
% 0.35/0.65 ! [F: int > int > int,X2: int,Y3: int] :
% 0.35/0.65 ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ X5 )
% 0.35/0.65 => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ Y4 )
% 0.35/0.65 => ( ! [X5: int,Y4: int,Z4: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X5 @ Y4 )
% 0.35/0.65 => ( ( ord_less_eq_int @ X5 @ Z4 )
% 0.35/0.65 => ( ord_less_eq_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 0.35/0.65 => ( ( inf_inf_int @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_unique
% 0.35/0.65 thf(fact_6938_le__iff__inf,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [X: set_nat,Y: set_nat] :
% 0.35/0.65 ( ( inf_inf_set_nat @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6939_le__iff__inf,axiom,
% 0.35/0.65 ( ord_le3146513528884898305at_nat
% 0.35/0.65 = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( inf_in2572325071724192079at_nat @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6940_le__iff__inf,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [X: set_int,Y: set_int] :
% 0.35/0.65 ( ( inf_inf_set_int @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6941_le__iff__inf,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [X: rat,Y: rat] :
% 0.35/0.65 ( ( inf_inf_rat @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6942_le__iff__inf,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [X: nat,Y: nat] :
% 0.35/0.65 ( ( inf_inf_nat @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6943_le__iff__inf,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [X: int,Y: int] :
% 0.35/0.65 ( ( inf_inf_int @ X @ Y )
% 0.35/0.65 = X ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_inf
% 0.35/0.65 thf(fact_6944_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6945_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6946_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_inf_set_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6947_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_inf_rat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6948_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_inf_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6949_inf_Oabsorb1,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ( ( inf_inf_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb1
% 0.35/0.65 thf(fact_6950_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6951_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6952_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_inf_set_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6953_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_inf_rat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6954_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_inf_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6955_inf_Oabsorb2,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( ( inf_inf_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb2
% 0.35/0.65 thf(fact_6956_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6957_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6958_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6959_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_inf_rat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6960_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_inf_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6961_inf__absorb1,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( inf_inf_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb1
% 0.35/0.65 thf(fact_6962_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: set_nat,X2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6963_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6964_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6965_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_inf_rat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6966_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_inf_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6967_inf__absorb2,axiom,
% 0.35/0.65 ! [Y3: int,X2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( inf_inf_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_absorb2
% 0.35/0.65 thf(fact_6968_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6969_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6970_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6971_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6972_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6973_inf_OboundedE,axiom,
% 0.35/0.65 ! [A2: int,B3: int,C: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 0.35/0.65 => ~ ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedE
% 0.35/0.65 thf(fact_6974_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6975_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6976_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int,C: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6977_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6978_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6979_inf_OboundedI,axiom,
% 0.35/0.65 ! [A2: int,B3: int,C: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ( ( ord_less_eq_int @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.boundedI
% 0.35/0.65 thf(fact_6980_inf__greatest,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ X2 @ Z )
% 0.35/0.65 => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6981_inf__greatest,axiom,
% 0.35/0.65 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_le3146513528884898305at_nat @ X2 @ Z )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6982_inf__greatest,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ X2 @ Z )
% 0.35/0.65 => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6983_inf__greatest,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat,Z: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ X2 @ Z )
% 0.35/0.65 => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6984_inf__greatest,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ X2 @ Z )
% 0.35/0.65 => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6985_inf__greatest,axiom,
% 0.35/0.65 ! [X2: int,Y3: int,Z: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( ord_less_eq_int @ X2 @ Z )
% 0.35/0.65 => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_greatest
% 0.35/0.65 thf(fact_6986_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6987_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_le3146513528884898305at_nat
% 0.35/0.65 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6988_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [A4: set_int,B4: set_int] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6989_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [A4: rat,B4: rat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_inf_rat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6990_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6991_inf_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [A4: int,B4: int] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( inf_inf_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.order_iff
% 0.35/0.65 thf(fact_6992_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6993_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6994_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6995_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6996_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6997_inf_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ A2 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded1
% 0.35/0.65 thf(fact_6998_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_6999_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_7000_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_7001_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_7002_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_7003_inf_Ocobounded2,axiom,
% 0.35/0.65 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ B3 ) ).
% 0.35/0.65
% 0.35/0.65 % inf.cobounded2
% 0.35/0.65 thf(fact_7004_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.35/0.65 ( ( inf_inf_set_nat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7005_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_le3146513528884898305at_nat
% 0.35/0.65 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7006_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [A4: set_int,B4: set_int] :
% 0.35/0.65 ( ( inf_inf_set_int @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7007_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [A4: rat,B4: rat] :
% 0.35/0.65 ( ( inf_inf_rat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7008_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.65 ( ( inf_inf_nat @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7009_inf_Oabsorb__iff1,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [A4: int,B4: int] :
% 0.35/0.65 ( ( inf_inf_int @ A4 @ B4 )
% 0.35/0.65 = A4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff1
% 0.35/0.65 thf(fact_7010_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.35/0.65 ( ( inf_inf_set_nat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7011_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_le3146513528884898305at_nat
% 0.35/0.65 = ( ^ [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7012_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [B4: set_int,A4: set_int] :
% 0.35/0.65 ( ( inf_inf_set_int @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7013_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.65 ( ( inf_inf_rat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7014_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.65 ( ( inf_inf_nat @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7015_inf_Oabsorb__iff2,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [B4: int,A4: int] :
% 0.35/0.65 ( ( inf_inf_int @ A4 @ B4 )
% 0.35/0.65 = B4 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.absorb_iff2
% 0.35/0.65 thf(fact_7016_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: set_nat,C: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7017_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7018_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: set_int,C: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7019_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7020_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7021_inf_OcoboundedI1,axiom,
% 0.35/0.65 ! [A2: int,C: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ C )
% 0.35/0.65 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI1
% 0.35/0.65 thf(fact_7022_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ C )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7023_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.65 ( ( ord_le3146513528884898305at_nat @ B3 @ C )
% 0.35/0.65 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7024_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: set_int,C: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ C )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7025_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ C )
% 0.35/0.65 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7026_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ C )
% 0.35/0.65 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7027_inf_OcoboundedI2,axiom,
% 0.35/0.65 ! [B3: int,C: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ C )
% 0.35/0.65 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf.coboundedI2
% 0.35/0.65 thf(fact_7028_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7029_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7030_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7031_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7032_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7033_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7034_inf__sup__ord_I4_J,axiom,
% 0.35/0.65 ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(4)
% 0.35/0.65 thf(fact_7035_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7036_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7037_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7038_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7039_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7040_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7041_inf__sup__ord_I3_J,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % inf_sup_ord(3)
% 0.35/0.65 thf(fact_7042_le__supE,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat,X2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_nat @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7043_le__supE,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_le3000389064537975527at_nat @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7044_le__supE,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_le1268244103169919719at_nat @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7045_le__supE,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int,X2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_int @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_int @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7046_le__supE,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat,X2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_rat @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_less_eq_rat @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7047_le__supE,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat,X2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_nat @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7048_le__supE,axiom,
% 0.35/0.65 ! [A2: int,B3: int,X2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_int @ A2 @ X2 )
% 0.35/0.65 => ~ ( ord_less_eq_int @ B3 @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supE
% 0.35/0.65 thf(fact_7049_le__supI,axiom,
% 0.35/0.65 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7050_le__supI,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7051_le__supI,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7052_le__supI,axiom,
% 0.35/0.65 ! [A2: set_int,X2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7053_le__supI,axiom,
% 0.35/0.65 ! [A2: rat,X2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7054_le__supI,axiom,
% 0.35/0.65 ! [A2: nat,X2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7055_le__supI,axiom,
% 0.35/0.65 ! [A2: int,X2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ B3 @ X2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI
% 0.35/0.65 thf(fact_7056_sup__ge1,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7057_sup__ge1,axiom,
% 0.35/0.65 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7058_sup__ge1,axiom,
% 0.35/0.65 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7059_sup__ge1,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7060_sup__ge1,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7061_sup__ge1,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7062_sup__ge1,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge1
% 0.35/0.65 thf(fact_7063_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7064_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7065_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7066_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7067_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7068_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7069_sup__ge2,axiom,
% 0.35/0.65 ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_ge2
% 0.35/0.65 thf(fact_7070_le__supI1,axiom,
% 0.35/0.65 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7071_le__supI1,axiom,
% 0.35/0.65 ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ X2 @ A2 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7072_le__supI1,axiom,
% 0.35/0.65 ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ X2 @ A2 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7073_le__supI1,axiom,
% 0.35/0.65 ! [X2: set_int,A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ A2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7074_le__supI1,axiom,
% 0.35/0.65 ! [X2: rat,A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ A2 )
% 0.35/0.65 => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7075_le__supI1,axiom,
% 0.35/0.65 ! [X2: nat,A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ A2 )
% 0.35/0.65 => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7076_le__supI1,axiom,
% 0.35/0.65 ! [X2: int,A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ A2 )
% 0.35/0.65 => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI1
% 0.35/0.65 thf(fact_7077_le__supI2,axiom,
% 0.35/0.65 ! [X2: set_nat,B3: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7078_le__supI2,axiom,
% 0.35/0.65 ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7079_le__supI2,axiom,
% 0.35/0.65 ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7080_le__supI2,axiom,
% 0.35/0.65 ! [X2: set_int,B3: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7081_le__supI2,axiom,
% 0.35/0.65 ! [X2: rat,B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7082_le__supI2,axiom,
% 0.35/0.65 ! [X2: nat,B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7083_le__supI2,axiom,
% 0.35/0.65 ! [X2: int,B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ B3 )
% 0.35/0.65 => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_supI2
% 0.35/0.65 thf(fact_7084_sup_Omono,axiom,
% 0.35/0.65 ! [C: set_nat,A2: set_nat,D: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7085_sup_Omono,axiom,
% 0.35/0.65 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ D @ B3 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ C @ D ) @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7086_sup_Omono,axiom,
% 0.35/0.65 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ D @ B3 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D ) @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7087_sup_Omono,axiom,
% 0.35/0.65 ! [C: set_int,A2: set_int,D: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D ) @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7088_sup_Omono,axiom,
% 0.35/0.65 ! [C: rat,A2: rat,D: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7089_sup_Omono,axiom,
% 0.35/0.65 ! [C: nat,A2: nat,D: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7090_sup_Omono,axiom,
% 0.35/0.65 ! [C: int,A2: int,D: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ D @ B3 )
% 0.35/0.65 => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A2 @ B3 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.mono
% 0.35/0.65 thf(fact_7091_sup__mono,axiom,
% 0.35/0.65 ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7092_sup__mono,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ B3 @ D )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ ( sup_su718114333110466843at_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7093_sup__mono,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ B3 @ D )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ ( sup_su5525570899277871387at_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7094_sup__mono,axiom,
% 0.35/0.65 ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ ( sup_sup_set_int @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7095_sup__mono,axiom,
% 0.35/0.65 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_rat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7096_sup__mono,axiom,
% 0.35/0.65 ! [A2: nat,C: nat,B3: nat,D: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_nat @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7097_sup__mono,axiom,
% 0.35/0.65 ! [A2: int,C: int,B3: int,D: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ C )
% 0.35/0.65 => ( ( ord_less_eq_int @ B3 @ D )
% 0.35/0.65 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_mono
% 0.35/0.65 thf(fact_7098_sup__least,axiom,
% 0.35/0.65 ! [Y3: set_nat,X2: set_nat,Z: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ Z @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7099_sup__least,axiom,
% 0.35/0.65 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ Z @ X2 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7100_sup__least,axiom,
% 0.35/0.65 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ Z @ X2 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7101_sup__least,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int,Z: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ Z @ X2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7102_sup__least,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat,Z: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ Z @ X2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( sup_sup_rat @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7103_sup__least,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat,Z: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ Z @ X2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7104_sup__least,axiom,
% 0.35/0.65 ! [Y3: int,X2: int,Z: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ Z @ X2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( sup_sup_int @ Y3 @ Z ) @ X2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_least
% 0.35/0.65 thf(fact_7105_le__iff__sup,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [X: set_nat,Y: set_nat] :
% 0.35/0.65 ( ( sup_sup_set_nat @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7106_le__iff__sup,axiom,
% 0.35/0.65 ( ord_le3000389064537975527at_nat
% 0.35/0.65 = ( ^ [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( sup_su718114333110466843at_nat @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7107_le__iff__sup,axiom,
% 0.35/0.65 ( ord_le1268244103169919719at_nat
% 0.35/0.65 = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( sup_su5525570899277871387at_nat @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7108_le__iff__sup,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [X: set_int,Y: set_int] :
% 0.35/0.65 ( ( sup_sup_set_int @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7109_le__iff__sup,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [X: rat,Y: rat] :
% 0.35/0.65 ( ( sup_sup_rat @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7110_le__iff__sup,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [X: nat,Y: nat] :
% 0.35/0.65 ( ( sup_sup_nat @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7111_le__iff__sup,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [X: int,Y: int] :
% 0.35/0.65 ( ( sup_sup_int @ X @ Y )
% 0.35/0.65 = Y ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % le_iff_sup
% 0.35/0.65 thf(fact_7112_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7113_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7114_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7115_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7116_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7117_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7118_sup_OorderE,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( A2
% 0.35/0.65 = ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderE
% 0.35/0.65 thf(fact_7119_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_sup_set_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7120_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7121_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7122_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_sup_set_int @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7123_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_sup_rat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7124_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_sup_nat @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7125_sup_OorderI,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( A2
% 0.35/0.65 = ( sup_sup_int @ A2 @ B3 ) )
% 0.35/0.65 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.orderI
% 0.35/0.65 thf(fact_7126_sup__unique,axiom,
% 0.35/0.65 ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
% 0.35/0.65 ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ Z4 @ X5 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7127_sup__unique,axiom,
% 0.35/0.65 ! [F: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat,Z4: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ Z4 @ X5 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7128_sup__unique,axiom,
% 0.35/0.65 ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ Z4 @ X5 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7129_sup__unique,axiom,
% 0.35/0.65 ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
% 0.35/0.65 ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ Z4 @ X5 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7130_sup__unique,axiom,
% 0.35/0.65 ! [F: rat > rat > rat,X2: rat,Y3: rat] :
% 0.35/0.65 ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: rat,Y4: rat,Z4: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ Z4 @ X5 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_sup_rat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7131_sup__unique,axiom,
% 0.35/0.65 ! [F: nat > nat > nat,X2: nat,Y3: nat] :
% 0.35/0.65 ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ Z4 @ X5 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_sup_nat @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7132_sup__unique,axiom,
% 0.35/0.65 ! [F: int > int > int,X2: int,Y3: int] :
% 0.35/0.65 ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ X5 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ Y4 @ ( F @ X5 @ Y4 ) )
% 0.35/0.65 => ( ! [X5: int,Y4: int,Z4: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Y4 @ X5 )
% 0.35/0.65 => ( ( ord_less_eq_int @ Z4 @ X5 )
% 0.35/0.65 => ( ord_less_eq_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 0.35/0.65 => ( ( sup_sup_int @ X2 @ Y3 )
% 0.35/0.65 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_unique
% 0.35/0.65 thf(fact_7133_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7134_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7135_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7136_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_sup_set_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7137_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_sup_rat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7138_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_sup_nat @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7139_sup_Oabsorb1,axiom,
% 0.35/0.65 ! [B3: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( ( sup_sup_int @ A2 @ B3 )
% 0.35/0.65 = A2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb1
% 0.35/0.65 thf(fact_7140_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7141_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7142_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7143_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: set_int,B3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_sup_set_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7144_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: rat,B3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_sup_rat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7145_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: nat,B3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_sup_nat @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7146_sup_Oabsorb2,axiom,
% 0.35/0.65 ! [A2: int,B3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ A2 @ B3 )
% 0.35/0.65 => ( ( sup_sup_int @ A2 @ B3 )
% 0.35/0.65 = B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.absorb2
% 0.35/0.65 thf(fact_7147_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: set_nat,X2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7148_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7149_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7150_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: set_int,X2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7151_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: rat,X2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_sup_rat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7152_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: nat,X2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_sup_nat @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7153_sup__absorb1,axiom,
% 0.35/0.65 ! [Y3: int,X2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ Y3 @ X2 )
% 0.35/0.65 => ( ( sup_sup_int @ X2 @ Y3 )
% 0.35/0.65 = X2 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb1
% 0.35/0.65 thf(fact_7154_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: set_nat,Y3: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7155_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7156_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7157_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: set_int,Y3: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7158_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: rat,Y3: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_sup_rat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7159_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: nat,Y3: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_sup_nat @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7160_sup__absorb2,axiom,
% 0.35/0.65 ! [X2: int,Y3: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.65 => ( ( sup_sup_int @ X2 @ Y3 )
% 0.35/0.65 = Y3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup_absorb2
% 0.35/0.65 thf(fact_7161_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7162_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7163_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7164_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: set_int,C: set_int,A2: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7165_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7166_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7167_sup_OboundedE,axiom,
% 0.35/0.65 ! [B3: int,C: int,A2: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 0.35/0.65 => ~ ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedE
% 0.35/0.65 thf(fact_7168_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: set_nat,A2: set_nat,C: set_nat] :
% 0.35/0.65 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_nat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7169_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 0.35/0.65 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7170_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 0.35/0.65 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7171_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: set_int,A2: set_int,C: set_int] :
% 0.35/0.65 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_set_int @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7172_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: rat,A2: rat,C: rat] :
% 0.35/0.65 ( ( ord_less_eq_rat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7173_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: nat,A2: nat,C: nat] :
% 0.35/0.65 ( ( ord_less_eq_nat @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7174_sup_OboundedI,axiom,
% 0.35/0.65 ! [B3: int,A2: int,C: int] :
% 0.35/0.65 ( ( ord_less_eq_int @ B3 @ A2 )
% 0.35/0.65 => ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.65 => ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.boundedI
% 0.35/0.65 thf(fact_7175_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_set_nat
% 0.35/0.65 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7176_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_le3000389064537975527at_nat
% 0.35/0.65 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7177_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_le1268244103169919719at_nat
% 0.35/0.65 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7178_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_set_int
% 0.35/0.65 = ( ^ [B4: set_int,A4: set_int] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_sup_set_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7179_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_rat
% 0.35/0.65 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_sup_rat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7180_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_nat
% 0.35/0.65 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7181_sup_Oorder__iff,axiom,
% 0.35/0.65 ( ord_less_eq_int
% 0.35/0.65 = ( ^ [B4: int,A4: int] :
% 0.35/0.65 ( A4
% 0.35/0.65 = ( sup_sup_int @ A4 @ B4 ) ) ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.order_iff
% 0.35/0.65 thf(fact_7182_sup_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% 0.35/0.65
% 0.35/0.65 % sup.cobounded1
% 0.35/0.65 thf(fact_7183_sup_Ocobounded1,axiom,
% 0.35/0.65 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7184_sup_Ocobounded1,axiom,
% 0.35/0.66 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7185_sup_Ocobounded1,axiom,
% 0.35/0.66 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7186_sup_Ocobounded1,axiom,
% 0.35/0.66 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( sup_sup_rat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7187_sup_Ocobounded1,axiom,
% 0.35/0.66 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7188_sup_Ocobounded1,axiom,
% 0.35/0.66 ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded1
% 0.35/0.66 thf(fact_7189_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7190_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B3 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7191_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B3 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7192_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: set_int,A2: set_int] : ( ord_less_eq_set_int @ B3 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7193_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( sup_sup_rat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7194_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7195_sup_Ocobounded2,axiom,
% 0.35/0.66 ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( sup_sup_int @ A2 @ B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.cobounded2
% 0.35/0.66 thf(fact_7196_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_less_eq_set_nat
% 0.35/0.66 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.35/0.66 ( ( sup_sup_set_nat @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7197_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_le3000389064537975527at_nat
% 0.35/0.66 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7198_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_le1268244103169919719at_nat
% 0.35/0.66 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7199_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_less_eq_set_int
% 0.35/0.66 = ( ^ [B4: set_int,A4: set_int] :
% 0.35/0.66 ( ( sup_sup_set_int @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7200_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_less_eq_rat
% 0.35/0.66 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.66 ( ( sup_sup_rat @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7201_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_less_eq_nat
% 0.35/0.66 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.66 ( ( sup_sup_nat @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7202_sup_Oabsorb__iff1,axiom,
% 0.35/0.66 ( ord_less_eq_int
% 0.35/0.66 = ( ^ [B4: int,A4: int] :
% 0.35/0.66 ( ( sup_sup_int @ A4 @ B4 )
% 0.35/0.66 = A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff1
% 0.35/0.66 thf(fact_7203_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_less_eq_set_nat
% 0.35/0.66 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.35/0.66 ( ( sup_sup_set_nat @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7204_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_le3000389064537975527at_nat
% 0.35/0.66 = ( ^ [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7205_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_le1268244103169919719at_nat
% 0.35/0.66 = ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7206_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_less_eq_set_int
% 0.35/0.66 = ( ^ [A4: set_int,B4: set_int] :
% 0.35/0.66 ( ( sup_sup_set_int @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7207_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_less_eq_rat
% 0.35/0.66 = ( ^ [A4: rat,B4: rat] :
% 0.35/0.66 ( ( sup_sup_rat @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7208_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_less_eq_nat
% 0.35/0.66 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.66 ( ( sup_sup_nat @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7209_sup_Oabsorb__iff2,axiom,
% 0.35/0.66 ( ord_less_eq_int
% 0.35/0.66 = ( ^ [A4: int,B4: int] :
% 0.35/0.66 ( ( sup_sup_int @ A4 @ B4 )
% 0.35/0.66 = B4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb_iff2
% 0.35/0.66 thf(fact_7210_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_eq_set_nat @ C @ A2 )
% 0.35/0.66 => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7211_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 0.35/0.66 => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7212_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 0.35/0.66 => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7213_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: set_int,A2: set_int,B3: set_int] :
% 0.35/0.66 ( ( ord_less_eq_set_int @ C @ A2 )
% 0.35/0.66 => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7214_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_eq_rat @ C @ A2 )
% 0.35/0.66 => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7215_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_eq_nat @ C @ A2 )
% 0.35/0.66 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7216_sup_OcoboundedI1,axiom,
% 0.35/0.66 ! [C: int,A2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_eq_int @ C @ A2 )
% 0.35/0.66 => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI1
% 0.35/0.66 thf(fact_7217_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: set_nat,B3: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_eq_set_nat @ C @ B3 )
% 0.35/0.66 => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7218_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le3000389064537975527at_nat @ C @ B3 )
% 0.35/0.66 => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7219_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le1268244103169919719at_nat @ C @ B3 )
% 0.35/0.66 => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7220_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: set_int,B3: set_int,A2: set_int] :
% 0.35/0.66 ( ( ord_less_eq_set_int @ C @ B3 )
% 0.35/0.66 => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7221_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_eq_rat @ C @ B3 )
% 0.35/0.66 => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7222_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_eq_nat @ C @ B3 )
% 0.35/0.66 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7223_sup_OcoboundedI2,axiom,
% 0.35/0.66 ! [C: int,B3: int,A2: int] :
% 0.35/0.66 ( ( ord_less_eq_int @ C @ B3 )
% 0.35/0.66 => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.coboundedI2
% 0.35/0.66 thf(fact_7224_less__infI1,axiom,
% 0.35/0.66 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ A2 @ X2 )
% 0.35/0.66 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7225_less__infI1,axiom,
% 0.35/0.66 ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ A2 @ X2 )
% 0.35/0.66 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7226_less__infI1,axiom,
% 0.35/0.66 ! [A2: real,X2: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ A2 @ X2 )
% 0.35/0.66 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7227_less__infI1,axiom,
% 0.35/0.66 ! [A2: rat,X2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ A2 @ X2 )
% 0.35/0.66 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7228_less__infI1,axiom,
% 0.35/0.66 ! [A2: nat,X2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ A2 @ X2 )
% 0.35/0.66 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7229_less__infI1,axiom,
% 0.35/0.66 ! [A2: int,X2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ X2 )
% 0.35/0.66 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI1
% 0.35/0.66 thf(fact_7230_less__infI2,axiom,
% 0.35/0.66 ! [B3: set_nat,X2: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ B3 @ X2 )
% 0.35/0.66 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7231_less__infI2,axiom,
% 0.35/0.66 ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ B3 @ X2 )
% 0.35/0.66 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7232_less__infI2,axiom,
% 0.35/0.66 ! [B3: real,X2: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ B3 @ X2 )
% 0.35/0.66 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7233_less__infI2,axiom,
% 0.35/0.66 ! [B3: rat,X2: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ B3 @ X2 )
% 0.35/0.66 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7234_less__infI2,axiom,
% 0.35/0.66 ! [B3: nat,X2: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ B3 @ X2 )
% 0.35/0.66 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7235_less__infI2,axiom,
% 0.35/0.66 ! [B3: int,X2: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ B3 @ X2 )
% 0.35/0.66 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_infI2
% 0.35/0.66 thf(fact_7236_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7237_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7238_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_inf_real @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7239_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_inf_rat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7240_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_inf_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7241_inf_Oabsorb3,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.66 => ( ( inf_inf_int @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb3
% 0.35/0.66 thf(fact_7242_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7243_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7244_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_inf_real @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7245_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_inf_rat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7246_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_inf_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7247_inf_Oabsorb4,axiom,
% 0.35/0.66 ! [B3: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.66 => ( ( inf_inf_int @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.absorb4
% 0.35/0.66 thf(fact_7248_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_less_set_nat @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7249_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_le7866589430770878221at_nat @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7250_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: real,B3: real,C: real] :
% 0.35/0.66 ( ( ord_less_real @ A2 @ ( inf_inf_real @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_less_real @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7251_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: rat,B3: rat,C: rat] :
% 0.35/0.66 ( ( ord_less_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_less_rat @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7252_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: nat,B3: nat,C: nat] :
% 0.35/0.66 ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7253_inf_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [A2: int,B3: int,C: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 0.35/0.66 => ~ ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.66 => ~ ( ord_less_int @ A2 @ C ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_boundedE
% 0.35/0.66 thf(fact_7254_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_set_nat
% 0.35/0.66 = ( ^ [A4: set_nat,B4: set_nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_inf_set_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7255_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_le7866589430770878221at_nat
% 0.35/0.66 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7256_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_real
% 0.35/0.66 = ( ^ [A4: real,B4: real] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_inf_real @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7257_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_rat
% 0.35/0.66 = ( ^ [A4: rat,B4: rat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_inf_rat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7258_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_nat
% 0.35/0.66 = ( ^ [A4: nat,B4: nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_inf_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7259_inf_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_int
% 0.35/0.66 = ( ^ [A4: int,B4: int] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( inf_inf_int @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_order_iff
% 0.35/0.66 thf(fact_7260_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: set_nat,C: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ A2 @ C )
% 0.35/0.66 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7261_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ A2 @ C )
% 0.35/0.66 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7262_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: real,C: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ A2 @ C )
% 0.35/0.66 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7263_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: rat,C: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ A2 @ C )
% 0.35/0.66 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7264_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: nat,C: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ A2 @ C )
% 0.35/0.66 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7265_inf_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [A2: int,C: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ C )
% 0.35/0.66 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI1
% 0.35/0.66 thf(fact_7266_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ B3 @ C )
% 0.35/0.66 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7267_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( ord_le7866589430770878221at_nat @ B3 @ C )
% 0.35/0.66 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7268_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: real,C: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ B3 @ C )
% 0.35/0.66 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7269_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ B3 @ C )
% 0.35/0.66 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7270_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ B3 @ C )
% 0.35/0.66 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7271_inf_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [B3: int,C: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ B3 @ C )
% 0.35/0.66 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 0.35/0.66
% 0.35/0.66 % inf.strict_coboundedI2
% 0.35/0.66 thf(fact_7272_less__supI1,axiom,
% 0.35/0.66 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ X2 @ A2 )
% 0.35/0.66 => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7273_less__supI1,axiom,
% 0.35/0.66 ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ X2 @ A2 )
% 0.35/0.66 => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7274_less__supI1,axiom,
% 0.35/0.66 ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ X2 @ A2 )
% 0.35/0.66 => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7275_less__supI1,axiom,
% 0.35/0.66 ! [X2: real,A2: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ X2 @ A2 )
% 0.35/0.66 => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7276_less__supI1,axiom,
% 0.35/0.66 ! [X2: rat,A2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ X2 @ A2 )
% 0.35/0.66 => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7277_less__supI1,axiom,
% 0.35/0.66 ! [X2: nat,A2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ X2 @ A2 )
% 0.35/0.66 => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7278_less__supI1,axiom,
% 0.35/0.66 ! [X2: int,A2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ X2 @ A2 )
% 0.35/0.66 => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI1
% 0.35/0.66 thf(fact_7279_less__supI2,axiom,
% 0.35/0.66 ! [X2: set_nat,B3: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ X2 @ B3 )
% 0.35/0.66 => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7280_less__supI2,axiom,
% 0.35/0.66 ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ X2 @ B3 )
% 0.35/0.66 => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7281_less__supI2,axiom,
% 0.35/0.66 ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ X2 @ B3 )
% 0.35/0.66 => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7282_less__supI2,axiom,
% 0.35/0.66 ! [X2: real,B3: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ X2 @ B3 )
% 0.35/0.66 => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7283_less__supI2,axiom,
% 0.35/0.66 ! [X2: rat,B3: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ X2 @ B3 )
% 0.35/0.66 => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7284_less__supI2,axiom,
% 0.35/0.66 ! [X2: nat,B3: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ X2 @ B3 )
% 0.35/0.66 => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7285_less__supI2,axiom,
% 0.35/0.66 ! [X2: int,B3: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ X2 @ B3 )
% 0.35/0.66 => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_supI2
% 0.35/0.66 thf(fact_7286_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7287_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7288_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7289_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_sup_real @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7290_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_sup_rat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7291_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_sup_nat @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7292_sup_Oabsorb3,axiom,
% 0.35/0.66 ! [B3: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.66 => ( ( sup_sup_int @ A2 @ B3 )
% 0.35/0.66 = A2 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb3
% 0.35/0.66 thf(fact_7293_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7294_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7295_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7296_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_sup_real @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7297_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_sup_rat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7298_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_sup_nat @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7299_sup_Oabsorb4,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ B3 )
% 0.35/0.66 => ( ( sup_sup_int @ A2 @ B3 )
% 0.35/0.66 = B3 ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.absorb4
% 0.35/0.66 thf(fact_7300_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_less_set_nat @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7301_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_le6428140832669894131at_nat @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7302_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_le2604355607129572851at_nat @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7303_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: real,C: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ ( sup_sup_real @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_less_real @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7304_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: rat,C: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_less_rat @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_less_rat @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7305_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: nat,C: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7306_sup_Ostrict__boundedE,axiom,
% 0.35/0.66 ! [B3: int,C: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 0.35/0.66 => ~ ( ( ord_less_int @ B3 @ A2 )
% 0.35/0.66 => ~ ( ord_less_int @ C @ A2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_boundedE
% 0.35/0.66 thf(fact_7307_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_set_nat
% 0.35/0.66 = ( ^ [B4: set_nat,A4: set_nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_sup_set_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7308_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_le6428140832669894131at_nat
% 0.35/0.66 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7309_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_le2604355607129572851at_nat
% 0.35/0.66 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7310_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_real
% 0.35/0.66 = ( ^ [B4: real,A4: real] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_sup_real @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7311_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_rat
% 0.35/0.66 = ( ^ [B4: rat,A4: rat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_sup_rat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7312_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_nat
% 0.35/0.66 = ( ^ [B4: nat,A4: nat] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_sup_nat @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7313_sup_Ostrict__order__iff,axiom,
% 0.35/0.66 ( ord_less_int
% 0.35/0.66 = ( ^ [B4: int,A4: int] :
% 0.35/0.66 ( ( A4
% 0.35/0.66 = ( sup_sup_int @ A4 @ B4 ) )
% 0.35/0.66 & ( A4 != B4 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_order_iff
% 0.35/0.66 thf(fact_7314_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: set_nat,A2: set_nat,B3: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ C @ A2 )
% 0.35/0.66 => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7315_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ C @ A2 )
% 0.35/0.66 => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7316_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ C @ A2 )
% 0.35/0.66 => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7317_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: real,A2: real,B3: real] :
% 0.35/0.66 ( ( ord_less_real @ C @ A2 )
% 0.35/0.66 => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7318_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: rat,A2: rat,B3: rat] :
% 0.35/0.66 ( ( ord_less_rat @ C @ A2 )
% 0.35/0.66 => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7319_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: nat,A2: nat,B3: nat] :
% 0.35/0.66 ( ( ord_less_nat @ C @ A2 )
% 0.35/0.66 => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7320_sup_Ostrict__coboundedI1,axiom,
% 0.35/0.66 ! [C: int,A2: int,B3: int] :
% 0.35/0.66 ( ( ord_less_int @ C @ A2 )
% 0.35/0.66 => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI1
% 0.35/0.66 thf(fact_7321_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: set_nat,B3: set_nat,A2: set_nat] :
% 0.35/0.66 ( ( ord_less_set_nat @ C @ B3 )
% 0.35/0.66 => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7322_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( ( ord_le6428140832669894131at_nat @ C @ B3 )
% 0.35/0.66 => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7323_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( ( ord_le2604355607129572851at_nat @ C @ B3 )
% 0.35/0.66 => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7324_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: real,B3: real,A2: real] :
% 0.35/0.66 ( ( ord_less_real @ C @ B3 )
% 0.35/0.66 => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7325_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: rat,B3: rat,A2: rat] :
% 0.35/0.66 ( ( ord_less_rat @ C @ B3 )
% 0.35/0.66 => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7326_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: nat,B3: nat,A2: nat] :
% 0.35/0.66 ( ( ord_less_nat @ C @ B3 )
% 0.35/0.66 => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7327_sup_Ostrict__coboundedI2,axiom,
% 0.35/0.66 ! [C: int,B3: int,A2: int] :
% 0.35/0.66 ( ( ord_less_int @ C @ B3 )
% 0.35/0.66 => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup.strict_coboundedI2
% 0.35/0.66 thf(fact_7328_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7329_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7330_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ ( inf_in4302113700860409141at_nat @ Y3 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ ( sup_su718114333110466843at_nat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7331_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ Y3 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7332_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) @ ( inf_inf_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ ( sup_sup_set_int @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7333_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ ( sup_sup_rat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7334_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7335_distrib__sup__le,axiom,
% 0.35/0.66 ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X2 @ Y3 ) @ ( sup_sup_int @ X2 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_sup_le
% 0.35/0.66 thf(fact_7336_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7337_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7338_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ ( inf_in4302113700860409141at_nat @ X2 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7339_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ X2 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7340_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ ( inf_inf_set_int @ X2 @ Z ) ) @ ( inf_inf_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7341_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ ( inf_inf_rat @ X2 @ Z ) ) @ ( inf_inf_rat @ X2 @ ( sup_sup_rat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7342_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7343_distrib__inf__le,axiom,
% 0.35/0.66 ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X2 @ Y3 ) @ ( inf_inf_int @ X2 @ Z ) ) @ ( inf_inf_int @ X2 @ ( sup_sup_int @ Y3 @ Z ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % distrib_inf_le
% 0.35/0.66 thf(fact_7344_lemma__termdiff3,axiom,
% 0.35/0.66 ! [H2: real,Z: real,K4: real,N: nat] :
% 0.35/0.66 ( ( H2 != zero_zero_real )
% 0.35/0.66 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
% 0.35/0.66 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K4 )
% 0.35/0.66 => ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % lemma_termdiff3
% 0.35/0.66 thf(fact_7345_lemma__termdiff3,axiom,
% 0.35/0.66 ! [H2: complex,Z: complex,K4: real,N: nat] :
% 0.35/0.66 ( ( H2 != zero_zero_complex )
% 0.35/0.66 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
% 0.35/0.66 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K4 )
% 0.35/0.66 => ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % lemma_termdiff3
% 0.35/0.66 thf(fact_7346_set__encode__insert,axiom,
% 0.35/0.66 ! [A3: set_nat,N: nat] :
% 0.35/0.66 ( ( finite_finite_nat @ A3 )
% 0.35/0.66 => ( ~ ( member_nat @ N @ A3 )
% 0.35/0.66 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 0.35/0.66 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_encode_insert
% 0.35/0.66 thf(fact_7347_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_int,Ys2: list_int] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7348_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7349_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_int,Ys2: list_o] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( product_Pair_int_o @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7350_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_int,Ys2: list_nat] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7351_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7352_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7353_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7354_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7355_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_o,Ys2: list_int] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7356_product__nth,axiom,
% 0.35/0.66 ! [N: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 0.35/0.66 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) @ N )
% 0.35/0.66 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % product_nth
% 0.35/0.66 thf(fact_7357_unset__bit__0,axiom,
% 0.35/0.66 ! [A2: code_integer] :
% 0.35/0.66 ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A2 )
% 0.35/0.66 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % unset_bit_0
% 0.35/0.66 thf(fact_7358_unset__bit__0,axiom,
% 0.35/0.66 ! [A2: nat] :
% 0.35/0.66 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
% 0.35/0.66 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % unset_bit_0
% 0.35/0.66 thf(fact_7359_unset__bit__0,axiom,
% 0.35/0.66 ! [A2: int] :
% 0.35/0.66 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
% 0.35/0.66 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % unset_bit_0
% 0.35/0.66 thf(fact_7360_succ__less__length__list,axiom,
% 0.35/0.66 ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_option_nat
% 0.35/0.66 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 != none_nat )
% 0.35/0.66 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ none_nat
% 0.35/0.66 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % succ_less_length_list
% 0.35/0.66 thf(fact_7361_succ__greatereq__min,axiom,
% 0.35/0.66 ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.35/0.66 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 != none_nat )
% 0.35/0.66 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ none_nat
% 0.35/0.66 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.66 @ none_nat ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % succ_greatereq_min
% 0.35/0.66 thf(fact_7362_set__vebt_H__def,axiom,
% 0.35/0.66 ( vEBT_VEBT_set_vebt
% 0.35/0.66 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_vebt'_def
% 0.35/0.66 thf(fact_7363_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: real > $o,Q: real > $o] :
% 0.35/0.66 ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.66 | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 0.35/0.66 => ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7364_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.66 ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.66 | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 0.35/0.66 => ( finite8100373058378681591st_nat
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7365_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.66 ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.66 | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 0.35/0.66 => ( finite1152437895449049373et_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7366_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.66 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.66 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.35/0.66 => ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7367_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: int > $o,Q: int > $o] :
% 0.35/0.66 ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.66 | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 0.35/0.66 => ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7368_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: complex > $o,Q: complex > $o] :
% 0.35/0.66 ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.66 | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 0.35/0.66 => ( finite3207457112153483333omplex
% 0.35/0.66 @ ( collect_complex
% 0.35/0.66 @ ^ [X: complex] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7369_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.66 ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.66 | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 0.35/0.66 => ( finite6177210948735845034at_nat
% 0.35/0.66 @ ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7370_finite__Collect__conjI,axiom,
% 0.35/0.66 ! [P: extended_enat > $o,Q: extended_enat > $o] :
% 0.35/0.66 ( ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 0.35/0.66 | ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) )
% 0.35/0.66 => ( finite4001608067531595151d_enat
% 0.35/0.66 @ ( collec4429806609662206161d_enat
% 0.35/0.66 @ ^ [X: extended_enat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 & ( Q @ X ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_conjI
% 0.35/0.66 thf(fact_7371_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: real > $o,Q: real > $o] :
% 0.35/0.66 ( ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.66 & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7372_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.66 ( ( finite8100373058378681591st_nat
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.66 & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7373_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.66 ( ( finite1152437895449049373et_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.66 & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7374_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.66 ( ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.66 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7375_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: int > $o,Q: int > $o] :
% 0.35/0.66 ( ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.66 & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7376_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: complex > $o,Q: complex > $o] :
% 0.35/0.66 ( ( finite3207457112153483333omplex
% 0.35/0.66 @ ( collect_complex
% 0.35/0.66 @ ^ [X: complex] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.66 & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7377_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.66 ( ( finite6177210948735845034at_nat
% 0.35/0.66 @ ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.66 & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7378_finite__Collect__disjI,axiom,
% 0.35/0.66 ! [P: extended_enat > $o,Q: extended_enat > $o] :
% 0.35/0.66 ( ( finite4001608067531595151d_enat
% 0.35/0.66 @ ( collec4429806609662206161d_enat
% 0.35/0.66 @ ^ [X: extended_enat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) ) )
% 0.35/0.66 = ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 0.35/0.66 & ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_disjI
% 0.35/0.66 thf(fact_7379_succ__empty,axiom,
% 0.35/0.66 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.66 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.66 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 0.35/0.66 = none_nat )
% 0.35/0.66 = ( ( collect_nat
% 0.35/0.66 @ ^ [Y: nat] :
% 0.35/0.66 ( ( vEBT_vebt_member @ T @ Y )
% 0.35/0.66 & ( ord_less_nat @ X2 @ Y ) ) )
% 0.35/0.66 = bot_bot_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % succ_empty
% 0.35/0.66 thf(fact_7380_pred__empty,axiom,
% 0.35/0.66 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 0.35/0.66 ( ( vEBT_invar_vebt @ T @ N )
% 0.35/0.66 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 0.35/0.66 = none_nat )
% 0.35/0.66 = ( ( collect_nat
% 0.35/0.66 @ ^ [Y: nat] :
% 0.35/0.66 ( ( vEBT_vebt_member @ T @ Y )
% 0.35/0.66 & ( ord_less_nat @ Y @ X2 ) ) )
% 0.35/0.66 = bot_bot_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_empty
% 0.35/0.66 thf(fact_7381_unset__bit__nonnegative__int__iff,axiom,
% 0.35/0.66 ! [N: nat,K: int] :
% 0.35/0.66 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 0.35/0.66 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.66
% 0.35/0.66 % unset_bit_nonnegative_int_iff
% 0.35/0.66 thf(fact_7382_unset__bit__negative__int__iff,axiom,
% 0.35/0.66 ! [N: nat,K: int] :
% 0.35/0.66 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.66 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.66
% 0.35/0.66 % unset_bit_negative_int_iff
% 0.35/0.66 thf(fact_7383_finite__nth__roots,axiom,
% 0.35/0.66 ! [N: nat,C: complex] :
% 0.35/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.66 => ( finite3207457112153483333omplex
% 0.35/0.66 @ ( collect_complex
% 0.35/0.66 @ ^ [Z2: complex] :
% 0.35/0.66 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.66 = C ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_nth_roots
% 0.35/0.66 thf(fact_7384_finite__Collect__subsets,axiom,
% 0.35/0.66 ! [A3: set_nat] :
% 0.35/0.66 ( ( finite_finite_nat @ A3 )
% 0.35/0.66 => ( finite1152437895449049373et_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_subsets
% 0.35/0.66 thf(fact_7385_finite__Collect__subsets,axiom,
% 0.35/0.66 ! [A3: set_complex] :
% 0.35/0.66 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.66 => ( finite6551019134538273531omplex
% 0.35/0.66 @ ( collect_set_complex
% 0.35/0.66 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_subsets
% 0.35/0.66 thf(fact_7386_finite__Collect__subsets,axiom,
% 0.35/0.66 ! [A3: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.66 => ( finite9047747110432174090at_nat
% 0.35/0.66 @ ( collec5514110066124741708at_nat
% 0.35/0.66 @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_subsets
% 0.35/0.66 thf(fact_7387_finite__Collect__subsets,axiom,
% 0.35/0.66 ! [A3: set_Extended_enat] :
% 0.35/0.66 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.66 => ( finite5468666774076196335d_enat
% 0.35/0.66 @ ( collec2260605976452661553d_enat
% 0.35/0.66 @ ^ [B6: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B6 @ A3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_subsets
% 0.35/0.66 thf(fact_7388_finite__Collect__subsets,axiom,
% 0.35/0.66 ! [A3: set_int] :
% 0.35/0.66 ( ( finite_finite_int @ A3 )
% 0.35/0.66 => ( finite6197958912794628473et_int
% 0.35/0.66 @ ( collect_set_int
% 0.35/0.66 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_subsets
% 0.35/0.66 thf(fact_7389_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: product_prod_nat_nat] :
% 0.35/0.66 ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ( ^ [Y6: product_prod_nat_nat,Z3: product_prod_nat_nat] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7390_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: list_nat] :
% 0.35/0.66 ( ( collect_list_nat
% 0.35/0.66 @ ( ^ [Y6: list_nat,Z3: list_nat] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7391_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: set_nat] :
% 0.35/0.66 ( ( collect_set_nat
% 0.35/0.66 @ ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7392_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: real] :
% 0.35/0.66 ( ( collect_real
% 0.35/0.66 @ ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7393_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: $o] :
% 0.35/0.66 ( ( collect_o
% 0.35/0.66 @ ( ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7394_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: nat] :
% 0.35/0.66 ( ( collect_nat
% 0.35/0.66 @ ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7395_singleton__conv2,axiom,
% 0.35/0.66 ! [A2: int] :
% 0.35/0.66 ( ( collect_int
% 0.35/0.66 @ ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 0.35/0.66 @ A2 ) )
% 0.35/0.66 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv2
% 0.35/0.66 thf(fact_7396_singleton__conv,axiom,
% 0.35/0.66 ! [A2: product_prod_nat_nat] :
% 0.35/0.66 ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] : ( X = A2 ) )
% 0.35/0.66 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7397_singleton__conv,axiom,
% 0.35/0.66 ! [A2: list_nat] :
% 0.35/0.66 ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] : ( X = A2 ) )
% 0.35/0.66 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7398_singleton__conv,axiom,
% 0.35/0.66 ! [A2: set_nat] :
% 0.35/0.66 ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] : ( X = A2 ) )
% 0.35/0.66 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7399_singleton__conv,axiom,
% 0.35/0.66 ! [A2: real] :
% 0.35/0.66 ( ( collect_real
% 0.35/0.66 @ ^ [X: real] : ( X = A2 ) )
% 0.35/0.66 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7400_singleton__conv,axiom,
% 0.35/0.66 ! [A2: $o] :
% 0.35/0.66 ( ( collect_o
% 0.35/0.66 @ ^ [X: $o] : ( X = A2 ) )
% 0.35/0.66 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7401_singleton__conv,axiom,
% 0.35/0.66 ! [A2: nat] :
% 0.35/0.66 ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] : ( X = A2 ) )
% 0.35/0.66 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7402_singleton__conv,axiom,
% 0.35/0.66 ! [A2: int] :
% 0.35/0.66 ( ( collect_int
% 0.35/0.66 @ ^ [X: int] : ( X = A2 ) )
% 0.35/0.66 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 0.35/0.66
% 0.35/0.66 % singleton_conv
% 0.35/0.66 thf(fact_7403_finite__Collect__less__nat,axiom,
% 0.35/0.66 ! [K: nat] :
% 0.35/0.66 ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_less_nat
% 0.35/0.66 thf(fact_7404_finite__Collect__le__nat,axiom,
% 0.35/0.66 ! [K: nat] :
% 0.35/0.66 ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_Collect_le_nat
% 0.35/0.66 thf(fact_7405_card__Collect__less__nat,axiom,
% 0.35/0.66 ! [N: nat] :
% 0.35/0.66 ( ( finite_card_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 0.35/0.66 = N ) ).
% 0.35/0.66
% 0.35/0.66 % card_Collect_less_nat
% 0.35/0.66 thf(fact_7406_finite__interval__int1,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [I4: int] :
% 0.35/0.66 ( ( ord_less_eq_int @ A2 @ I4 )
% 0.35/0.66 & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_interval_int1
% 0.35/0.66 thf(fact_7407_finite__interval__int4,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [I4: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ I4 )
% 0.35/0.66 & ( ord_less_int @ I4 @ B3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_interval_int4
% 0.35/0.66 thf(fact_7408_norm__eq__zero,axiom,
% 0.35/0.66 ! [X2: real] :
% 0.35/0.66 ( ( ( real_V7735802525324610683m_real @ X2 )
% 0.35/0.66 = zero_zero_real )
% 0.35/0.66 = ( X2 = zero_zero_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % norm_eq_zero
% 0.35/0.66 thf(fact_7409_norm__eq__zero,axiom,
% 0.35/0.66 ! [X2: complex] :
% 0.35/0.66 ( ( ( real_V1022390504157884413omplex @ X2 )
% 0.35/0.66 = zero_zero_real )
% 0.35/0.66 = ( X2 = zero_zero_complex ) ) ).
% 0.35/0.66
% 0.35/0.66 % norm_eq_zero
% 0.35/0.66 thf(fact_7410_norm__zero,axiom,
% 0.35/0.66 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.35/0.66 = zero_zero_real ) ).
% 0.35/0.66
% 0.35/0.66 % norm_zero
% 0.35/0.66 thf(fact_7411_norm__zero,axiom,
% 0.35/0.66 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.35/0.66 = zero_zero_real ) ).
% 0.35/0.66
% 0.35/0.66 % norm_zero
% 0.35/0.66 thf(fact_7412_set__encode__empty,axiom,
% 0.35/0.66 ( ( nat_set_encode @ bot_bot_set_nat )
% 0.35/0.66 = zero_zero_nat ) ).
% 0.35/0.66
% 0.35/0.66 % set_encode_empty
% 0.35/0.66 thf(fact_7413_card__Collect__le__nat,axiom,
% 0.35/0.66 ! [N: nat] :
% 0.35/0.66 ( ( finite_card_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 0.35/0.66 = ( suc @ N ) ) ).
% 0.35/0.66
% 0.35/0.66 % card_Collect_le_nat
% 0.35/0.66 thf(fact_7414_finite__interval__int2,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [I4: int] :
% 0.35/0.66 ( ( ord_less_eq_int @ A2 @ I4 )
% 0.35/0.66 & ( ord_less_int @ I4 @ B3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_interval_int2
% 0.35/0.66 thf(fact_7415_finite__interval__int3,axiom,
% 0.35/0.66 ! [A2: int,B3: int] :
% 0.35/0.66 ( finite_finite_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [I4: int] :
% 0.35/0.66 ( ( ord_less_int @ A2 @ I4 )
% 0.35/0.66 & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_interval_int3
% 0.35/0.66 thf(fact_7416_zero__less__norm__iff,axiom,
% 0.35/0.66 ! [X2: real] :
% 0.35/0.66 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
% 0.35/0.66 = ( X2 != zero_zero_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % zero_less_norm_iff
% 0.35/0.66 thf(fact_7417_zero__less__norm__iff,axiom,
% 0.35/0.66 ! [X2: complex] :
% 0.35/0.66 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
% 0.35/0.66 = ( X2 != zero_zero_complex ) ) ).
% 0.35/0.66
% 0.35/0.66 % zero_less_norm_iff
% 0.35/0.66 thf(fact_7418_norm__le__zero__iff,axiom,
% 0.35/0.66 ! [X2: real] :
% 0.35/0.66 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
% 0.35/0.66 = ( X2 = zero_zero_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % norm_le_zero_iff
% 0.35/0.66 thf(fact_7419_norm__le__zero__iff,axiom,
% 0.35/0.66 ! [X2: complex] :
% 0.35/0.66 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
% 0.35/0.66 = ( X2 = zero_zero_complex ) ) ).
% 0.35/0.66
% 0.35/0.66 % norm_le_zero_iff
% 0.35/0.66 thf(fact_7420_del__x__not__mia,axiom,
% 0.35/0.66 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ( ord_less_nat @ Mi @ X2 )
% 0.35/0.66 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Mi
% 0.35/0.66 @ ( if_nat @ ( X2 = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Mi
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_not_mia
% 0.35/0.66 thf(fact_7421_del__x__not__mi__new__node__nil,axiom,
% 0.35/0.66 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.66 ( ( ( ord_less_nat @ Mi @ X2 )
% 0.35/0.66 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( Newnode
% 0.35/0.66 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( Sn
% 0.35/0.66 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 => ( ( Newlist
% 0.35/0.66 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Mi
% 0.35/0.66 @ ( if_nat @ ( X2 = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ Sn )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Mi
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ Newlist
% 0.35/0.66 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_not_mi_new_node_nil
% 0.35/0.66 thf(fact_7422_del__x__not__mi,axiom,
% 0.35/0.66 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ( ord_less_nat @ Mi @ X2 )
% 0.35/0.66 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( Newnode
% 0.35/0.66 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 => ( ( Newlist
% 0.35/0.66 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Mi
% 0.35/0.66 @ ( if_nat @ ( X2 = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Mi
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ Newlist
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.35/0.66 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_not_mi
% 0.35/0.66 thf(fact_7423_del__in__range,axiom,
% 0.35/0.66 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ( ord_less_eq_nat @ Mi @ X2 )
% 0.35/0.66 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( ( X2 = Mi )
% 0.35/0.66 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 = Ma ) )
% 0.35/0.66 & ( ( X2 != Mi )
% 0.35/0.66 => ( X2 = Ma ) ) )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( ( X2 = Mi )
% 0.35/0.66 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 = Ma ) )
% 0.35/0.66 & ( ( X2 != Mi )
% 0.35/0.66 => ( X2 = Ma ) ) )
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ Summary ) )
% 0.35/0.66 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_in_range
% 0.35/0.66 thf(fact_7424_del__x__mi,axiom,
% 0.35/0.66 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
% 0.35/0.66 ( ( ( X2 = Mi )
% 0.35/0.66 & ( ord_less_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( Xn
% 0.35/0.66 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Xn
% 0.35/0.66 @ ( if_nat @ ( Xn = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Xn
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_mi
% 0.35/0.66 thf(fact_7425_del__x__mi__lets__in,axiom,
% 0.35/0.66 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 0.35/0.66 ( ( ( X2 = Mi )
% 0.35/0.66 & ( ord_less_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( Xn
% 0.35/0.66 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( Newnode
% 0.35/0.66 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 => ( ( Newlist
% 0.35/0.66 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.66 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Xn
% 0.35/0.66 @ ( if_nat @ ( Xn = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Xn
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ Newlist
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 0.35/0.66 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_mi_lets_in
% 0.35/0.66 thf(fact_7426_del__x__mi__lets__in__minNull,axiom,
% 0.35/0.66 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 0.35/0.66 ( ( ( X2 = Mi )
% 0.35/0.66 & ( ord_less_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = H2 )
% 0.35/0.66 => ( ( Xn
% 0.35/0.66 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 0.35/0.66 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.66 = L )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( Newnode
% 0.35/0.66 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 0.35/0.66 => ( ( Newlist
% 0.35/0.66 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 0.35/0.66 => ( ( vEBT_VEBT_minNull @ Newnode )
% 0.35/0.66 => ( ( Sn
% 0.35/0.66 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ Xn
% 0.35/0.66 @ ( if_nat @ ( Xn = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ Sn )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ Xn
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ Newlist
% 0.35/0.66 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_mi_lets_in_minNull
% 0.35/0.66 thf(fact_7427_del__x__mia,axiom,
% 0.35/0.66 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ( X2 = Mi )
% 0.35/0.66 & ( ord_less_nat @ X2 @ Ma ) )
% 0.35/0.66 => ( ( Mi != Ma )
% 0.35/0.66 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 = Ma )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( vEBT_Node
% 0.35/0.66 @ ( some_P7363390416028606310at_nat
% 0.35/0.66 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 @ ( if_nat
% 0.35/0.66 @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.66 = Ma )
% 0.35/0.66 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ Ma ) ) )
% 0.35/0.66 @ Deg
% 0.35/0.66 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ Summary ) )
% 0.35/0.66 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % del_x_mia
% 0.35/0.66 thf(fact_7428_pred__less__length__list,axiom,
% 0.35/0.66 ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 0.35/0.66 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_option_nat
% 0.35/0.66 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 != none_nat )
% 0.35/0.66 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.66 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_less_length_list
% 0.35/0.66 thf(fact_7429_pred__lesseq__max,axiom,
% 0.35/0.66 ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.66 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.35/0.66 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 0.35/0.66 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.66 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 != none_nat )
% 0.35/0.66 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.66 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.66 @ ( if_option_nat
% 0.35/0.66 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.66 = none_nat )
% 0.35/0.66 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.66 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.66 @ none_nat ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_lesseq_max
% 0.35/0.66 thf(fact_7430_max__def__raw,axiom,
% 0.35/0.66 ( ord_max_set_int
% 0.35/0.66 = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % max_def_raw
% 0.35/0.66 thf(fact_7431_max__def__raw,axiom,
% 0.35/0.66 ( ord_max_rat
% 0.35/0.66 = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % max_def_raw
% 0.35/0.66 thf(fact_7432_max__def__raw,axiom,
% 0.35/0.66 ( ord_max_num
% 0.35/0.66 = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % max_def_raw
% 0.35/0.66 thf(fact_7433_max__def__raw,axiom,
% 0.35/0.66 ( ord_max_nat
% 0.35/0.66 = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % max_def_raw
% 0.35/0.66 thf(fact_7434_max__def__raw,axiom,
% 0.35/0.66 ( ord_max_int
% 0.35/0.66 = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % max_def_raw
% 0.35/0.66 thf(fact_7435_finite__M__bounded__by__nat,axiom,
% 0.35/0.66 ! [P: nat > $o,I: nat] :
% 0.35/0.66 ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [K3: nat] :
% 0.35/0.66 ( ( P @ K3 )
% 0.35/0.66 & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_M_bounded_by_nat
% 0.35/0.66 thf(fact_7436_card__nth__roots,axiom,
% 0.35/0.66 ! [C: complex,N: nat] :
% 0.35/0.66 ( ( C != zero_zero_complex )
% 0.35/0.66 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.66 => ( ( finite_card_complex
% 0.35/0.66 @ ( collect_complex
% 0.35/0.66 @ ^ [Z2: complex] :
% 0.35/0.66 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.66 = C ) ) )
% 0.35/0.66 = N ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % card_nth_roots
% 0.35/0.66 thf(fact_7437_card__roots__unity__eq,axiom,
% 0.35/0.66 ! [N: nat] :
% 0.35/0.66 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.66 => ( ( finite_card_complex
% 0.35/0.66 @ ( collect_complex
% 0.35/0.66 @ ^ [Z2: complex] :
% 0.35/0.66 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.66 = one_one_complex ) ) )
% 0.35/0.66 = N ) ) ).
% 0.35/0.66
% 0.35/0.66 % card_roots_unity_eq
% 0.35/0.66 thf(fact_7438_minus__set__def,axiom,
% 0.35/0.66 ( minus_minus_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ( minus_minus_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7439_minus__set__def,axiom,
% 0.35/0.66 ( minus_minus_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ( minus_minus_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7440_minus__set__def,axiom,
% 0.35/0.66 ( minus_7954133019191499631st_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ( minus_1139252259498527702_nat_o
% 0.35/0.66 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7441_minus__set__def,axiom,
% 0.35/0.66 ( minus_2163939370556025621et_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ( minus_6910147592129066416_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7442_minus__set__def,axiom,
% 0.35/0.66 ( minus_minus_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ( minus_minus_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7443_minus__set__def,axiom,
% 0.35/0.66 ( minus_minus_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ( minus_minus_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % minus_set_def
% 0.35/0.66 thf(fact_7444_set__diff__eq,axiom,
% 0.35/0.66 ( minus_minus_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( member_o @ X @ A6 )
% 0.35/0.66 & ~ ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7445_set__diff__eq,axiom,
% 0.35/0.66 ( minus_minus_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( member_real @ X @ A6 )
% 0.35/0.66 & ~ ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7446_set__diff__eq,axiom,
% 0.35/0.66 ( minus_7954133019191499631st_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( member_list_nat @ X @ A6 )
% 0.35/0.66 & ~ ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7447_set__diff__eq,axiom,
% 0.35/0.66 ( minus_2163939370556025621et_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( member_set_nat @ X @ A6 )
% 0.35/0.66 & ~ ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7448_set__diff__eq,axiom,
% 0.35/0.66 ( minus_minus_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( member_int @ X @ A6 )
% 0.35/0.66 & ~ ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7449_set__diff__eq,axiom,
% 0.35/0.66 ( minus_minus_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( member_nat @ X @ A6 )
% 0.35/0.66 & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % set_diff_eq
% 0.35/0.66 thf(fact_7450_lambda__zero,axiom,
% 0.35/0.66 ( ( ^ [H: complex] : zero_zero_complex )
% 0.35/0.66 = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.35/0.66
% 0.35/0.66 % lambda_zero
% 0.35/0.66 thf(fact_7451_lambda__zero,axiom,
% 0.35/0.66 ( ( ^ [H: real] : zero_zero_real )
% 0.35/0.66 = ( times_times_real @ zero_zero_real ) ) ).
% 0.35/0.66
% 0.35/0.66 % lambda_zero
% 0.35/0.66 thf(fact_7452_lambda__zero,axiom,
% 0.35/0.66 ( ( ^ [H: rat] : zero_zero_rat )
% 0.35/0.66 = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.35/0.66
% 0.35/0.66 % lambda_zero
% 0.35/0.66 thf(fact_7453_lambda__zero,axiom,
% 0.35/0.66 ( ( ^ [H: nat] : zero_zero_nat )
% 0.35/0.66 = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.35/0.66
% 0.35/0.66 % lambda_zero
% 0.35/0.66 thf(fact_7454_lambda__zero,axiom,
% 0.35/0.66 ( ( ^ [H: int] : zero_zero_int )
% 0.35/0.66 = ( times_times_int @ zero_zero_int ) ) ).
% 0.35/0.66
% 0.35/0.66 % lambda_zero
% 0.35/0.66 thf(fact_7455_less__set__def,axiom,
% 0.35/0.66 ( ord_less_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( ord_less_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_set_def
% 0.35/0.66 thf(fact_7456_less__set__def,axiom,
% 0.35/0.66 ( ord_less_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( ord_less_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_set_def
% 0.35/0.66 thf(fact_7457_less__set__def,axiom,
% 0.35/0.66 ( ord_less_set_set_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( ord_less_set_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_set_def
% 0.35/0.66 thf(fact_7458_less__set__def,axiom,
% 0.35/0.66 ( ord_less_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( ord_less_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_set_def
% 0.35/0.66 thf(fact_7459_less__set__def,axiom,
% 0.35/0.66 ( ord_less_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( ord_less_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_set_def
% 0.35/0.66 thf(fact_7460_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( member_o @ X @ A6 )
% 0.35/0.66 | ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7461_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( member_real @ X @ A6 )
% 0.35/0.66 | ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7462_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_list_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( member_list_nat @ X @ A6 )
% 0.35/0.66 | ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7463_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_set_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( member_set_nat @ X @ A6 )
% 0.35/0.66 | ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7464_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( member_int @ X @ A6 )
% 0.35/0.66 | ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7465_Un__def,axiom,
% 0.35/0.66 ( sup_sup_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( member_nat @ X @ A6 )
% 0.35/0.66 | ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7466_Un__def,axiom,
% 0.35/0.66 ( sup_su718114333110466843at_nat
% 0.35/0.66 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( collec7088162979684241874at_nat
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] :
% 0.35/0.66 ( ( member8206827879206165904at_nat @ X @ A6 )
% 0.35/0.66 | ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7467_Un__def,axiom,
% 0.35/0.66 ( sup_su5525570899277871387at_nat
% 0.35/0.66 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( collec6321179662152712658at_nat
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] :
% 0.35/0.66 ( ( member8757157785044589968at_nat @ X @ A6 )
% 0.35/0.66 | ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Un_def
% 0.35/0.66 thf(fact_7468_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ( sup_sup_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7469_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ( sup_sup_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7470_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_list_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ( sup_sup_list_nat_o
% 0.35/0.66 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7471_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_set_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ( sup_sup_set_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7472_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ( sup_sup_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7473_sup__set__def,axiom,
% 0.35/0.66 ( sup_sup_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ( sup_sup_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7474_sup__set__def,axiom,
% 0.35/0.66 ( sup_su718114333110466843at_nat
% 0.35/0.66 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 0.35/0.66 ( collec7088162979684241874at_nat
% 0.35/0.66 @ ( sup_su8986005896011022210_nat_o
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7475_sup__set__def,axiom,
% 0.35/0.66 ( sup_su5525570899277871387at_nat
% 0.35/0.66 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 0.35/0.66 ( collec6321179662152712658at_nat
% 0.35/0.66 @ ( sup_su2080679670758317954_nat_o
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % sup_set_def
% 0.35/0.66 thf(fact_7476_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: real > $o,Q: real > $o] :
% 0.35/0.66 ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7477_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.66 ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7478_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.66 ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7479_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: int > $o,Q: int > $o] :
% 0.35/0.66 ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7480_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.66 ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7481_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
% 0.35/0.66 ( ( collec7088162979684241874at_nat
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_su718114333110466843at_nat @ ( collec7088162979684241874at_nat @ P ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7482_Collect__disj__eq,axiom,
% 0.35/0.66 ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
% 0.35/0.66 ( ( collec6321179662152712658at_nat
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 | ( Q @ X ) ) )
% 0.35/0.66 = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_disj_eq
% 0.35/0.66 thf(fact_7483_insert__def,axiom,
% 0.35/0.66 ( insert8211810215607154385at_nat
% 0.35/0.66 = ( ^ [A4: product_prod_nat_nat] :
% 0.35/0.66 ( sup_su6327502436637775413at_nat
% 0.35/0.66 @ ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7484_insert__def,axiom,
% 0.35/0.66 ( insert_o
% 0.35/0.66 = ( ^ [A4: $o] :
% 0.35/0.66 ( sup_sup_set_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [X: $o] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7485_insert__def,axiom,
% 0.35/0.66 ( insert_real
% 0.35/0.66 = ( ^ [A4: real] :
% 0.35/0.66 ( sup_sup_set_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7486_insert__def,axiom,
% 0.35/0.66 ( insert_list_nat
% 0.35/0.66 = ( ^ [A4: list_nat] :
% 0.35/0.66 ( sup_sup_set_list_nat
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7487_insert__def,axiom,
% 0.35/0.66 ( insert_set_nat
% 0.35/0.66 = ( ^ [A4: set_nat] :
% 0.35/0.66 ( sup_sup_set_set_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7488_insert__def,axiom,
% 0.35/0.66 ( insert_int
% 0.35/0.66 = ( ^ [A4: int] :
% 0.35/0.66 ( sup_sup_set_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7489_insert__def,axiom,
% 0.35/0.66 ( insert_nat
% 0.35/0.66 = ( ^ [A4: nat] :
% 0.35/0.66 ( sup_sup_set_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7490_insert__def,axiom,
% 0.35/0.66 ( insert5050368324300391991at_nat
% 0.35/0.66 = ( ^ [A4: produc859450856879609959at_nat] :
% 0.35/0.66 ( sup_su718114333110466843at_nat
% 0.35/0.66 @ ( collec7088162979684241874at_nat
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7491_insert__def,axiom,
% 0.35/0.66 ( insert9069300056098147895at_nat
% 0.35/0.66 = ( ^ [A4: produc3843707927480180839at_nat] :
% 0.35/0.66 ( sup_su5525570899277871387at_nat
% 0.35/0.66 @ ( collec6321179662152712658at_nat
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] : ( X = A4 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_def
% 0.35/0.66 thf(fact_7492_Compl__eq,axiom,
% 0.35/0.66 ( uminus_uminus_set_o
% 0.35/0.66 = ( ^ [A6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ~ ( member_o @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7493_Compl__eq,axiom,
% 0.35/0.66 ( uminus612125837232591019t_real
% 0.35/0.66 = ( ^ [A6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ~ ( member_real @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7494_Compl__eq,axiom,
% 0.35/0.66 ( uminus3195874150345416415st_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ~ ( member_list_nat @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7495_Compl__eq,axiom,
% 0.35/0.66 ( uminus613421341184616069et_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ~ ( member_set_nat @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7496_Compl__eq,axiom,
% 0.35/0.66 ( uminus5710092332889474511et_nat
% 0.35/0.66 = ( ^ [A6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ~ ( member_nat @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7497_Compl__eq,axiom,
% 0.35/0.66 ( uminus1532241313380277803et_int
% 0.35/0.66 = ( ^ [A6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ~ ( member_int @ X @ A6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Compl_eq
% 0.35/0.66 thf(fact_7498_Collect__neg__eq,axiom,
% 0.35/0.66 ! [P: real > $o] :
% 0.35/0.66 ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ~ ( P @ X ) )
% 0.35/0.66 = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_neg_eq
% 0.35/0.66 thf(fact_7499_Collect__neg__eq,axiom,
% 0.35/0.66 ! [P: list_nat > $o] :
% 0.35/0.66 ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ~ ( P @ X ) )
% 0.35/0.66 = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_neg_eq
% 0.35/0.66 thf(fact_7500_Collect__neg__eq,axiom,
% 0.35/0.66 ! [P: set_nat > $o] :
% 0.35/0.66 ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ~ ( P @ X ) )
% 0.35/0.66 = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_neg_eq
% 0.35/0.66 thf(fact_7501_Collect__neg__eq,axiom,
% 0.35/0.66 ! [P: nat > $o] :
% 0.35/0.66 ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ~ ( P @ X ) )
% 0.35/0.66 = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_neg_eq
% 0.35/0.66 thf(fact_7502_Collect__neg__eq,axiom,
% 0.35/0.66 ! [P: int > $o] :
% 0.35/0.66 ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ~ ( P @ X ) )
% 0.35/0.66 = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_neg_eq
% 0.35/0.66 thf(fact_7503_uminus__set__def,axiom,
% 0.35/0.66 ( uminus_uminus_set_o
% 0.35/0.66 = ( ^ [A6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ( uminus_uminus_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7504_uminus__set__def,axiom,
% 0.35/0.66 ( uminus612125837232591019t_real
% 0.35/0.66 = ( ^ [A6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ( uminus_uminus_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7505_uminus__set__def,axiom,
% 0.35/0.66 ( uminus3195874150345416415st_nat
% 0.35/0.66 = ( ^ [A6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ( uminus5770388063884162150_nat_o
% 0.35/0.66 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7506_uminus__set__def,axiom,
% 0.35/0.66 ( uminus613421341184616069et_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ( uminus6401447641752708672_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7507_uminus__set__def,axiom,
% 0.35/0.66 ( uminus5710092332889474511et_nat
% 0.35/0.66 = ( ^ [A6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ( uminus_uminus_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7508_uminus__set__def,axiom,
% 0.35/0.66 ( uminus1532241313380277803et_int
% 0.35/0.66 = ( ^ [A6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ( uminus_uminus_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ A6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % uminus_set_def
% 0.35/0.66 thf(fact_7509_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: real > $o] :
% 0.35/0.66 ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 0.35/0.66 => ? [X_1: real] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7510_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: list_nat > $o] :
% 0.35/0.66 ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.35/0.66 => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7511_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: set_nat > $o] :
% 0.35/0.66 ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.35/0.66 => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7512_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: nat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.66 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7513_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: int > $o] :
% 0.35/0.66 ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 0.35/0.66 => ? [X_1: int] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7514_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: complex > $o] :
% 0.35/0.66 ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.35/0.66 => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7515_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: product_prod_nat_nat > $o] :
% 0.35/0.66 ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.66 => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7516_not__finite__existsD,axiom,
% 0.35/0.66 ! [P: extended_enat > $o] :
% 0.35/0.66 ( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 0.35/0.66 => ? [X_1: extended_enat] : ( P @ X_1 ) ) ).
% 0.35/0.66
% 0.35/0.66 % not_finite_existsD
% 0.35/0.66 thf(fact_7517_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_o,B2: set_nat,R: $o > nat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_o @ A3 )
% 0.35/0.66 => ( ( finite_finite_nat @ B2 )
% 0.35/0.66 => ( ! [X5: $o] :
% 0.35/0.66 ( ( member_o @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: nat] :
% 0.35/0.66 ( ( member_nat @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: nat] :
% 0.35/0.66 ( ( member_nat @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [A4: $o] :
% 0.35/0.66 ( ( member_o @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7518_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_real,B2: set_nat,R: real > nat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_real @ A3 )
% 0.35/0.66 => ( ( finite_finite_nat @ B2 )
% 0.35/0.66 => ( ! [X5: real] :
% 0.35/0.66 ( ( member_real @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: nat] :
% 0.35/0.66 ( ( member_nat @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: nat] :
% 0.35/0.66 ( ( member_nat @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [A4: real] :
% 0.35/0.66 ( ( member_real @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7519_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_o,B2: set_int,R: $o > int > $o] :
% 0.35/0.66 ( ~ ( finite_finite_o @ A3 )
% 0.35/0.66 => ( ( finite_finite_int @ B2 )
% 0.35/0.66 => ( ! [X5: $o] :
% 0.35/0.66 ( ( member_o @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: int] :
% 0.35/0.66 ( ( member_int @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: int] :
% 0.35/0.66 ( ( member_int @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [A4: $o] :
% 0.35/0.66 ( ( member_o @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7520_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_real,B2: set_int,R: real > int > $o] :
% 0.35/0.66 ( ~ ( finite_finite_real @ A3 )
% 0.35/0.66 => ( ( finite_finite_int @ B2 )
% 0.35/0.66 => ( ! [X5: real] :
% 0.35/0.66 ( ( member_real @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: int] :
% 0.35/0.66 ( ( member_int @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: int] :
% 0.35/0.66 ( ( member_int @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [A4: real] :
% 0.35/0.66 ( ( member_real @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7521_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_o,B2: set_complex,R: $o > complex > $o] :
% 0.35/0.66 ( ~ ( finite_finite_o @ A3 )
% 0.35/0.66 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.66 => ( ! [X5: $o] :
% 0.35/0.66 ( ( member_o @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: complex] :
% 0.35/0.66 ( ( member_complex @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: complex] :
% 0.35/0.66 ( ( member_complex @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [A4: $o] :
% 0.35/0.66 ( ( member_o @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7522_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_real,B2: set_complex,R: real > complex > $o] :
% 0.35/0.66 ( ~ ( finite_finite_real @ A3 )
% 0.35/0.66 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.66 => ( ! [X5: real] :
% 0.35/0.66 ( ( member_real @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: complex] :
% 0.35/0.66 ( ( member_complex @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: complex] :
% 0.35/0.66 ( ( member_complex @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [A4: real] :
% 0.35/0.66 ( ( member_real @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7523_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_o,B2: set_Extended_enat,R: $o > extended_enat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_o @ A3 )
% 0.35/0.66 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.66 => ( ! [X5: $o] :
% 0.35/0.66 ( ( member_o @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: extended_enat] :
% 0.35/0.66 ( ( member_Extended_enat @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: extended_enat] :
% 0.35/0.66 ( ( member_Extended_enat @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [A4: $o] :
% 0.35/0.66 ( ( member_o @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7524_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_real,B2: set_Extended_enat,R: real > extended_enat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_real @ A3 )
% 0.35/0.66 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.66 => ( ! [X5: real] :
% 0.35/0.66 ( ( member_real @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: extended_enat] :
% 0.35/0.66 ( ( member_Extended_enat @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: extended_enat] :
% 0.35/0.66 ( ( member_Extended_enat @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [A4: real] :
% 0.35/0.66 ( ( member_real @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7525_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_nat,B2: set_nat,R: nat > nat > $o] :
% 0.35/0.66 ( ~ ( finite_finite_nat @ A3 )
% 0.35/0.66 => ( ( finite_finite_nat @ B2 )
% 0.35/0.66 => ( ! [X5: nat] :
% 0.35/0.66 ( ( member_nat @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: nat] :
% 0.35/0.66 ( ( member_nat @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: nat] :
% 0.35/0.66 ( ( member_nat @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [A4: nat] :
% 0.35/0.66 ( ( member_nat @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7526_pigeonhole__infinite__rel,axiom,
% 0.35/0.66 ! [A3: set_nat,B2: set_int,R: nat > int > $o] :
% 0.35/0.66 ( ~ ( finite_finite_nat @ A3 )
% 0.35/0.66 => ( ( finite_finite_int @ B2 )
% 0.35/0.66 => ( ! [X5: nat] :
% 0.35/0.66 ( ( member_nat @ X5 @ A3 )
% 0.35/0.66 => ? [Xa: int] :
% 0.35/0.66 ( ( member_int @ Xa @ B2 )
% 0.35/0.66 & ( R @ X5 @ Xa ) ) )
% 0.35/0.66 => ? [X5: int] :
% 0.35/0.66 ( ( member_int @ X5 @ B2 )
% 0.35/0.66 & ~ ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [A4: nat] :
% 0.35/0.66 ( ( member_nat @ A4 @ A3 )
% 0.35/0.66 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % pigeonhole_infinite_rel
% 0.35/0.66 thf(fact_7527_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7528_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: list_nat > $o,A2: list_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_list_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7529_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: set_nat > $o,A2: set_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7530_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: real > $o,A2: real] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_real ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7531_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: $o > $o,A2: $o] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_o ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7532_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: nat > $o,A2: nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7533_Collect__conv__if,axiom,
% 0.35/0.66 ! [P: int > $o,A2: int] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( X = A2 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_int ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if
% 0.35/0.66 thf(fact_7534_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7535_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: list_nat > $o,A2: list_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_list_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7536_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: set_nat > $o,A2: set_nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7537_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: real > $o,A2: real] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_real ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7538_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: $o > $o,A2: $o] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_o ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7539_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: nat > $o,A2: nat] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7540_Collect__conv__if2,axiom,
% 0.35/0.66 ! [P: int > $o,A2: int] :
% 0.35/0.66 ( ( ( P @ A2 )
% 0.35/0.66 => ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.66 & ( ~ ( P @ A2 )
% 0.35/0.66 => ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( A2 = X )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 = bot_bot_set_int ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_conv_if2
% 0.35/0.66 thf(fact_7541_insert__Collect,axiom,
% 0.35/0.66 ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.66 ( ( insert8211810215607154385at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
% 0.35/0.66 = ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [U2: product_prod_nat_nat] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7542_insert__Collect,axiom,
% 0.35/0.66 ! [A2: $o,P: $o > $o] :
% 0.35/0.66 ( ( insert_o @ A2 @ ( collect_o @ P ) )
% 0.35/0.66 = ( collect_o
% 0.35/0.66 @ ^ [U2: $o] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7543_insert__Collect,axiom,
% 0.35/0.66 ! [A2: real,P: real > $o] :
% 0.35/0.66 ( ( insert_real @ A2 @ ( collect_real @ P ) )
% 0.35/0.66 = ( collect_real
% 0.35/0.66 @ ^ [U2: real] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7544_insert__Collect,axiom,
% 0.35/0.66 ! [A2: list_nat,P: list_nat > $o] :
% 0.35/0.66 ( ( insert_list_nat @ A2 @ ( collect_list_nat @ P ) )
% 0.35/0.66 = ( collect_list_nat
% 0.35/0.66 @ ^ [U2: list_nat] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7545_insert__Collect,axiom,
% 0.35/0.66 ! [A2: set_nat,P: set_nat > $o] :
% 0.35/0.66 ( ( insert_set_nat @ A2 @ ( collect_set_nat @ P ) )
% 0.35/0.66 = ( collect_set_nat
% 0.35/0.66 @ ^ [U2: set_nat] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7546_insert__Collect,axiom,
% 0.35/0.66 ! [A2: nat,P: nat > $o] :
% 0.35/0.66 ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
% 0.35/0.66 = ( collect_nat
% 0.35/0.66 @ ^ [U2: nat] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7547_insert__Collect,axiom,
% 0.35/0.66 ! [A2: int,P: int > $o] :
% 0.35/0.66 ( ( insert_int @ A2 @ ( collect_int @ P ) )
% 0.35/0.66 = ( collect_int
% 0.35/0.66 @ ^ [U2: int] :
% 0.35/0.66 ( ( U2 != A2 )
% 0.35/0.66 => ( P @ U2 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_Collect
% 0.35/0.66 thf(fact_7548_insert__compr,axiom,
% 0.35/0.66 ( insert8211810215607154385at_nat
% 0.35/0.66 = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.66 ( collec3392354462482085612at_nat
% 0.35/0.66 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7549_insert__compr,axiom,
% 0.35/0.66 ( insert_o
% 0.35/0.66 = ( ^ [A4: $o,B6: set_o] :
% 0.35/0.66 ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7550_insert__compr,axiom,
% 0.35/0.66 ( insert_real
% 0.35/0.66 = ( ^ [A4: real,B6: set_real] :
% 0.35/0.66 ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7551_insert__compr,axiom,
% 0.35/0.66 ( insert_list_nat
% 0.35/0.66 = ( ^ [A4: list_nat,B6: set_list_nat] :
% 0.35/0.66 ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7552_insert__compr,axiom,
% 0.35/0.66 ( insert_set_nat
% 0.35/0.66 = ( ^ [A4: set_nat,B6: set_set_nat] :
% 0.35/0.66 ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7553_insert__compr,axiom,
% 0.35/0.66 ( insert_nat
% 0.35/0.66 = ( ^ [A4: nat,B6: set_nat] :
% 0.35/0.66 ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7554_insert__compr,axiom,
% 0.35/0.66 ( insert_int
% 0.35/0.66 = ( ^ [A4: int,B6: set_int] :
% 0.35/0.66 ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( X = A4 )
% 0.35/0.66 | ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % insert_compr
% 0.35/0.66 thf(fact_7555_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_list_nat
% 0.35/0.66 = ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7556_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_set_nat
% 0.35/0.66 = ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7557_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_real
% 0.35/0.66 = ( collect_real
% 0.35/0.66 @ ^ [X: real] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7558_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_o
% 0.35/0.66 = ( collect_o
% 0.35/0.66 @ ^ [X: $o] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7559_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_nat
% 0.35/0.66 = ( collect_nat
% 0.35/0.66 @ ^ [X: nat] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7560_empty__def,axiom,
% 0.35/0.66 ( bot_bot_set_int
% 0.35/0.66 = ( collect_int
% 0.35/0.66 @ ^ [X: int] : $false ) ) ).
% 0.35/0.66
% 0.35/0.66 % empty_def
% 0.35/0.66 thf(fact_7561_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: real > $o,Q: real > $o] :
% 0.35/0.66 ( ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) @ ( collect_real @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7562_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.66 ( ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7563_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.66 ( ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7564_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: int > $o,Q: int > $o] :
% 0.35/0.66 ( ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7565_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.66 ( ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7566_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
% 0.35/0.66 ( ( collec7088162979684241874at_nat
% 0.35/0.66 @ ^ [X: produc859450856879609959at_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ ( collec7088162979684241874at_nat @ P ) ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7567_Collect__imp__eq,axiom,
% 0.35/0.66 ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
% 0.35/0.66 ( ( collec6321179662152712658at_nat
% 0.35/0.66 @ ^ [X: produc3843707927480180839at_nat] :
% 0.35/0.66 ( ( P @ X )
% 0.35/0.66 => ( Q @ X ) ) )
% 0.35/0.66 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_imp_eq
% 0.35/0.66 thf(fact_7568_finite__less__ub,axiom,
% 0.35/0.66 ! [F: nat > nat,U: nat] :
% 0.35/0.66 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 0.35/0.66 => ( finite_finite_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % finite_less_ub
% 0.35/0.66 thf(fact_7569_bot__empty__eq2,axiom,
% 0.35/0.66 ( bot_bo5043116465536727218_nat_o
% 0.35/0.66 = ( ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ bot_bo232370072503712749on_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % bot_empty_eq2
% 0.35/0.66 thf(fact_7570_bot__empty__eq2,axiom,
% 0.35/0.66 ( bot_bo3364206721330744218_nat_o
% 0.35/0.66 = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ bot_bo4948859079157340979at_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % bot_empty_eq2
% 0.35/0.66 thf(fact_7571_bot__empty__eq2,axiom,
% 0.35/0.66 ( bot_bo394778441745866138_nat_o
% 0.35/0.66 = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % bot_empty_eq2
% 0.35/0.66 thf(fact_7572_bot__empty__eq2,axiom,
% 0.35/0.66 ( bot_bot_nat_nat_o
% 0.35/0.66 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % bot_empty_eq2
% 0.35/0.66 thf(fact_7573_bot__empty__eq2,axiom,
% 0.35/0.66 ( bot_bot_int_int_o
% 0.35/0.66 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % bot_empty_eq2
% 0.35/0.66 thf(fact_7574_pred__subset__eq,axiom,
% 0.35/0.66 ! [R: set_real,S: set_real] :
% 0.35/0.66 ( ( ord_less_eq_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ R )
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ S ) )
% 0.35/0.66 = ( ord_less_eq_set_real @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq
% 0.35/0.66 thf(fact_7575_pred__subset__eq,axiom,
% 0.35/0.66 ! [R: set_o,S: set_o] :
% 0.35/0.66 ( ( ord_less_eq_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ R )
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ S ) )
% 0.35/0.66 = ( ord_less_eq_set_o @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq
% 0.35/0.66 thf(fact_7576_pred__subset__eq,axiom,
% 0.35/0.66 ! [R: set_set_nat,S: set_set_nat] :
% 0.35/0.66 ( ( ord_le3964352015994296041_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
% 0.35/0.66 = ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq
% 0.35/0.66 thf(fact_7577_pred__subset__eq,axiom,
% 0.35/0.66 ! [R: set_nat,S: set_nat] :
% 0.35/0.66 ( ( ord_less_eq_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ R )
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ S ) )
% 0.35/0.66 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq
% 0.35/0.66 thf(fact_7578_pred__subset__eq,axiom,
% 0.35/0.66 ! [R: set_int,S: set_int] :
% 0.35/0.66 ( ( ord_less_eq_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ R )
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ S ) )
% 0.35/0.66 = ( ord_less_eq_set_int @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq
% 0.35/0.66 thf(fact_7579_prop__restrict,axiom,
% 0.35/0.66 ! [X2: $o,Z6: set_o,X6: set_o,P: $o > $o] :
% 0.35/0.66 ( ( member_o @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_less_eq_set_o @ Z6
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( member_o @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7580_prop__restrict,axiom,
% 0.35/0.66 ! [X2: real,Z6: set_real,X6: set_real,P: real > $o] :
% 0.35/0.66 ( ( member_real @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_less_eq_set_real @ Z6
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( member_real @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7581_prop__restrict,axiom,
% 0.35/0.66 ! [X2: list_nat,Z6: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
% 0.35/0.66 ( ( member_list_nat @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_le6045566169113846134st_nat @ Z6
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( member_list_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7582_prop__restrict,axiom,
% 0.35/0.66 ! [X2: set_nat,Z6: set_set_nat,X6: set_set_nat,P: set_nat > $o] :
% 0.35/0.66 ( ( member_set_nat @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_le6893508408891458716et_nat @ Z6
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( member_set_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7583_prop__restrict,axiom,
% 0.35/0.66 ! [X2: nat,Z6: set_nat,X6: set_nat,P: nat > $o] :
% 0.35/0.66 ( ( member_nat @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_less_eq_set_nat @ Z6
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( member_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7584_prop__restrict,axiom,
% 0.35/0.66 ! [X2: int,Z6: set_int,X6: set_int,P: int > $o] :
% 0.35/0.66 ( ( member_int @ X2 @ Z6 )
% 0.35/0.66 => ( ( ord_less_eq_set_int @ Z6
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( member_int @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) ) )
% 0.35/0.66 => ( P @ X2 ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % prop_restrict
% 0.35/0.66 thf(fact_7585_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_o,P: $o > $o] :
% 0.35/0.66 ( ord_less_eq_set_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( member_o @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7586_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_real,P: real > $o] :
% 0.35/0.66 ( ord_less_eq_set_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( member_real @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7587_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_list_nat,P: list_nat > $o] :
% 0.35/0.66 ( ord_le6045566169113846134st_nat
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( member_list_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7588_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_set_nat,P: set_nat > $o] :
% 0.35/0.66 ( ord_le6893508408891458716et_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( member_set_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7589_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_nat,P: nat > $o] :
% 0.35/0.66 ( ord_less_eq_set_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( member_nat @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7590_Collect__restrict,axiom,
% 0.35/0.66 ! [X6: set_int,P: int > $o] :
% 0.35/0.66 ( ord_less_eq_set_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( member_int @ X @ X6 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ X6 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_restrict
% 0.35/0.66 thf(fact_7591_less__eq__set__def,axiom,
% 0.35/0.66 ( ord_less_eq_set_real
% 0.35/0.66 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.66 ( ord_less_eq_real_o
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.35/0.66 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_eq_set_def
% 0.35/0.66 thf(fact_7592_less__eq__set__def,axiom,
% 0.35/0.66 ( ord_less_eq_set_o
% 0.35/0.66 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.66 ( ord_less_eq_o_o
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 0.35/0.66 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_eq_set_def
% 0.35/0.66 thf(fact_7593_less__eq__set__def,axiom,
% 0.35/0.66 ( ord_le6893508408891458716et_nat
% 0.35/0.66 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.66 ( ord_le3964352015994296041_nat_o
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_eq_set_def
% 0.35/0.66 thf(fact_7594_less__eq__set__def,axiom,
% 0.35/0.66 ( ord_less_eq_set_nat
% 0.35/0.66 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.66 ( ord_less_eq_nat_o
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.35/0.66 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_eq_set_def
% 0.35/0.66 thf(fact_7595_less__eq__set__def,axiom,
% 0.35/0.66 ( ord_less_eq_set_int
% 0.35/0.66 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.66 ( ord_less_eq_int_o
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.35/0.66 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 0.35/0.66
% 0.35/0.66 % less_eq_set_def
% 0.35/0.66 thf(fact_7596_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_o,P: $o > $o] :
% 0.35/0.66 ( ord_less_eq_set_o
% 0.35/0.66 @ ( collect_o
% 0.35/0.66 @ ^ [X: $o] :
% 0.35/0.66 ( ( member_o @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7597_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_real,P: real > $o] :
% 0.35/0.66 ( ord_less_eq_set_real
% 0.35/0.66 @ ( collect_real
% 0.35/0.66 @ ^ [X: real] :
% 0.35/0.66 ( ( member_real @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7598_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_list_nat,P: list_nat > $o] :
% 0.35/0.66 ( ord_le6045566169113846134st_nat
% 0.35/0.66 @ ( collect_list_nat
% 0.35/0.66 @ ^ [X: list_nat] :
% 0.35/0.66 ( ( member_list_nat @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7599_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_set_nat,P: set_nat > $o] :
% 0.35/0.66 ( ord_le6893508408891458716et_nat
% 0.35/0.66 @ ( collect_set_nat
% 0.35/0.66 @ ^ [X: set_nat] :
% 0.35/0.66 ( ( member_set_nat @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7600_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_nat,P: nat > $o] :
% 0.35/0.66 ( ord_less_eq_set_nat
% 0.35/0.66 @ ( collect_nat
% 0.35/0.66 @ ^ [X: nat] :
% 0.35/0.66 ( ( member_nat @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7601_Collect__subset,axiom,
% 0.35/0.66 ! [A3: set_int,P: int > $o] :
% 0.35/0.66 ( ord_less_eq_set_int
% 0.35/0.66 @ ( collect_int
% 0.35/0.66 @ ^ [X: int] :
% 0.35/0.66 ( ( member_int @ X @ A3 )
% 0.35/0.66 & ( P @ X ) ) )
% 0.35/0.66 @ A3 ) ).
% 0.35/0.66
% 0.35/0.66 % Collect_subset
% 0.35/0.66 thf(fact_7602_pred__subset__eq2,axiom,
% 0.35/0.66 ! [R: set_Pr6588086440996610945on_nat,S: set_Pr6588086440996610945on_nat] :
% 0.35/0.66 ( ( ord_le8905833333647802342_nat_o
% 0.35/0.66 @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ R )
% 0.35/0.66 @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ S ) )
% 0.35/0.66 = ( ord_le6406482658798684961on_nat @ R @ S ) ) ).
% 0.35/0.66
% 0.35/0.66 % pred_subset_eq2
% 0.35/0.66 thf(fact_7603_pred__subset__eq2,axiom,
% 0.35/0.66 ! [R: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
% 0.35/0.66 ( ( ord_le3072208448688395470_nat_o
% 0.35/0.66 @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
% 0.35/0.67 @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S ) )
% 0.35/0.67 = ( ord_le5997549366648089703at_nat @ R @ S ) ) ).
% 0.35/0.67
% 0.35/0.67 % pred_subset_eq2
% 0.35/0.67 thf(fact_7604_pred__subset__eq2,axiom,
% 0.35/0.67 ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 0.35/0.67 ( ( ord_le3935385432712749774_nat_o
% 0.35/0.67 @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
% 0.35/0.67 @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S ) )
% 0.35/0.67 = ( ord_le1268244103169919719at_nat @ R @ S ) ) ).
% 0.35/0.67
% 0.35/0.67 % pred_subset_eq2
% 0.35/0.67 thf(fact_7605_pred__subset__eq2,axiom,
% 0.35/0.67 ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.35/0.67 ( ( ord_le2646555220125990790_nat_o
% 0.35/0.67 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 0.35/0.67 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
% 0.35/0.67 = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% 0.35/0.67
% 0.35/0.67 % pred_subset_eq2
% 0.35/0.67 thf(fact_7606_pred__subset__eq2,axiom,
% 0.35/0.67 ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.35/0.67 ( ( ord_le6741204236512500942_int_o
% 0.35/0.67 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 0.35/0.67 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) )
% 0.35/0.67 = ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% 0.35/0.67
% 0.35/0.67 % pred_subset_eq2
% 0.35/0.67 thf(fact_7607_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_o
% 0.35/0.67 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.67 ( collect_o
% 0.35/0.67 @ ^ [X: $o] :
% 0.35/0.67 ( ( member_o @ X @ A6 )
% 0.35/0.67 & ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7608_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_real
% 0.35/0.67 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.67 ( collect_real
% 0.35/0.67 @ ^ [X: real] :
% 0.35/0.67 ( ( member_real @ X @ A6 )
% 0.35/0.67 & ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7609_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_list_nat
% 0.35/0.67 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.67 ( collect_list_nat
% 0.35/0.67 @ ^ [X: list_nat] :
% 0.35/0.67 ( ( member_list_nat @ X @ A6 )
% 0.35/0.67 & ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7610_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_set_nat
% 0.35/0.67 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.67 ( collect_set_nat
% 0.35/0.67 @ ^ [X: set_nat] :
% 0.35/0.67 ( ( member_set_nat @ X @ A6 )
% 0.35/0.67 & ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7611_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_int
% 0.35/0.67 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.67 ( collect_int
% 0.35/0.67 @ ^ [X: int] :
% 0.35/0.67 ( ( member_int @ X @ A6 )
% 0.35/0.67 & ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7612_Int__def,axiom,
% 0.35/0.67 ( inf_inf_set_nat
% 0.35/0.67 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.67 ( collect_nat
% 0.35/0.67 @ ^ [X: nat] :
% 0.35/0.67 ( ( member_nat @ X @ A6 )
% 0.35/0.67 & ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7613_Int__def,axiom,
% 0.35/0.67 ( inf_in2572325071724192079at_nat
% 0.35/0.67 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.67 ( collec3392354462482085612at_nat
% 0.35/0.67 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.67 ( ( member8440522571783428010at_nat @ X @ A6 )
% 0.35/0.67 & ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_def
% 0.35/0.67 thf(fact_7614_Int__Collect,axiom,
% 0.35/0.67 ! [X2: $o,A3: set_o,P: $o > $o] :
% 0.35/0.67 ( ( member_o @ X2 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
% 0.35/0.67 = ( ( member_o @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7615_Int__Collect,axiom,
% 0.35/0.67 ! [X2: real,A3: set_real,P: real > $o] :
% 0.35/0.67 ( ( member_real @ X2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) )
% 0.35/0.67 = ( ( member_real @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7616_Int__Collect,axiom,
% 0.35/0.67 ! [X2: list_nat,A3: set_list_nat,P: list_nat > $o] :
% 0.35/0.67 ( ( member_list_nat @ X2 @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) )
% 0.35/0.67 = ( ( member_list_nat @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7617_Int__Collect,axiom,
% 0.35/0.67 ! [X2: set_nat,A3: set_set_nat,P: set_nat > $o] :
% 0.35/0.67 ( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) )
% 0.35/0.67 = ( ( member_set_nat @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7618_Int__Collect,axiom,
% 0.35/0.67 ! [X2: int,A3: set_int,P: int > $o] :
% 0.35/0.67 ( ( member_int @ X2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) )
% 0.35/0.67 = ( ( member_int @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7619_Int__Collect,axiom,
% 0.35/0.67 ! [X2: nat,A3: set_nat,P: nat > $o] :
% 0.35/0.67 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
% 0.35/0.67 = ( ( member_nat @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7620_Int__Collect,axiom,
% 0.35/0.67 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.67 ( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) )
% 0.35/0.67 = ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 0.35/0.67 & ( P @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Int_Collect
% 0.35/0.67 thf(fact_7621_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_o
% 0.35/0.67 = ( ^ [A6: set_o,B6: set_o] :
% 0.35/0.67 ( collect_o
% 0.35/0.67 @ ( inf_inf_o_o
% 0.35/0.67 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 0.35/0.67 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7622_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_real
% 0.35/0.67 = ( ^ [A6: set_real,B6: set_real] :
% 0.35/0.67 ( collect_real
% 0.35/0.67 @ ( inf_inf_real_o
% 0.35/0.67 @ ^ [X: real] : ( member_real @ X @ A6 )
% 0.35/0.67 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7623_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_list_nat
% 0.35/0.67 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 0.35/0.67 ( collect_list_nat
% 0.35/0.67 @ ( inf_inf_list_nat_o
% 0.35/0.67 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 0.35/0.67 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7624_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_set_nat
% 0.35/0.67 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 0.35/0.67 ( collect_set_nat
% 0.35/0.67 @ ( inf_inf_set_nat_o
% 0.35/0.67 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 0.35/0.67 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7625_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_int
% 0.35/0.67 = ( ^ [A6: set_int,B6: set_int] :
% 0.35/0.67 ( collect_int
% 0.35/0.67 @ ( inf_inf_int_o
% 0.35/0.67 @ ^ [X: int] : ( member_int @ X @ A6 )
% 0.35/0.67 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7626_inf__set__def,axiom,
% 0.35/0.67 ( inf_inf_set_nat
% 0.35/0.67 = ( ^ [A6: set_nat,B6: set_nat] :
% 0.35/0.67 ( collect_nat
% 0.35/0.67 @ ( inf_inf_nat_o
% 0.35/0.67 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 0.35/0.67 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7627_inf__set__def,axiom,
% 0.35/0.67 ( inf_in2572325071724192079at_nat
% 0.35/0.67 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 0.35/0.67 ( collec3392354462482085612at_nat
% 0.35/0.67 @ ( inf_in5163264567034779214_nat_o
% 0.35/0.67 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 0.35/0.67 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % inf_set_def
% 0.35/0.67 thf(fact_7628_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: real > $o,Q: real > $o] :
% 0.35/0.67 ( ( collect_real
% 0.35/0.67 @ ^ [X: real] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_inf_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7629_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.35/0.67 ( ( collect_list_nat
% 0.35/0.67 @ ^ [X: list_nat] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7630_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.35/0.67 ( ( collect_set_nat
% 0.35/0.67 @ ^ [X: set_nat] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7631_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: int > $o,Q: int > $o] :
% 0.35/0.67 ( ( collect_int
% 0.35/0.67 @ ^ [X: int] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7632_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: nat > $o,Q: nat > $o] :
% 0.35/0.67 ( ( collect_nat
% 0.35/0.67 @ ^ [X: nat] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7633_Collect__conj__eq,axiom,
% 0.35/0.67 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.35/0.67 ( ( collec3392354462482085612at_nat
% 0.35/0.67 @ ^ [X: product_prod_nat_nat] :
% 0.35/0.67 ( ( P @ X )
% 0.35/0.67 & ( Q @ X ) ) )
% 0.35/0.67 = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Collect_conj_eq
% 0.35/0.67 thf(fact_7634_set__vebt__def,axiom,
% 0.35/0.67 ( vEBT_set_vebt
% 0.35/0.67 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % set_vebt_def
% 0.35/0.67 thf(fact_7635_finite__int__segment,axiom,
% 0.35/0.67 ! [A2: real,B3: real] :
% 0.35/0.67 ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [X: real] :
% 0.35/0.67 ( ( member_real @ X @ ring_1_Ints_real )
% 0.35/0.67 & ( ord_less_eq_real @ A2 @ X )
% 0.35/0.67 & ( ord_less_eq_real @ X @ B3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_int_segment
% 0.35/0.67 thf(fact_7636_finite__int__segment,axiom,
% 0.35/0.67 ! [A2: rat,B3: rat] :
% 0.35/0.67 ( finite_finite_rat
% 0.35/0.67 @ ( collect_rat
% 0.35/0.67 @ ^ [X: rat] :
% 0.35/0.67 ( ( member_rat @ X @ ring_1_Ints_rat )
% 0.35/0.67 & ( ord_less_eq_rat @ A2 @ X )
% 0.35/0.67 & ( ord_less_eq_rat @ X @ B3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_int_segment
% 0.35/0.67 thf(fact_7637_nat__less__as__int,axiom,
% 0.35/0.67 ( ord_less_nat
% 0.35/0.67 = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % nat_less_as_int
% 0.35/0.67 thf(fact_7638_nat__leq__as__int,axiom,
% 0.35/0.67 ( ord_less_eq_nat
% 0.35/0.67 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % nat_leq_as_int
% 0.35/0.67 thf(fact_7639_unset__bit__less__eq,axiom,
% 0.35/0.67 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 0.35/0.67
% 0.35/0.67 % unset_bit_less_eq
% 0.35/0.67 thf(fact_7640_finite__abs__int__segment,axiom,
% 0.35/0.67 ! [A2: real] :
% 0.35/0.67 ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [K3: real] :
% 0.35/0.67 ( ( member_real @ K3 @ ring_1_Ints_real )
% 0.35/0.67 & ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_abs_int_segment
% 0.35/0.67 thf(fact_7641_finite__abs__int__segment,axiom,
% 0.35/0.67 ! [A2: rat] :
% 0.35/0.67 ( finite_finite_rat
% 0.35/0.67 @ ( collect_rat
% 0.35/0.67 @ ^ [K3: rat] :
% 0.35/0.67 ( ( member_rat @ K3 @ ring_1_Ints_rat )
% 0.35/0.67 & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_abs_int_segment
% 0.35/0.67 thf(fact_7642_card__less,axiom,
% 0.35/0.67 ! [M5: set_nat,I: nat] :
% 0.35/0.67 ( ( member_nat @ zero_zero_nat @ M5 )
% 0.35/0.67 => ( ( finite_card_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [K3: nat] :
% 0.35/0.67 ( ( member_nat @ K3 @ M5 )
% 0.35/0.67 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.35/0.67 != zero_zero_nat ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_less
% 0.35/0.67 thf(fact_7643_card__less__Suc,axiom,
% 0.35/0.67 ! [M5: set_nat,I: nat] :
% 0.35/0.67 ( ( member_nat @ zero_zero_nat @ M5 )
% 0.35/0.67 => ( ( suc
% 0.35/0.67 @ ( finite_card_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [K3: nat] :
% 0.35/0.67 ( ( member_nat @ ( suc @ K3 ) @ M5 )
% 0.35/0.67 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 0.35/0.67 = ( finite_card_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [K3: nat] :
% 0.35/0.67 ( ( member_nat @ K3 @ M5 )
% 0.35/0.67 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_less_Suc
% 0.35/0.67 thf(fact_7644_card__less__Suc2,axiom,
% 0.35/0.67 ! [M5: set_nat,I: nat] :
% 0.35/0.67 ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 0.35/0.67 => ( ( finite_card_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [K3: nat] :
% 0.35/0.67 ( ( member_nat @ ( suc @ K3 ) @ M5 )
% 0.35/0.67 & ( ord_less_nat @ K3 @ I ) ) ) )
% 0.35/0.67 = ( finite_card_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [K3: nat] :
% 0.35/0.67 ( ( member_nat @ K3 @ M5 )
% 0.35/0.67 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_less_Suc2
% 0.35/0.67 thf(fact_7645_norm__not__less__zero,axiom,
% 0.35/0.67 ! [X2: complex] :
% 0.35/0.67 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).
% 0.35/0.67
% 0.35/0.67 % norm_not_less_zero
% 0.35/0.67 thf(fact_7646_norm__ge__zero,axiom,
% 0.35/0.67 ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_ge_zero
% 0.35/0.67 thf(fact_7647_complex__mod__minus__le__complex__mod,axiom,
% 0.35/0.67 ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % complex_mod_minus_le_complex_mod
% 0.35/0.67 thf(fact_7648_complex__mod__triangle__ineq2,axiom,
% 0.35/0.67 ! [B3: complex,A2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B3 @ A2 ) ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ A2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % complex_mod_triangle_ineq2
% 0.35/0.67 thf(fact_7649_set__encode__eq,axiom,
% 0.35/0.67 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( ( finite_finite_nat @ B2 )
% 0.35/0.67 => ( ( ( nat_set_encode @ A3 )
% 0.35/0.67 = ( nat_set_encode @ B2 ) )
% 0.35/0.67 = ( A3 = B2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % set_encode_eq
% 0.35/0.67 thf(fact_7650_finite__roots__unity,axiom,
% 0.35/0.67 ! [N: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.67 => ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [Z2: real] :
% 0.35/0.67 ( ( power_power_real @ Z2 @ N )
% 0.35/0.67 = one_one_real ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_roots_unity
% 0.35/0.67 thf(fact_7651_finite__roots__unity,axiom,
% 0.35/0.67 ! [N: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.67 => ( finite3207457112153483333omplex
% 0.35/0.67 @ ( collect_complex
% 0.35/0.67 @ ^ [Z2: complex] :
% 0.35/0.67 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.67 = one_one_complex ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_roots_unity
% 0.35/0.67 thf(fact_7652_card__roots__unity,axiom,
% 0.35/0.67 ! [N: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.67 => ( ord_less_eq_nat
% 0.35/0.67 @ ( finite_card_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [Z2: real] :
% 0.35/0.67 ( ( power_power_real @ Z2 @ N )
% 0.35/0.67 = one_one_real ) ) )
% 0.35/0.67 @ N ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_roots_unity
% 0.35/0.67 thf(fact_7653_card__roots__unity,axiom,
% 0.35/0.67 ! [N: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 0.35/0.67 => ( ord_less_eq_nat
% 0.35/0.67 @ ( finite_card_complex
% 0.35/0.67 @ ( collect_complex
% 0.35/0.67 @ ^ [Z2: complex] :
% 0.35/0.67 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.67 = one_one_complex ) ) )
% 0.35/0.67 @ N ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_roots_unity
% 0.35/0.67 thf(fact_7654_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_complex,N: nat] :
% 0.35/0.67 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.67 => ( finite8712137658972009173omplex
% 0.35/0.67 @ ( collect_list_complex
% 0.35/0.67 @ ^ [Xs2: list_complex] :
% 0.35/0.67 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7655_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 0.35/0.67 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.67 => ( finite500796754983035824at_nat
% 0.35/0.67 @ ( collec3343600615725829874at_nat
% 0.35/0.67 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.35/0.67 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7656_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_Extended_enat,N: nat] :
% 0.35/0.67 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.67 => ( finite1862508098717546133d_enat
% 0.35/0.67 @ ( collec8433460942617342167d_enat
% 0.35/0.67 @ ^ [Xs2: list_Extended_enat] :
% 0.35/0.67 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3941691890525107288d_enat @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7657_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_VEBT_VEBT,N: nat] :
% 0.35/0.67 ( ( finite5795047828879050333T_VEBT @ A3 )
% 0.35/0.67 => ( finite3004134309566078307T_VEBT
% 0.35/0.67 @ ( collec5608196760682091941T_VEBT
% 0.35/0.67 @ ^ [Xs2: list_VEBT_VEBT] :
% 0.35/0.67 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7658_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_o,N: nat] :
% 0.35/0.67 ( ( finite_finite_o @ A3 )
% 0.35/0.67 => ( finite_finite_list_o
% 0.35/0.67 @ ( collect_list_o
% 0.35/0.67 @ ^ [Xs2: list_o] :
% 0.35/0.67 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_o @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7659_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_nat,N: nat] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( finite8100373058378681591st_nat
% 0.35/0.67 @ ( collect_list_nat
% 0.35/0.67 @ ^ [Xs2: list_nat] :
% 0.35/0.67 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7660_finite__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_int,N: nat] :
% 0.35/0.67 ( ( finite_finite_int @ A3 )
% 0.35/0.67 => ( finite3922522038869484883st_int
% 0.35/0.67 @ ( collect_list_int
% 0.35/0.67 @ ^ [Xs2: list_int] :
% 0.35/0.67 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_int @ Xs2 )
% 0.35/0.67 = N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_eq
% 0.35/0.67 thf(fact_7661_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_list_nat,N: nat] :
% 0.35/0.67 ( ( finite8100373058378681591st_nat @ A3 )
% 0.35/0.67 => ( ( finite7325466520557071688st_nat
% 0.35/0.67 @ ( collec5989764272469232197st_nat
% 0.35/0.67 @ ^ [Xs2: list_list_nat] :
% 0.35/0.67 ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3023201423986296836st_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_list_nat @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7662_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_set_nat,N: nat] :
% 0.35/0.67 ( ( finite1152437895449049373et_nat @ A3 )
% 0.35/0.67 => ( ( finite5631907774883551598et_nat
% 0.35/0.67 @ ( collect_list_set_nat
% 0.35/0.67 @ ^ [Xs2: list_set_nat] :
% 0.35/0.67 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3254054031482475050et_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_set_nat @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7663_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_complex,N: nat] :
% 0.35/0.67 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.67 => ( ( finite5120063068150530198omplex
% 0.35/0.67 @ ( collect_list_complex
% 0.35/0.67 @ ^ [Xs2: list_complex] :
% 0.35/0.67 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_complex @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7664_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 0.35/0.67 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.67 => ( ( finite249151656366948015at_nat
% 0.35/0.67 @ ( collec3343600615725829874at_nat
% 0.35/0.67 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.35/0.67 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite711546835091564841at_nat @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7665_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_Extended_enat,N: nat] :
% 0.35/0.67 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.67 => ( ( finite7441382602597825044d_enat
% 0.35/0.67 @ ( collec8433460942617342167d_enat
% 0.35/0.67 @ ^ [Xs2: list_Extended_enat] :
% 0.35/0.67 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s3941691890525107288d_enat @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite121521170596916366d_enat @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7666_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_VEBT_VEBT,N: nat] :
% 0.35/0.67 ( ( finite5795047828879050333T_VEBT @ A3 )
% 0.35/0.67 => ( ( finite5915292604075114978T_VEBT
% 0.35/0.67 @ ( collec5608196760682091941T_VEBT
% 0.35/0.67 @ ^ [Xs2: list_VEBT_VEBT] :
% 0.35/0.67 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7667_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_o,N: nat] :
% 0.35/0.67 ( ( finite_finite_o @ A3 )
% 0.35/0.67 => ( ( finite_card_list_o
% 0.35/0.67 @ ( collect_list_o
% 0.35/0.67 @ ^ [Xs2: list_o] :
% 0.35/0.67 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_o @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_o @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7668_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_nat,N: nat] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( ( finite_card_list_nat
% 0.35/0.67 @ ( collect_list_nat
% 0.35/0.67 @ ^ [Xs2: list_nat] :
% 0.35/0.67 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_nat @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_nat @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7669_card__lists__length__eq,axiom,
% 0.35/0.67 ! [A3: set_int,N: nat] :
% 0.35/0.67 ( ( finite_finite_int @ A3 )
% 0.35/0.67 => ( ( finite_card_list_int
% 0.35/0.67 @ ( collect_list_int
% 0.35/0.67 @ ^ [Xs2: list_int] :
% 0.35/0.67 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ( size_size_list_int @ Xs2 )
% 0.35/0.67 = N ) ) ) )
% 0.35/0.67 = ( power_power_nat @ ( finite_card_int @ A3 ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % card_lists_length_eq
% 0.35/0.67 thf(fact_7670_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_complex,N: nat] :
% 0.35/0.67 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.67 => ( finite8712137658972009173omplex
% 0.35/0.67 @ ( collect_list_complex
% 0.35/0.67 @ ^ [Xs2: list_complex] :
% 0.35/0.67 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7671_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 0.35/0.67 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.67 => ( finite500796754983035824at_nat
% 0.35/0.67 @ ( collec3343600615725829874at_nat
% 0.35/0.67 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.35/0.67 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7672_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_Extended_enat,N: nat] :
% 0.35/0.67 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.67 => ( finite1862508098717546133d_enat
% 0.35/0.67 @ ( collec8433460942617342167d_enat
% 0.35/0.67 @ ^ [Xs2: list_Extended_enat] :
% 0.35/0.67 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7673_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_VEBT_VEBT,N: nat] :
% 0.35/0.67 ( ( finite5795047828879050333T_VEBT @ A3 )
% 0.35/0.67 => ( finite3004134309566078307T_VEBT
% 0.35/0.67 @ ( collec5608196760682091941T_VEBT
% 0.35/0.67 @ ^ [Xs2: list_VEBT_VEBT] :
% 0.35/0.67 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7674_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_o,N: nat] :
% 0.35/0.67 ( ( finite_finite_o @ A3 )
% 0.35/0.67 => ( finite_finite_list_o
% 0.35/0.67 @ ( collect_list_o
% 0.35/0.67 @ ^ [Xs2: list_o] :
% 0.35/0.67 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7675_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_nat,N: nat] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( finite8100373058378681591st_nat
% 0.35/0.67 @ ( collect_list_nat
% 0.35/0.67 @ ^ [Xs2: list_nat] :
% 0.35/0.67 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7676_finite__lists__length__le,axiom,
% 0.35/0.67 ! [A3: set_int,N: nat] :
% 0.35/0.67 ( ( finite_finite_int @ A3 )
% 0.35/0.67 => ( finite3922522038869484883st_int
% 0.35/0.67 @ ( collect_list_int
% 0.35/0.67 @ ^ [Xs2: list_int] :
% 0.35/0.67 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 0.35/0.67 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % finite_lists_length_le
% 0.35/0.67 thf(fact_7677_nonzero__norm__divide,axiom,
% 0.35/0.67 ! [B3: real,A2: real] :
% 0.35/0.67 ( ( B3 != zero_zero_real )
% 0.35/0.67 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B3 ) )
% 0.35/0.67 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % nonzero_norm_divide
% 0.35/0.67 thf(fact_7678_nonzero__norm__divide,axiom,
% 0.35/0.67 ! [B3: complex,A2: complex] :
% 0.35/0.67 ( ( B3 != zero_zero_complex )
% 0.35/0.67 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 0.35/0.67 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % nonzero_norm_divide
% 0.35/0.67 thf(fact_7679_power__eq__imp__eq__norm,axiom,
% 0.35/0.67 ! [W2: real,N: nat,Z: real] :
% 0.35/0.67 ( ( ( power_power_real @ W2 @ N )
% 0.35/0.67 = ( power_power_real @ Z @ N ) )
% 0.35/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.67 => ( ( real_V7735802525324610683m_real @ W2 )
% 0.35/0.67 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % power_eq_imp_eq_norm
% 0.35/0.67 thf(fact_7680_power__eq__imp__eq__norm,axiom,
% 0.35/0.67 ! [W2: complex,N: nat,Z: complex] :
% 0.35/0.67 ( ( ( power_power_complex @ W2 @ N )
% 0.35/0.67 = ( power_power_complex @ Z @ N ) )
% 0.35/0.67 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.67 => ( ( real_V1022390504157884413omplex @ W2 )
% 0.35/0.67 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % power_eq_imp_eq_norm
% 0.35/0.67 thf(fact_7681_norm__mult__less,axiom,
% 0.35/0.67 ! [X2: real,R2: real,Y3: real,S2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_mult_less
% 0.35/0.67 thf(fact_7682_norm__mult__less,axiom,
% 0.35/0.67 ! [X2: complex,R2: real,Y3: complex,S2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_mult_less
% 0.35/0.67 thf(fact_7683_norm__mult__ineq,axiom,
% 0.35/0.67 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_mult_ineq
% 0.35/0.67 thf(fact_7684_norm__mult__ineq,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_mult_ineq
% 0.35/0.67 thf(fact_7685_norm__add__less,axiom,
% 0.35/0.67 ! [X2: real,R2: real,Y3: real,S2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_add_less
% 0.35/0.67 thf(fact_7686_norm__add__less,axiom,
% 0.35/0.67 ! [X2: complex,R2: real,Y3: complex,S2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_add_less
% 0.35/0.67 thf(fact_7687_norm__triangle__lt,axiom,
% 0.35/0.67 ! [X2: real,Y3: real,E2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_lt
% 0.35/0.67 thf(fact_7688_norm__triangle__lt,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex,E2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_lt
% 0.35/0.67 thf(fact_7689_norm__power__ineq,axiom,
% 0.35/0.67 ! [X2: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_power_ineq
% 0.35/0.67 thf(fact_7690_norm__power__ineq,axiom,
% 0.35/0.67 ! [X2: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_power_ineq
% 0.35/0.67 thf(fact_7691_norm__add__leD,axiom,
% 0.35/0.67 ! [A2: real,B3: real,C: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ C )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ C ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_add_leD
% 0.35/0.67 thf(fact_7692_norm__add__leD,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex,C: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ C )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ C ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_add_leD
% 0.35/0.67 thf(fact_7693_norm__triangle__le,axiom,
% 0.35/0.67 ! [X2: real,Y3: real,E2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_le
% 0.35/0.67 thf(fact_7694_norm__triangle__le,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex,E2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_le
% 0.35/0.67 thf(fact_7695_norm__triangle__ineq,axiom,
% 0.35/0.67 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq
% 0.35/0.67 thf(fact_7696_norm__triangle__ineq,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq
% 0.35/0.67 thf(fact_7697_norm__triangle__mono,axiom,
% 0.35/0.67 ! [A2: real,R2: real,B3: real,S2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ S2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_mono
% 0.35/0.67 thf(fact_7698_norm__triangle__mono,axiom,
% 0.35/0.67 ! [A2: complex,R2: real,B3: complex,S2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A2 ) @ R2 )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ S2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_mono
% 0.35/0.67 thf(fact_7699_norm__diff__triangle__less,axiom,
% 0.35/0.67 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.35/0.67 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_less
% 0.35/0.67 thf(fact_7700_norm__diff__triangle__less,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.35/0.67 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_less
% 0.35/0.67 thf(fact_7701_norm__triangle__sub,axiom,
% 0.35/0.67 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_sub
% 0.35/0.67 thf(fact_7702_norm__triangle__sub,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_sub
% 0.35/0.67 thf(fact_7703_norm__triangle__ineq4,axiom,
% 0.35/0.67 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq4
% 0.35/0.67 thf(fact_7704_norm__triangle__ineq4,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq4
% 0.35/0.67 thf(fact_7705_norm__diff__triangle__le,axiom,
% 0.35/0.67 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_le
% 0.35/0.67 thf(fact_7706_norm__diff__triangle__le,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_le
% 0.35/0.67 thf(fact_7707_norm__triangle__le__diff,axiom,
% 0.35/0.67 ! [X2: real,Y3: real,E2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_le_diff
% 0.35/0.67 thf(fact_7708_norm__triangle__le__diff,axiom,
% 0.35/0.67 ! [X2: complex,Y3: complex,E2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_le_diff
% 0.35/0.67 thf(fact_7709_norm__diff__ineq,axiom,
% 0.35/0.67 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_ineq
% 0.35/0.67 thf(fact_7710_norm__diff__ineq,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_ineq
% 0.35/0.67 thf(fact_7711_norm__triangle__ineq2,axiom,
% 0.35/0.67 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq2
% 0.35/0.67 thf(fact_7712_norm__triangle__ineq2,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq2
% 0.35/0.67 thf(fact_7713_norm__exp,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_exp
% 0.35/0.67 thf(fact_7714_norm__exp,axiom,
% 0.35/0.67 ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_exp
% 0.35/0.67 thf(fact_7715_set__encode__inf,axiom,
% 0.35/0.67 ! [A3: set_nat] :
% 0.35/0.67 ( ~ ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( ( nat_set_encode @ A3 )
% 0.35/0.67 = zero_zero_nat ) ) ).
% 0.35/0.67
% 0.35/0.67 % set_encode_inf
% 0.35/0.67 thf(fact_7716_power__eq__1__iff,axiom,
% 0.35/0.67 ! [W2: real,N: nat] :
% 0.35/0.67 ( ( ( power_power_real @ W2 @ N )
% 0.35/0.67 = one_one_real )
% 0.35/0.67 => ( ( ( real_V7735802525324610683m_real @ W2 )
% 0.35/0.67 = one_one_real )
% 0.35/0.67 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % power_eq_1_iff
% 0.35/0.67 thf(fact_7717_power__eq__1__iff,axiom,
% 0.35/0.67 ! [W2: complex,N: nat] :
% 0.35/0.67 ( ( ( power_power_complex @ W2 @ N )
% 0.35/0.67 = one_one_complex )
% 0.35/0.67 => ( ( ( real_V1022390504157884413omplex @ W2 )
% 0.35/0.67 = one_one_real )
% 0.35/0.67 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % power_eq_1_iff
% 0.35/0.67 thf(fact_7718_norm__diff__triangle__ineq,axiom,
% 0.35/0.67 ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_ineq
% 0.35/0.67 thf(fact_7719_norm__diff__triangle__ineq,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B3 @ D ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_diff_triangle_ineq
% 0.35/0.67 thf(fact_7720_norm__sgn,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ( X2 = zero_zero_real )
% 0.35/0.67 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
% 0.35/0.67 = zero_zero_real ) )
% 0.35/0.67 & ( ( X2 != zero_zero_real )
% 0.35/0.67 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
% 0.35/0.67 = one_one_real ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_sgn
% 0.35/0.67 thf(fact_7721_norm__sgn,axiom,
% 0.35/0.67 ! [X2: complex] :
% 0.35/0.67 ( ( ( X2 = zero_zero_complex )
% 0.35/0.67 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
% 0.35/0.67 = zero_zero_real ) )
% 0.35/0.67 & ( ( X2 != zero_zero_complex )
% 0.35/0.67 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
% 0.35/0.67 = one_one_real ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_sgn
% 0.35/0.67 thf(fact_7722_norm__triangle__ineq3,axiom,
% 0.35/0.67 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq3
% 0.35/0.67 thf(fact_7723_norm__triangle__ineq3,axiom,
% 0.35/0.67 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_triangle_ineq3
% 0.35/0.67 thf(fact_7724_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.35/0.67 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 0.35/0.67 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S2 ) @ X2 )
% 0.35/0.67 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.simps(3)
% 0.35/0.67 thf(fact_7725_norm__power__diff,axiom,
% 0.35/0.67 ! [Z: real,W2: real,M: nat] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W2 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_power_diff
% 0.35/0.67 thf(fact_7726_norm__power__diff,axiom,
% 0.35/0.67 ! [Z: complex,W2: complex,M: nat] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W2 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % norm_power_diff
% 0.35/0.67 thf(fact_7727_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.35/0.67 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
% 0.35/0.67 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X2 )
% 0.35/0.67 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.simps(5)
% 0.35/0.67 thf(fact_7728_vebt__member_Osimps_I5_J,axiom,
% 0.35/0.67 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.35/0.67 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( ( X2 != Mi )
% 0.35/0.67 => ( ( X2 != Ma )
% 0.35/0.67 => ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.simps(5)
% 0.35/0.67 thf(fact_7729_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.35/0.67 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 0.35/0.67 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X2 )
% 0.35/0.67 = ( ( X2 = Mi )
% 0.35/0.67 | ( X2 = Ma )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.simps(4)
% 0.35/0.67 thf(fact_7730_exp__bound__half,axiom,
% 0.35/0.67 ! [Z: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % exp_bound_half
% 0.35/0.67 thf(fact_7731_exp__bound__half,axiom,
% 0.35/0.67 ! [Z: complex] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % exp_bound_half
% 0.35/0.67 thf(fact_7732_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.35/0.67 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [S3: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.elims(1)
% 0.35/0.67 thf(fact_7733_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [S3: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.elims(2)
% 0.35/0.67 thf(fact_7734_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [S3: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.elims(3)
% 0.35/0.67 thf(fact_7735_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat] :
% 0.35/0.67 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.elims(2)
% 0.35/0.67 thf(fact_7736_vebt__member_Oelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Summary2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.elims(2)
% 0.35/0.67 thf(fact_7737_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat] :
% 0.35/0.67 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.elims(3)
% 0.35/0.67 thf(fact_7738_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat] :
% 0.35/0.67 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vc2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Vd2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.elims(1)
% 0.35/0.67 thf(fact_7739_vebt__insert_Osimps_I5_J,axiom,
% 0.35/0.67 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 0.35/0.67 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( if_VEBT_VEBT
% 0.35/0.67 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 & ~ ( ( X2 = Mi )
% 0.35/0.67 | ( X2 = Ma ) ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_insert.simps(5)
% 0.35/0.67 thf(fact_7740_vebt__member_Oelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.35/0.67 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.67 => Y3 )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Summary2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( ~ ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.elims(1)
% 0.35/0.67 thf(fact_7741_vebt__member_Oelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 0.35/0.67 ( ? [Summary2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.elims(3)
% 0.35/0.67 thf(fact_7742_exp__bound__lemma,axiom,
% 0.35/0.67 ! [Z: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % exp_bound_lemma
% 0.35/0.67 thf(fact_7743_exp__bound__lemma,axiom,
% 0.35/0.67 ! [Z: complex] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % exp_bound_lemma
% 0.35/0.67 thf(fact_7744_vebt__insert_Oelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.67 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ $true @ B ) ) )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.35/0.67 & ( ( Xa2 != one_one_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) )
% 0.35/0.67 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) ) )
% 0.35/0.67 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
% 0.35/0.67 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( if_VEBT_VEBT
% 0.35/0.67 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 & ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_insert.elims
% 0.35/0.67 thf(fact_7745_vebt__succ_Osimps_I6_J,axiom,
% 0.35/0.67 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.67 ( ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( some_nat @ Mi ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ none_nat
% 0.35/0.67 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_succ.simps(6)
% 0.35/0.67 thf(fact_7746_vebt__pred_Osimps_I7_J,axiom,
% 0.35/0.67 ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.67 ( ( ( ord_less_nat @ Ma @ X2 )
% 0.35/0.67 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( some_nat @ Ma ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 0.35/0.67 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_pred.simps(7)
% 0.35/0.67 thf(fact_7747_vebt__delete_Osimps_I7_J,axiom,
% 0.35/0.67 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.35/0.67 ( ( ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 | ( ord_less_nat @ Ma @ X2 ) )
% 0.35/0.67 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 0.35/0.67 & ( ~ ( ( ord_less_nat @ X2 @ Mi )
% 0.35/0.67 | ( ord_less_nat @ Ma @ X2 ) )
% 0.35/0.67 => ( ( ( ( X2 = Mi )
% 0.35/0.67 & ( X2 = Ma ) )
% 0.35/0.67 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 0.35/0.67 & ( ~ ( ( X2 = Mi )
% 0.35/0.67 & ( X2 = Ma ) )
% 0.35/0.67 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 0.35/0.67 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.35/0.67 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( X2 = Mi )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.67 = Ma ) )
% 0.35/0.67 & ( ( X2 != Mi )
% 0.35/0.67 => ( X2 = Ma ) ) )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( X2 = Mi )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 0.35/0.67 = Ma ) )
% 0.35/0.67 & ( ( X2 != Mi )
% 0.35/0.67 => ( X2 = Ma ) ) )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ Summary ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_delete.simps(7)
% 0.35/0.67 thf(fact_7748_vebt__delete_Oelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.67 ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Leaf @ $false @ B ) ) ) )
% 0.35/0.67 => ( ! [A: $o] :
% 0.35/0.67 ( ? [B: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Xa2
% 0.35/0.67 = ( suc @ zero_zero_nat ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Leaf @ A @ $false ) ) ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ? [N3: nat] :
% 0.35/0.67 ( Xa2
% 0.35/0.67 = ( suc @ ( suc @ N3 ) ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Leaf @ A @ B ) ) ) )
% 0.35/0.67 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.35/0.67 => ( ( ( ( Xa2 = Mi2 )
% 0.35/0.67 & ( Xa2 = Ma2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 & ( ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 & ( Xa2 = Ma2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( Xa2 = Mi2 )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.35/0.67 = Ma2 ) )
% 0.35/0.67 & ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma2 ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( Xa2 = Mi2 )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.35/0.67 = Ma2 ) )
% 0.35/0.67 & ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma2 ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ Summary2 ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_delete.elims
% 0.35/0.67 thf(fact_7749_vebt__pred_Oelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.35/0.67 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( Y3 != none_nat ) ) )
% 0.35/0.67 => ( ! [A: $o] :
% 0.35/0.67 ( ? [Uw2: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ Uw2 ) )
% 0.35/0.67 => ( ( Xa2
% 0.35/0.67 = ( suc @ zero_zero_nat ) )
% 0.35/0.67 => ~ ( ( A
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.67 & ( ~ A
% 0.35/0.67 => ( Y3 = none_nat ) ) ) ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ? [Va2: nat] :
% 0.35/0.67 ( Xa2
% 0.35/0.67 = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.67 => ~ ( ( B
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.67 & ( ~ B
% 0.35/0.67 => ( ( A
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.67 & ( ~ A
% 0.35/0.67 => ( Y3 = none_nat ) ) ) ) ) ) )
% 0.35/0.67 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ Ma2 ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_pred.elims
% 0.35/0.67 thf(fact_7750_vebt__succ_Oelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.35/0.67 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ! [Uu2: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ B ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ~ ( ( B
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.67 & ( ~ B
% 0.35/0.67 => ( Y3 = none_nat ) ) ) ) )
% 0.35/0.67 => ( ( ? [Uv2: $o,Uw2: $o] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ( ? [N3: nat] :
% 0.35/0.67 ( Xa2
% 0.35/0.67 = ( suc @ N3 ) )
% 0.35/0.67 => ( Y3 != none_nat ) ) )
% 0.35/0.67 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.35/0.67 ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.35/0.67 => ( Y3 != none_nat ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ Mi2 ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ none_nat
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_succ.elims
% 0.35/0.67 thf(fact_7751_of__int__code__if,axiom,
% 0.35/0.67 ( ring_1_of_int_int
% 0.35/0.67 = ( ^ [K3: int] :
% 0.35/0.67 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 0.35/0.67 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.67 @ ( if_int
% 0.35/0.67 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_code_if
% 0.35/0.67 thf(fact_7752_of__int__code__if,axiom,
% 0.35/0.67 ( ring_1_of_int_real
% 0.35/0.67 = ( ^ [K3: int] :
% 0.35/0.67 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 0.35/0.67 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.67 @ ( if_real
% 0.35/0.67 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_code_if
% 0.35/0.67 thf(fact_7753_of__int__code__if,axiom,
% 0.35/0.67 ( ring_1_of_int_rat
% 0.35/0.67 = ( ^ [K3: int] :
% 0.35/0.67 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 0.35/0.67 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.67 @ ( if_rat
% 0.35/0.67 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_code_if
% 0.35/0.67 thf(fact_7754_of__int__code__if,axiom,
% 0.35/0.67 ( ring_18347121197199848620nteger
% 0.35/0.67 = ( ^ [K3: int] :
% 0.35/0.67 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.35/0.67 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.67 @ ( if_Code_integer
% 0.35/0.67 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_code_if
% 0.35/0.67 thf(fact_7755_of__int__code__if,axiom,
% 0.35/0.67 ( ring_17405671764205052669omplex
% 0.35/0.67 = ( ^ [K3: int] :
% 0.35/0.67 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 0.35/0.67 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.67 @ ( if_complex
% 0.35/0.67 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_code_if
% 0.35/0.67 thf(fact_7756_vebt__succ_Opelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.35/0.67 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [Uu2: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ B ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( ( ( B
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.67 & ( ~ B
% 0.35/0.67 => ( Y3 = none_nat ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.67 => ( ! [Uv2: $o,Uw2: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ! [N3: nat] :
% 0.35/0.67 ( ( Xa2
% 0.35/0.67 = ( suc @ N3 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
% 0.35/0.67 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ Mi2 ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ none_nat
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_succ.pelims
% 0.35/0.67 thf(fact_7757_vebt__pred_Opelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 0.35/0.67 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.67 => ( ! [A: $o,Uw2: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ Uw2 ) )
% 0.35/0.67 => ( ( Xa2
% 0.35/0.67 = ( suc @ zero_zero_nat ) )
% 0.35/0.67 => ( ( ( A
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.67 & ( ~ A
% 0.35/0.67 => ( Y3 = none_nat ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ! [Va2: nat] :
% 0.35/0.67 ( ( Xa2
% 0.35/0.67 = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.67 => ( ( ( B
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ one_one_nat ) ) )
% 0.35/0.67 & ( ~ B
% 0.35/0.67 => ( ( A
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ zero_zero_nat ) ) )
% 0.35/0.67 & ( ~ A
% 0.35/0.67 => ( Y3 = none_nat ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 0.35/0.67 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 0.35/0.67 => ( ( Y3 = none_nat )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( some_nat @ Ma2 ) ) )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 != none_nat )
% 0.35/0.67 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( if_option_nat
% 0.35/0.67 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 0.35/0.67 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ none_nat ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_pred.pelims
% 0.35/0.67 thf(fact_7758_vebt__delete_Opelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.67 ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ $false @ B ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Xa2
% 0.35/0.67 = ( suc @ zero_zero_nat ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ $false ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ! [N3: nat] :
% 0.35/0.67 ( ( Xa2
% 0.35/0.67 = ( suc @ ( suc @ N3 ) ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
% 0.35/0.67 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 0.35/0.67 => ( ( ( ( Xa2 = Mi2 )
% 0.35/0.67 & ( Xa2 = Ma2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 & ( ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 & ( Xa2 = Ma2 ) )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( Xa2 = Mi2 )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.35/0.67 = Ma2 ) )
% 0.35/0.67 & ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 = none_nat )
% 0.35/0.67 @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma2 ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ ( vEBT_Node
% 0.35/0.67 @ ( some_P7363390416028606310at_nat
% 0.35/0.67 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 0.35/0.67 @ ( if_nat
% 0.35/0.67 @ ( ( ( Xa2 = Mi2 )
% 0.35/0.67 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 0.35/0.67 = Ma2 ) )
% 0.35/0.67 & ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 0.35/0.67 @ Ma2 ) ) )
% 0.35/0.67 @ ( suc @ ( suc @ Va2 ) )
% 0.35/0.67 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 @ Summary2 ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_delete.pelims
% 0.35/0.67 thf(fact_7759_monoseq__arctan__series,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.67 => ( topolo6980174941875973593q_real
% 0.35/0.67 @ ^ [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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % monoseq_arctan_series
% 0.35/0.67 thf(fact_7760_vebt__insert_Opelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.67 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ $true @ B ) ) )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ $true ) ) )
% 0.35/0.67 & ( ( Xa2 != one_one_nat )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( if_VEBT_VEBT
% 0.35/0.67 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 & ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 0.35/0.67 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_insert.pelims
% 0.35/0.67 thf(fact_7761_monoseq__realpow,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.67 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.67 => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % monoseq_realpow
% 0.35/0.67 thf(fact_7762_VEBT__internal_Oinsert_H_Opelims,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 0.35/0.67 ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
% 0.35/0.67 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 0.35/0.67 => ( Y3
% 0.35/0.67 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.insert'.pelims
% 0.35/0.67 thf(fact_7763_vebt__member_Opelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.pelims(1)
% 0.35/0.67 thf(fact_7764_vebt__member_Opelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 0.35/0.67 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.67 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.35/0.67 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
% 0.35/0.67 => ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.pelims(3)
% 0.35/0.67 thf(fact_7765_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 0.35/0.67 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.67 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.pelims(3)
% 0.35/0.67 thf(fact_7766_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.pelims(2)
% 0.35/0.67 thf(fact_7767_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.naive_member.pelims(1)
% 0.35/0.67 thf(fact_7768_vebt__member_Opelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [A: $o,B: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ A @ B ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.67 => A )
% 0.35/0.67 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.67 => ( ( ( Xa2 = one_one_nat )
% 0.35/0.67 => B )
% 0.35/0.67 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.35/0.67 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( Xa2 != Mi2 )
% 0.35/0.67 => ( ( Xa2 != Ma2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.35/0.67 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % vebt_member.pelims(2)
% 0.35/0.67 thf(fact_7769_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.35/0.67 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.35/0.67 => ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
% 0.35/0.67 => ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
% 0.35/0.67 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.pelims(3)
% 0.35/0.67 thf(fact_7770_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.67 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 = Y3 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.35/0.67 => ( ~ Y3
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ( ( Y3
% 0.35/0.67 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
% 0.35/0.67 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.pelims(1)
% 0.35/0.67 thf(fact_7771_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.35/0.67 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.67 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 ) ) ) )
% 0.35/0.67 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( Xa2 = Mi2 )
% 0.35/0.67 | ( Xa2 = Ma2 )
% 0.35/0.67 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
% 0.35/0.67 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 0.35/0.67 ( ( X2
% 0.35/0.67 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 0.35/0.67 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
% 0.35/0.67 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 0.35/0.67 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.67 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % VEBT_internal.membermima.pelims(2)
% 0.35/0.67 thf(fact_7772_ln__series,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.67 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.67 => ( ( ln_ln_real @ X2 )
% 0.35/0.67 = ( suminf_real
% 0.35/0.67 @ ^ [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 @ X2 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % ln_series
% 0.35/0.67 thf(fact_7773_signed__take__bit__rec,axiom,
% 0.35/0.67 ( bit_ri6519982836138164636nteger
% 0.35/0.67 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_rec
% 0.35/0.67 thf(fact_7774_signed__take__bit__rec,axiom,
% 0.35/0.67 ( bit_ri631733984087533419it_int
% 0.35/0.67 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_rec
% 0.35/0.67 thf(fact_7775_arctan__series,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.67 => ( ( arctan @ X2 )
% 0.35/0.67 = ( suminf_real
% 0.35/0.67 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % arctan_series
% 0.35/0.67 thf(fact_7776_signed__take__bit__of__0,axiom,
% 0.35/0.67 ! [N: nat] :
% 0.35/0.67 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 0.35/0.67 = zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_of_0
% 0.35/0.67 thf(fact_7777_powser__zero,axiom,
% 0.35/0.67 ! [F: nat > complex] :
% 0.35/0.67 ( ( suminf_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 0.35/0.67 = ( F @ zero_zero_nat ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_zero
% 0.35/0.67 thf(fact_7778_powser__zero,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 0.35/0.67 = ( F @ zero_zero_nat ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_zero
% 0.35/0.67 thf(fact_7779_signed__take__bit__0,axiom,
% 0.35/0.67 ! [A2: code_integer] :
% 0.35/0.67 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A2 )
% 0.35/0.67 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_0
% 0.35/0.67 thf(fact_7780_signed__take__bit__0,axiom,
% 0.35/0.67 ! [A2: int] :
% 0.35/0.67 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A2 )
% 0.35/0.67 = ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_0
% 0.35/0.67 thf(fact_7781_signed__take__bit__int__less__exp,axiom,
% 0.35/0.67 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_less_exp
% 0.35/0.67 thf(fact_7782_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.35/0.67 ! [K: int,N: nat] :
% 0.35/0.67 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.35/0.67 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_greater_eq_self_iff
% 0.35/0.67 thf(fact_7783_signed__take__bit__int__less__self__iff,axiom,
% 0.35/0.67 ! [N: nat,K: int] :
% 0.35/0.67 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.35/0.67 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_less_self_iff
% 0.35/0.67 thf(fact_7784_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 0.35/0.67 ! [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 ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_greater_eq_minus_exp
% 0.35/0.67 thf(fact_7785_signed__take__bit__int__less__eq__self__iff,axiom,
% 0.35/0.67 ! [N: nat,K: int] :
% 0.35/0.67 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 0.35/0.67 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_less_eq_self_iff
% 0.35/0.67 thf(fact_7786_signed__take__bit__int__greater__self__iff,axiom,
% 0.35/0.67 ! [K: int,N: nat] :
% 0.35/0.67 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.35/0.67 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_greater_self_iff
% 0.35/0.67 thf(fact_7787_signed__take__bit__int__less__eq,axiom,
% 0.35/0.67 ! [N: nat,K: int] :
% 0.35/0.67 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.35/0.67 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_less_eq
% 0.35/0.67 thf(fact_7788_signed__take__bit__int__eq__self,axiom,
% 0.35/0.67 ! [N: nat,K: int] :
% 0.35/0.67 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.35/0.67 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.67 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.35/0.67 = K ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_eq_self
% 0.35/0.67 thf(fact_7789_signed__take__bit__int__eq__self__iff,axiom,
% 0.35/0.67 ! [N: nat,K: int] :
% 0.35/0.67 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 0.35/0.67 = K )
% 0.35/0.67 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 0.35/0.67 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_eq_self_iff
% 0.35/0.67 thf(fact_7790_signed__take__bit__int__greater__eq,axiom,
% 0.35/0.67 ! [K: int,N: nat] :
% 0.35/0.67 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.67 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % signed_take_bit_int_greater_eq
% 0.35/0.67 thf(fact_7791_suminf__geometric,axiom,
% 0.35/0.67 ! [C: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.67 => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.35/0.67 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_geometric
% 0.35/0.67 thf(fact_7792_suminf__geometric,axiom,
% 0.35/0.67 ! [C: complex] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.67 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.35/0.67 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_geometric
% 0.35/0.67 thf(fact_7793_suminf__zero,axiom,
% 0.35/0.67 ( ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_real )
% 0.35/0.67 = zero_zero_real ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_zero
% 0.35/0.67 thf(fact_7794_suminf__zero,axiom,
% 0.35/0.67 ( ( suminf_nat
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_nat )
% 0.35/0.67 = zero_zero_nat ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_zero
% 0.35/0.67 thf(fact_7795_suminf__zero,axiom,
% 0.35/0.67 ( ( suminf_int
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_int )
% 0.35/0.67 = zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_zero
% 0.35/0.67 thf(fact_7796_round__unique,axiom,
% 0.35/0.67 ! [X2: real,Y3: int] :
% 0.35/0.67 ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 0.35/0.67 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 => ( ( archim8280529875227126926d_real @ X2 )
% 0.35/0.67 = Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_unique
% 0.35/0.67 thf(fact_7797_round__unique,axiom,
% 0.35/0.67 ! [X2: rat,Y3: int] :
% 0.35/0.67 ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 0.35/0.67 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.67 => ( ( archim7778729529865785530nd_rat @ X2 )
% 0.35/0.67 = Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_unique
% 0.35/0.67 thf(fact_7798_summable__arctan__series,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_arctan_series
% 0.35/0.67 thf(fact_7799_tanh__ln__real,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.67 => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.67 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_ln_real
% 0.35/0.67 thf(fact_7800_round__unique_H,axiom,
% 0.35/0.67 ! [X2: rat,N: int] :
% 0.35/0.67 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ( archim7778729529865785530nd_rat @ X2 )
% 0.35/0.67 = N ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_unique'
% 0.35/0.67 thf(fact_7801_round__unique_H,axiom,
% 0.35/0.67 ! [X2: real,N: int] :
% 0.35/0.67 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.67 => ( ( archim8280529875227126926d_real @ X2 )
% 0.35/0.67 = N ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_unique'
% 0.35/0.67 thf(fact_7802_tanh__0,axiom,
% 0.35/0.67 ( ( tanh_real @ zero_zero_real )
% 0.35/0.67 = zero_zero_real ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_0
% 0.35/0.67 thf(fact_7803_tanh__real__less__iff,axiom,
% 0.35/0.67 ! [X2: real,Y3: real] :
% 0.35/0.67 ( ( ord_less_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 0.35/0.67 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_less_iff
% 0.35/0.67 thf(fact_7804_tanh__real__le__iff,axiom,
% 0.35/0.67 ! [X2: real,Y3: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 0.35/0.67 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_le_iff
% 0.35/0.67 thf(fact_7805_summable__zero,axiom,
% 0.35/0.67 ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_real ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero
% 0.35/0.67 thf(fact_7806_summable__zero,axiom,
% 0.35/0.67 ( summable_nat
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_nat ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero
% 0.35/0.67 thf(fact_7807_summable__zero,axiom,
% 0.35/0.67 ( summable_int
% 0.35/0.67 @ ^ [N2: nat] : zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero
% 0.35/0.67 thf(fact_7808_summable__single,axiom,
% 0.35/0.67 ! [I: nat,F: nat > real] :
% 0.35/0.67 ( summable_real
% 0.35/0.67 @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_single
% 0.35/0.67 thf(fact_7809_summable__single,axiom,
% 0.35/0.67 ! [I: nat,F: nat > nat] :
% 0.35/0.67 ( summable_nat
% 0.35/0.67 @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_single
% 0.35/0.67 thf(fact_7810_summable__single,axiom,
% 0.35/0.67 ! [I: nat,F: nat > int] :
% 0.35/0.67 ( summable_int
% 0.35/0.67 @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_single
% 0.35/0.67 thf(fact_7811_round__0,axiom,
% 0.35/0.67 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.35/0.67 = zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % round_0
% 0.35/0.67 thf(fact_7812_round__0,axiom,
% 0.35/0.67 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.35/0.67 = zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % round_0
% 0.35/0.67 thf(fact_7813_tanh__real__neg__iff,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 0.35/0.67 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_neg_iff
% 0.35/0.67 thf(fact_7814_tanh__real__pos__iff,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 0.35/0.67 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_pos_iff
% 0.35/0.67 thf(fact_7815_tanh__real__nonneg__iff,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 0.35/0.67 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_nonneg_iff
% 0.35/0.67 thf(fact_7816_tanh__real__nonpos__iff,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 0.35/0.67 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_nonpos_iff
% 0.35/0.67 thf(fact_7817_summable__cmult__iff,axiom,
% 0.35/0.67 ! [C: complex,F: nat > complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.35/0.67 = ( ( C = zero_zero_complex )
% 0.35/0.67 | ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_cmult_iff
% 0.35/0.67 thf(fact_7818_summable__cmult__iff,axiom,
% 0.35/0.67 ! [C: real,F: nat > real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.35/0.67 = ( ( C = zero_zero_real )
% 0.35/0.67 | ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_cmult_iff
% 0.35/0.67 thf(fact_7819_summable__divide__iff,axiom,
% 0.35/0.67 ! [F: nat > real,C: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 0.35/0.67 = ( ( C = zero_zero_real )
% 0.35/0.67 | ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_divide_iff
% 0.35/0.67 thf(fact_7820_summable__divide__iff,axiom,
% 0.35/0.67 ! [F: nat > complex,C: complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 0.35/0.67 = ( ( C = zero_zero_complex )
% 0.35/0.67 | ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_divide_iff
% 0.35/0.67 thf(fact_7821_summable__If__finite__set,axiom,
% 0.35/0.67 ! [A3: set_nat,F: nat > real] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite_set
% 0.35/0.67 thf(fact_7822_summable__If__finite__set,axiom,
% 0.35/0.67 ! [A3: set_nat,F: nat > nat] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( summable_nat
% 0.35/0.67 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite_set
% 0.35/0.67 thf(fact_7823_summable__If__finite__set,axiom,
% 0.35/0.67 ! [A3: set_nat,F: nat > int] :
% 0.35/0.67 ( ( finite_finite_nat @ A3 )
% 0.35/0.67 => ( summable_int
% 0.35/0.67 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite_set
% 0.35/0.67 thf(fact_7824_summable__If__finite,axiom,
% 0.35/0.67 ! [P: nat > $o,F: nat > real] :
% 0.35/0.67 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite
% 0.35/0.67 thf(fact_7825_summable__If__finite,axiom,
% 0.35/0.67 ! [P: nat > $o,F: nat > nat] :
% 0.35/0.67 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.67 => ( summable_nat
% 0.35/0.67 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite
% 0.35/0.67 thf(fact_7826_summable__If__finite,axiom,
% 0.35/0.67 ! [P: nat > $o,F: nat > int] :
% 0.35/0.67 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.67 => ( summable_int
% 0.35/0.67 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_If_finite
% 0.35/0.67 thf(fact_7827_summable__geometric__iff,axiom,
% 0.35/0.67 ! [C: real] :
% 0.35/0.67 ( ( summable_real @ ( power_power_real @ C ) )
% 0.35/0.67 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_geometric_iff
% 0.35/0.67 thf(fact_7828_summable__geometric__iff,axiom,
% 0.35/0.67 ! [C: complex] :
% 0.35/0.67 ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.35/0.67 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_geometric_iff
% 0.35/0.67 thf(fact_7829_summable__const__iff,axiom,
% 0.35/0.67 ! [C: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [Uu3: nat] : C )
% 0.35/0.67 = ( C = zero_zero_real ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_const_iff
% 0.35/0.67 thf(fact_7830_summable__comparison__test_H,axiom,
% 0.35/0.67 ! [G3: nat > real,N6: nat,F: nat > real] :
% 0.35/0.67 ( ( summable_real @ G3 )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_comparison_test'
% 0.35/0.67 thf(fact_7831_summable__comparison__test_H,axiom,
% 0.35/0.67 ! [G3: nat > real,N6: nat,F: nat > complex] :
% 0.35/0.67 ( ( summable_real @ G3 )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_comparison_test'
% 0.35/0.67 thf(fact_7832_summable__comparison__test,axiom,
% 0.35/0.67 ! [F: nat > real,G3: nat > real] :
% 0.35/0.67 ( ? [N8: nat] :
% 0.35/0.67 ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( ( summable_real @ G3 )
% 0.35/0.67 => ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_comparison_test
% 0.35/0.67 thf(fact_7833_summable__comparison__test,axiom,
% 0.35/0.67 ! [F: nat > complex,G3: nat > real] :
% 0.35/0.67 ( ? [N8: nat] :
% 0.35/0.67 ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( ( summable_real @ G3 )
% 0.35/0.67 => ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_comparison_test
% 0.35/0.67 thf(fact_7834_suminf__le,axiom,
% 0.35/0.67 ! [F: nat > real,G3: nat > real] :
% 0.35/0.67 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.67 => ( ( summable_real @ F )
% 0.35/0.67 => ( ( summable_real @ G3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G3 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_le
% 0.35/0.67 thf(fact_7835_suminf__le,axiom,
% 0.35/0.67 ! [F: nat > nat,G3: nat > nat] :
% 0.35/0.67 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.67 => ( ( summable_nat @ F )
% 0.35/0.67 => ( ( summable_nat @ G3 )
% 0.35/0.67 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G3 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_le
% 0.35/0.67 thf(fact_7836_suminf__le,axiom,
% 0.35/0.67 ! [F: nat > int,G3: nat > int] :
% 0.35/0.67 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.67 => ( ( summable_int @ F )
% 0.35/0.67 => ( ( summable_int @ G3 )
% 0.35/0.67 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G3 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_le
% 0.35/0.67 thf(fact_7837_summable__finite,axiom,
% 0.35/0.67 ! [N6: set_nat,F: nat > real] :
% 0.35/0.67 ( ( finite_finite_nat @ N6 )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.67 => ( ( F @ N3 )
% 0.35/0.67 = zero_zero_real ) )
% 0.35/0.67 => ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_finite
% 0.35/0.67 thf(fact_7838_summable__finite,axiom,
% 0.35/0.67 ! [N6: set_nat,F: nat > nat] :
% 0.35/0.67 ( ( finite_finite_nat @ N6 )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.67 => ( ( F @ N3 )
% 0.35/0.67 = zero_zero_nat ) )
% 0.35/0.67 => ( summable_nat @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_finite
% 0.35/0.67 thf(fact_7839_summable__finite,axiom,
% 0.35/0.67 ! [N6: set_nat,F: nat > int] :
% 0.35/0.67 ( ( finite_finite_nat @ N6 )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.67 => ( ( F @ N3 )
% 0.35/0.67 = zero_zero_int ) )
% 0.35/0.67 => ( summable_int @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_finite
% 0.35/0.67 thf(fact_7840_summable__mult__D,axiom,
% 0.35/0.67 ! [C: complex,F: nat > complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 0.35/0.67 => ( ( C != zero_zero_complex )
% 0.35/0.67 => ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_mult_D
% 0.35/0.67 thf(fact_7841_summable__mult__D,axiom,
% 0.35/0.67 ! [C: real,F: nat > real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 0.35/0.67 => ( ( C != zero_zero_real )
% 0.35/0.67 => ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_mult_D
% 0.35/0.67 thf(fact_7842_summable__zero__power,axiom,
% 0.35/0.67 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power
% 0.35/0.67 thf(fact_7843_summable__zero__power,axiom,
% 0.35/0.67 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power
% 0.35/0.67 thf(fact_7844_summable__zero__power,axiom,
% 0.35/0.67 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power
% 0.35/0.67 thf(fact_7845_powser__insidea,axiom,
% 0.35/0.67 ! [F: nat > real,X2: real,Z: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_insidea
% 0.35/0.67 thf(fact_7846_powser__insidea,axiom,
% 0.35/0.67 ! [F: nat > complex,X2: complex,Z: complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_insidea
% 0.35/0.67 thf(fact_7847_suminf__nonneg,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_nonneg
% 0.35/0.67 thf(fact_7848_suminf__nonneg,axiom,
% 0.35/0.67 ! [F: nat > nat] :
% 0.35/0.67 ( ( summable_nat @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_nonneg
% 0.35/0.67 thf(fact_7849_suminf__nonneg,axiom,
% 0.35/0.67 ! [F: nat > int] :
% 0.35/0.67 ( ( summable_int @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_nonneg
% 0.35/0.67 thf(fact_7850_suminf__eq__zero__iff,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ( suminf_real @ F )
% 0.35/0.67 = zero_zero_real )
% 0.35/0.67 = ( ! [N2: nat] :
% 0.35/0.67 ( ( F @ N2 )
% 0.35/0.67 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_eq_zero_iff
% 0.35/0.67 thf(fact_7851_suminf__eq__zero__iff,axiom,
% 0.35/0.67 ! [F: nat > nat] :
% 0.35/0.67 ( ( summable_nat @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ( suminf_nat @ F )
% 0.35/0.67 = zero_zero_nat )
% 0.35/0.67 = ( ! [N2: nat] :
% 0.35/0.67 ( ( F @ N2 )
% 0.35/0.67 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_eq_zero_iff
% 0.35/0.67 thf(fact_7852_suminf__eq__zero__iff,axiom,
% 0.35/0.67 ! [F: nat > int] :
% 0.35/0.67 ( ( summable_int @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ( suminf_int @ F )
% 0.35/0.67 = zero_zero_int )
% 0.35/0.67 = ( ! [N2: nat] :
% 0.35/0.67 ( ( F @ N2 )
% 0.35/0.67 = zero_zero_int ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_eq_zero_iff
% 0.35/0.67 thf(fact_7853_suminf__pos,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos
% 0.35/0.67 thf(fact_7854_suminf__pos,axiom,
% 0.35/0.67 ! [F: nat > nat] :
% 0.35/0.67 ( ( summable_nat @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos
% 0.35/0.67 thf(fact_7855_suminf__pos,axiom,
% 0.35/0.67 ! [F: nat > int] :
% 0.35/0.67 ( ( summable_int @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.67 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos
% 0.35/0.67 thf(fact_7856_summable__zero__power_H,axiom,
% 0.35/0.67 ! [F: nat > complex] :
% 0.35/0.67 ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power'
% 0.35/0.67 thf(fact_7857_summable__zero__power_H,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power'
% 0.35/0.67 thf(fact_7858_summable__zero__power_H,axiom,
% 0.35/0.67 ! [F: nat > int] :
% 0.35/0.67 ( summable_int
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_zero_power'
% 0.35/0.67 thf(fact_7859_summable__0__powser,axiom,
% 0.35/0.67 ! [F: nat > complex] :
% 0.35/0.67 ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_0_powser
% 0.35/0.67 thf(fact_7860_summable__0__powser,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_0_powser
% 0.35/0.67 thf(fact_7861_summable__norm__comparison__test,axiom,
% 0.35/0.67 ! [F: nat > complex,G3: nat > real] :
% 0.35/0.67 ( ? [N8: nat] :
% 0.35/0.67 ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( ( summable_real @ G3 )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_norm_comparison_test
% 0.35/0.67 thf(fact_7862_summable__rabs__comparison__test,axiom,
% 0.35/0.67 ! [F: nat > real,G3: nat > real] :
% 0.35/0.67 ( ? [N8: nat] :
% 0.35/0.67 ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 0.35/0.67 => ( ( summable_real @ G3 )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_rabs_comparison_test
% 0.35/0.67 thf(fact_7863_summable__rabs,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_rabs
% 0.35/0.67 thf(fact_7864_tanh__real__lt__1,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_lt_1
% 0.35/0.67 thf(fact_7865_suminf__pos__iff,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.35/0.67 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos_iff
% 0.35/0.67 thf(fact_7866_suminf__pos__iff,axiom,
% 0.35/0.67 ! [F: nat > nat] :
% 0.35/0.67 ( ( summable_nat @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.35/0.67 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos_iff
% 0.35/0.67 thf(fact_7867_suminf__pos__iff,axiom,
% 0.35/0.67 ! [F: nat > int] :
% 0.35/0.67 ( ( summable_int @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.35/0.67 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos_iff
% 0.35/0.67 thf(fact_7868_suminf__pos2,axiom,
% 0.35/0.67 ! [F: nat > real,I: nat] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.67 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos2
% 0.35/0.67 thf(fact_7869_suminf__pos2,axiom,
% 0.35/0.67 ! [F: nat > nat,I: nat] :
% 0.35/0.67 ( ( summable_nat @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.67 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos2
% 0.35/0.67 thf(fact_7870_suminf__pos2,axiom,
% 0.35/0.67 ! [F: nat > int,I: nat] :
% 0.35/0.67 ( ( summable_int @ F )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.67 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.35/0.67 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_pos2
% 0.35/0.67 thf(fact_7871_summable__geometric,axiom,
% 0.35/0.67 ! [C: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.67 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_geometric
% 0.35/0.67 thf(fact_7872_summable__geometric,axiom,
% 0.35/0.67 ! [C: complex] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.67 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_geometric
% 0.35/0.67 thf(fact_7873_complete__algebra__summable__geometric,axiom,
% 0.35/0.67 ! [X2: real] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
% 0.35/0.67 => ( summable_real @ ( power_power_real @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % complete_algebra_summable_geometric
% 0.35/0.67 thf(fact_7874_complete__algebra__summable__geometric,axiom,
% 0.35/0.67 ! [X2: complex] :
% 0.35/0.67 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
% 0.35/0.67 => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % complete_algebra_summable_geometric
% 0.35/0.67 thf(fact_7875_suminf__split__head,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real @ F )
% 0.35/0.67 => ( ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 0.35/0.67 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_split_head
% 0.35/0.67 thf(fact_7876_round__mono,axiom,
% 0.35/0.67 ! [X2: rat,Y3: rat] :
% 0.35/0.67 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 0.35/0.67 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_mono
% 0.35/0.67 thf(fact_7877_summable__norm,axiom,
% 0.35/0.67 ! [F: nat > real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_norm
% 0.35/0.67 thf(fact_7878_summable__norm,axiom,
% 0.35/0.67 ! [F: nat > complex] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_norm
% 0.35/0.67 thf(fact_7879_ceiling__ge__round,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % ceiling_ge_round
% 0.35/0.67 thf(fact_7880_tanh__real__gt__neg1,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).
% 0.35/0.67
% 0.35/0.67 % tanh_real_gt_neg1
% 0.35/0.67 thf(fact_7881_powser__inside,axiom,
% 0.35/0.67 ! [F: nat > real,X2: real,Z: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_inside
% 0.35/0.67 thf(fact_7882_powser__inside,axiom,
% 0.35/0.67 ! [F: nat > complex,X2: complex,Z: complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 0.35/0.67 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 0.35/0.67 => ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_inside
% 0.35/0.67 thf(fact_7883_powser__split__head_I1_J,axiom,
% 0.35/0.67 ! [F: nat > complex,Z: complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 => ( ( suminf_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.35/0.67 @ ( times_times_complex
% 0.35/0.67 @ ( suminf_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 @ Z ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_split_head(1)
% 0.35/0.67 thf(fact_7884_powser__split__head_I1_J,axiom,
% 0.35/0.67 ! [F: nat > real,Z: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 => ( ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.35/0.67 @ ( times_times_real
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 @ Z ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_split_head(1)
% 0.35/0.67 thf(fact_7885_powser__split__head_I2_J,axiom,
% 0.35/0.67 ! [F: nat > complex,Z: complex] :
% 0.35/0.67 ( ( summable_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 => ( ( times_times_complex
% 0.35/0.67 @ ( suminf_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 @ Z )
% 0.35/0.67 = ( minus_minus_complex
% 0.35/0.67 @ ( suminf_complex
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 0.35/0.67 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_split_head(2)
% 0.35/0.67 thf(fact_7886_powser__split__head_I2_J,axiom,
% 0.35/0.67 ! [F: nat > real,Z: real] :
% 0.35/0.67 ( ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 => ( ( times_times_real
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 @ Z )
% 0.35/0.67 = ( minus_minus_real
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 0.35/0.67 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % powser_split_head(2)
% 0.35/0.67 thf(fact_7887_suminf__exist__split,axiom,
% 0.35/0.67 ! [R2: real,F: nat > real] :
% 0.35/0.67 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.67 => ( ( summable_real @ F )
% 0.35/0.67 => ? [N9: nat] :
% 0.35/0.67 ! [N5: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N9 @ N5 )
% 0.35/0.67 => ( ord_less_real
% 0.35/0.67 @ ( real_V7735802525324610683m_real
% 0.35/0.67 @ ( suminf_real
% 0.35/0.67 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 0.35/0.67 @ R2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_exist_split
% 0.35/0.67 thf(fact_7888_suminf__exist__split,axiom,
% 0.35/0.67 ! [R2: real,F: nat > complex] :
% 0.35/0.67 ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.35/0.67 => ( ( summable_complex @ F )
% 0.35/0.67 => ? [N9: nat] :
% 0.35/0.67 ! [N5: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N9 @ N5 )
% 0.35/0.67 => ( ord_less_real
% 0.35/0.67 @ ( real_V1022390504157884413omplex
% 0.35/0.67 @ ( suminf_complex
% 0.35/0.67 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 0.35/0.67 @ R2 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % suminf_exist_split
% 0.35/0.67 thf(fact_7889_summable__power__series,axiom,
% 0.35/0.67 ! [F: nat > real,Z: real] :
% 0.35/0.67 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 0.35/0.67 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 0.35/0.67 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 0.35/0.67 => ( ( ord_less_real @ Z @ one_one_real )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_power_series
% 0.35/0.67 thf(fact_7890_Abel__lemma,axiom,
% 0.35/0.67 ! [R2: real,R0: real,A2: nat > complex,M5: real] :
% 0.35/0.67 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.35/0.67 => ( ( ord_less_real @ R2 @ R0 )
% 0.35/0.67 => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M5 )
% 0.35/0.67 => ( summable_real
% 0.35/0.67 @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % Abel_lemma
% 0.35/0.67 thf(fact_7891_summable__ratio__test,axiom,
% 0.35/0.67 ! [C: real,N6: nat,F: nat > real] :
% 0.35/0.67 ( ( ord_less_real @ C @ one_one_real )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 0.35/0.67 => ( summable_real @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_ratio_test
% 0.35/0.67 thf(fact_7892_summable__ratio__test,axiom,
% 0.35/0.67 ! [C: real,N6: nat,F: nat > complex] :
% 0.35/0.67 ( ( ord_less_real @ C @ one_one_real )
% 0.35/0.67 => ( ! [N3: nat] :
% 0.35/0.67 ( ( ord_less_eq_nat @ N6 @ N3 )
% 0.35/0.67 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 0.35/0.67 => ( summable_complex @ F ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % summable_ratio_test
% 0.35/0.67 thf(fact_7893_round__diff__minimal,axiom,
% 0.35/0.67 ! [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 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_diff_minimal
% 0.35/0.67 thf(fact_7894_round__diff__minimal,axiom,
% 0.35/0.67 ! [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 ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % round_diff_minimal
% 0.35/0.67 thf(fact_7895_of__int__round__le,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_le
% 0.35/0.67 thf(fact_7896_of__int__round__le,axiom,
% 0.35/0.67 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_le
% 0.35/0.67 thf(fact_7897_of__int__round__ge,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_ge
% 0.35/0.67 thf(fact_7898_of__int__round__ge,axiom,
% 0.35/0.67 ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_ge
% 0.35/0.67 thf(fact_7899_of__int__round__gt,axiom,
% 0.35/0.67 ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_gt
% 0.35/0.67 thf(fact_7900_of__int__round__gt,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_gt
% 0.35/0.67 thf(fact_7901_of__int__round__abs__le,axiom,
% 0.35/0.67 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_abs_le
% 0.35/0.67 thf(fact_7902_of__int__round__abs__le,axiom,
% 0.35/0.67 ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % of_int_round_abs_le
% 0.35/0.67 thf(fact_7903_accp__subset,axiom,
% 0.35/0.67 ! [R1: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,R22: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o] :
% 0.35/0.67 ( ( ord_le7862513914298786254_nat_o @ R1 @ R22 )
% 0.35/0.67 => ( ord_le8126618931240741628_nat_o @ ( accp_P8646395344606611882on_nat @ R22 ) @ ( accp_P8646395344606611882on_nat @ R1 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % accp_subset
% 0.35/0.67 thf(fact_7904_accp__subset,axiom,
% 0.35/0.67 ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.35/0.67 ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 0.35/0.67 => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % accp_subset
% 0.35/0.67 thf(fact_7905_accp__subset,axiom,
% 0.35/0.67 ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 0.35/0.67 ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 0.35/0.67 => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % accp_subset
% 0.35/0.67 thf(fact_7906_accp__subset,axiom,
% 0.35/0.67 ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 0.35/0.67 ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 0.35/0.67 => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % accp_subset
% 0.35/0.67 thf(fact_7907_accp__subset,axiom,
% 0.35/0.67 ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 0.35/0.67 ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 0.35/0.67 => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % accp_subset
% 0.35/0.67 thf(fact_7908_prod_Ofinite__Collect__op,axiom,
% 0.35/0.67 ! [I5: set_o,X2: $o > complex,Y3: $o > complex] :
% 0.35/0.67 ( ( finite_finite_o
% 0.35/0.67 @ ( collect_o
% 0.35/0.67 @ ^ [I4: $o] :
% 0.35/0.67 ( ( member_o @ I4 @ I5 )
% 0.35/0.67 & ( ( X2 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( ( finite_finite_o
% 0.35/0.67 @ ( collect_o
% 0.35/0.67 @ ^ [I4: $o] :
% 0.35/0.67 ( ( member_o @ I4 @ I5 )
% 0.35/0.67 & ( ( Y3 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( finite_finite_o
% 0.35/0.67 @ ( collect_o
% 0.35/0.67 @ ^ [I4: $o] :
% 0.35/0.67 ( ( member_o @ I4 @ I5 )
% 0.35/0.67 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.67 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % prod.finite_Collect_op
% 0.35/0.67 thf(fact_7909_prod_Ofinite__Collect__op,axiom,
% 0.35/0.67 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 0.35/0.67 ( ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [I4: real] :
% 0.35/0.67 ( ( member_real @ I4 @ I5 )
% 0.35/0.67 & ( ( X2 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [I4: real] :
% 0.35/0.67 ( ( member_real @ I4 @ I5 )
% 0.35/0.67 & ( ( Y3 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( finite_finite_real
% 0.35/0.67 @ ( collect_real
% 0.35/0.67 @ ^ [I4: real] :
% 0.35/0.67 ( ( member_real @ I4 @ I5 )
% 0.35/0.67 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.67 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % prod.finite_Collect_op
% 0.35/0.67 thf(fact_7910_prod_Ofinite__Collect__op,axiom,
% 0.35/0.67 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 0.35/0.67 ( ( finite_finite_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [I4: nat] :
% 0.35/0.67 ( ( member_nat @ I4 @ I5 )
% 0.35/0.67 & ( ( X2 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( ( finite_finite_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [I4: nat] :
% 0.35/0.67 ( ( member_nat @ I4 @ I5 )
% 0.35/0.67 & ( ( Y3 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( finite_finite_nat
% 0.35/0.67 @ ( collect_nat
% 0.35/0.67 @ ^ [I4: nat] :
% 0.35/0.67 ( ( member_nat @ I4 @ I5 )
% 0.35/0.67 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.67 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.67
% 0.35/0.67 % prod.finite_Collect_op
% 0.35/0.67 thf(fact_7911_prod_Ofinite__Collect__op,axiom,
% 0.35/0.67 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 0.35/0.67 ( ( finite_finite_int
% 0.35/0.67 @ ( collect_int
% 0.35/0.67 @ ^ [I4: int] :
% 0.35/0.67 ( ( member_int @ I4 @ I5 )
% 0.35/0.67 & ( ( X2 @ I4 )
% 0.35/0.67 != one_one_complex ) ) ) )
% 0.35/0.67 => ( ( finite_finite_int
% 0.35/0.67 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_complex ) ) ) )
% 0.35/0.68 => ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7912_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 0.35/0.68 ( ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_complex ) ) ) )
% 0.35/0.68 => ( ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_complex ) ) ) )
% 0.35/0.68 => ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7913_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_Extended_enat,X2: extended_enat > complex,Y3: extended_enat > complex] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_complex ) ) ) )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_complex ) ) ) )
% 0.35/0.68 => ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_complex ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7914_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_o,X2: $o > real,Y3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7915_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 0.35/0.68 ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7916_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 0.35/0.68 ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7917_prod_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 0.35/0.68 ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != one_one_real ) ) ) )
% 0.35/0.68 => ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != one_one_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % prod.finite_Collect_op
% 0.35/0.68 thf(fact_7918_sum__gp,axiom,
% 0.35/0.68 ! [N: nat,M: nat,X2: rat] :
% 0.35/0.68 ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = zero_zero_rat ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( ( X2 = one_one_rat )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68 & ( ( X2 != one_one_rat )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_gp
% 0.35/0.68 thf(fact_7919_sum__gp,axiom,
% 0.35/0.68 ! [N: nat,M: nat,X2: complex] :
% 0.35/0.68 ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = zero_zero_complex ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( ( X2 = one_one_complex )
% 0.35/0.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68 & ( ( X2 != one_one_complex )
% 0.35/0.68 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_gp
% 0.35/0.68 thf(fact_7920_sum__gp,axiom,
% 0.35/0.68 ! [N: nat,M: nat,X2: real] :
% 0.35/0.68 ( ( ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ N @ M )
% 0.35/0.68 => ( ( ( X2 = one_one_real )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 0.35/0.68 & ( ( X2 != one_one_real )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.68 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_gp
% 0.35/0.68 thf(fact_7921_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_o,X2: $o > real,Y3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7922_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 0.35/0.68 ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7923_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 0.35/0.68 ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7924_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 0.35/0.68 ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7925_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite3207457112153483333omplex
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [I4: complex] :
% 0.35/0.68 ( ( member_complex @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7926_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_Extended_enat,X2: extended_enat > real,Y3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_real ) ) ) )
% 0.35/0.68 => ( finite4001608067531595151d_enat
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [I4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7927_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_o,X2: $o > rat,Y3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( finite_finite_o
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [I4: $o] :
% 0.35/0.68 ( ( member_o @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7928_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( finite_finite_real
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [I4: real] :
% 0.35/0.68 ( ( member_real @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7929_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( finite_finite_nat
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [I4: nat] :
% 0.35/0.68 ( ( member_nat @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7930_sum_Ofinite__Collect__op,axiom,
% 0.35/0.68 ! [I5: set_int,X2: int > rat,Y3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( X2 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( Y3 @ I4 )
% 0.35/0.68 != zero_zero_rat ) ) ) )
% 0.35/0.68 => ( finite_finite_int
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [I4: int] :
% 0.35/0.68 ( ( member_int @ I4 @ I5 )
% 0.35/0.68 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 0.35/0.68 != zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.finite_Collect_op
% 0.35/0.68 thf(fact_7931_geometric__deriv__sums,axiom,
% 0.35/0.68 ! [Z: real] :
% 0.35/0.68 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 0.35/0.68 => ( sums_real
% 0.35/0.68 @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 0.35/0.68 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % geometric_deriv_sums
% 0.35/0.68 thf(fact_7932_geometric__deriv__sums,axiom,
% 0.35/0.68 ! [Z: complex] :
% 0.35/0.68 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 0.35/0.68 => ( sums_complex
% 0.35/0.68 @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 0.35/0.68 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % geometric_deriv_sums
% 0.35/0.68 thf(fact_7933_log__base__10__eq1,axiom,
% 0.35/0.68 ! [X2: real] :
% 0.35/0.68 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.68 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.68 = ( 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 @ X2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % log_base_10_eq1
% 0.35/0.68 thf(fact_7934_semiring__norm_I80_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.68 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(80)
% 0.35/0.68 thf(fact_7935_semiring__norm_I73_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.68 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(73)
% 0.35/0.68 thf(fact_7936_sum_Oneutral__const,axiom,
% 0.35/0.68 ! [A3: set_nat] :
% 0.35/0.68 ( ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Uu3: nat] : zero_zero_nat
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_nat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral_const
% 0.35/0.68 thf(fact_7937_sum_Oneutral__const,axiom,
% 0.35/0.68 ! [A3: set_complex] :
% 0.35/0.68 ( ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Uu3: complex] : zero_zero_complex
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_complex ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral_const
% 0.35/0.68 thf(fact_7938_sum_Oneutral__const,axiom,
% 0.35/0.68 ! [A3: set_int] :
% 0.35/0.68 ( ( groups4538972089207619220nt_int
% 0.35/0.68 @ ^ [Uu3: int] : zero_zero_int
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_int ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral_const
% 0.35/0.68 thf(fact_7939_sum_Oneutral__const,axiom,
% 0.35/0.68 ! [A3: set_nat] :
% 0.35/0.68 ( ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [Uu3: nat] : zero_zero_real
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_real ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral_const
% 0.35/0.68 thf(fact_7940_sums__zero,axiom,
% 0.35/0.68 ( sums_real
% 0.35/0.68 @ ^ [N2: nat] : zero_zero_real
% 0.35/0.68 @ zero_zero_real ) ).
% 0.35/0.68
% 0.35/0.68 % sums_zero
% 0.35/0.68 thf(fact_7941_sums__zero,axiom,
% 0.35/0.68 ( sums_nat
% 0.35/0.68 @ ^ [N2: nat] : zero_zero_nat
% 0.35/0.68 @ zero_zero_nat ) ).
% 0.35/0.68
% 0.35/0.68 % sums_zero
% 0.35/0.68 thf(fact_7942_sums__zero,axiom,
% 0.35/0.68 ( sums_int
% 0.35/0.68 @ ^ [N2: nat] : zero_zero_int
% 0.35/0.68 @ zero_zero_int ) ).
% 0.35/0.68
% 0.35/0.68 % sums_zero
% 0.35/0.68 thf(fact_7943_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: real > real] :
% 0.35/0.68 ( ( groups8097168146408367636l_real @ G3 @ bot_bot_set_real )
% 0.35/0.68 = zero_zero_real ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7944_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: real > rat] :
% 0.35/0.68 ( ( groups1300246762558778688al_rat @ G3 @ bot_bot_set_real )
% 0.35/0.68 = zero_zero_rat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7945_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: real > nat] :
% 0.35/0.68 ( ( groups1935376822645274424al_nat @ G3 @ bot_bot_set_real )
% 0.35/0.68 = zero_zero_nat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7946_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: real > int] :
% 0.35/0.68 ( ( groups1932886352136224148al_int @ G3 @ bot_bot_set_real )
% 0.35/0.68 = zero_zero_int ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7947_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: $o > real] :
% 0.35/0.68 ( ( groups8691415230153176458o_real @ G3 @ bot_bot_set_o )
% 0.35/0.68 = zero_zero_real ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7948_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: $o > rat] :
% 0.35/0.68 ( ( groups7872700643590313910_o_rat @ G3 @ bot_bot_set_o )
% 0.35/0.68 = zero_zero_rat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7949_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: $o > nat] :
% 0.35/0.68 ( ( groups8507830703676809646_o_nat @ G3 @ bot_bot_set_o )
% 0.35/0.68 = zero_zero_nat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7950_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: $o > int] :
% 0.35/0.68 ( ( groups8505340233167759370_o_int @ G3 @ bot_bot_set_o )
% 0.35/0.68 = zero_zero_int ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7951_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: nat > rat] :
% 0.35/0.68 ( ( groups2906978787729119204at_rat @ G3 @ bot_bot_set_nat )
% 0.35/0.68 = zero_zero_rat ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7952_sum_Oempty,axiom,
% 0.35/0.68 ! [G3: nat > int] :
% 0.35/0.68 ( ( groups3539618377306564664at_int @ G3 @ bot_bot_set_nat )
% 0.35/0.68 = zero_zero_int ) ).
% 0.35/0.68
% 0.35/0.68 % sum.empty
% 0.35/0.68 thf(fact_7953_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > real] :
% 0.35/0.68 ( ~ ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7954_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > real] :
% 0.35/0.68 ( ~ ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7955_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.68 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7956_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_nat,G3: nat > rat] :
% 0.35/0.68 ( ~ ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7957_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > rat] :
% 0.35/0.68 ( ~ ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7958_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > rat] :
% 0.35/0.68 ( ~ ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7959_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.68 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7960_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > nat] :
% 0.35/0.68 ( ~ ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7961_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > nat] :
% 0.35/0.68 ( ~ ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7962_sum_Oinfinite,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.68 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.infinite
% 0.35/0.68 thf(fact_7963_sum__eq__0__iff,axiom,
% 0.35/0.68 ! [F2: set_int,F: int > nat] :
% 0.35/0.68 ( ( finite_finite_int @ F2 )
% 0.35/0.68 => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 = ( ! [X: int] :
% 0.35/0.68 ( ( member_int @ X @ F2 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_eq_0_iff
% 0.35/0.68 thf(fact_7964_sum__eq__0__iff,axiom,
% 0.35/0.68 ! [F2: set_complex,F: complex > nat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ F2 )
% 0.35/0.68 => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 = ( ! [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ F2 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_eq_0_iff
% 0.35/0.68 thf(fact_7965_sum__eq__0__iff,axiom,
% 0.35/0.68 ! [F2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.68 ( ( finite6177210948735845034at_nat @ F2 )
% 0.35/0.68 => ( ( ( groups977919841031483927at_nat @ F @ F2 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 = ( ! [X: product_prod_nat_nat] :
% 0.35/0.68 ( ( member8440522571783428010at_nat @ X @ F2 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_eq_0_iff
% 0.35/0.68 thf(fact_7966_sum__eq__0__iff,axiom,
% 0.35/0.68 ! [F2: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ F2 )
% 0.35/0.68 => ( ( ( groups2027974829824023292at_nat @ F @ F2 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 = ( ! [X: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X @ F2 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_eq_0_iff
% 0.35/0.68 thf(fact_7967_sum__eq__0__iff,axiom,
% 0.35/0.68 ! [F2: set_nat,F: nat > nat] :
% 0.35/0.68 ( ( finite_finite_nat @ F2 )
% 0.35/0.68 => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 = ( ! [X: nat] :
% 0.35/0.68 ( ( member_nat @ X @ F2 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_nat ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_eq_0_iff
% 0.35/0.68 thf(fact_7968_semiring__norm_I81_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.68 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(81)
% 0.35/0.68 thf(fact_7969_semiring__norm_I72_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.68 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(72)
% 0.35/0.68 thf(fact_7970_semiring__norm_I77_J,axiom,
% 0.35/0.68 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(77)
% 0.35/0.68 thf(fact_7971_semiring__norm_I70_J,axiom,
% 0.35/0.68 ! [M: num] :
% 0.35/0.68 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(70)
% 0.35/0.68 thf(fact_7972_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_real,A2: real,B3: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ S )
% 0.35/0.68 => ( ( ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups8097168146408367636l_real
% 0.35/0.68 @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups8097168146408367636l_real
% 0.35/0.68 @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7973_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_o,A2: $o,B3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ S )
% 0.35/0.68 => ( ( ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups8691415230153176458o_real
% 0.35/0.68 @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups8691415230153176458o_real
% 0.35/0.68 @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7974_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_int,A2: int,B3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ( ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups8778361861064173332t_real
% 0.35/0.68 @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups8778361861064173332t_real
% 0.35/0.68 @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7975_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_complex,A2: complex,B3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7976_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.68 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.68 => ( ( groups4148127829035722712t_real
% 0.35/0.68 @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.68 => ( ( groups4148127829035722712t_real
% 0.35/0.68 @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7977_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_real,A2: real,B3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ S )
% 0.35/0.68 => ( ( ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat
% 0.35/0.68 @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat
% 0.35/0.68 @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7978_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_o,A2: $o,B3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ S )
% 0.35/0.68 => ( ( ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat
% 0.35/0.68 @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat
% 0.35/0.68 @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7979_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_nat,A2: nat,B3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ S )
% 0.35/0.68 => ( ( ( member_nat @ A2 @ S )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_nat @ A2 @ S )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7980_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_int,A2: int,B3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ( ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat
% 0.35/0.68 @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat
% 0.35/0.68 @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7981_sum_Odelta_H,axiom,
% 0.35/0.68 ! [S: set_complex,A2: complex,B3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat
% 0.35/0.68 @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat
% 0.35/0.68 @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta'
% 0.35/0.68 thf(fact_7982_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_real,A2: real,B3: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ S )
% 0.35/0.68 => ( ( ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups8097168146408367636l_real
% 0.35/0.68 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups8097168146408367636l_real
% 0.35/0.68 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7983_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_o,A2: $o,B3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ S )
% 0.35/0.68 => ( ( ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups8691415230153176458o_real
% 0.35/0.68 @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups8691415230153176458o_real
% 0.35/0.68 @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7984_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_int,A2: int,B3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ( ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups8778361861064173332t_real
% 0.35/0.68 @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups8778361861064173332t_real
% 0.35/0.68 @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7985_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_complex,A2: complex,B3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7986_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.68 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.68 => ( ( groups4148127829035722712t_real
% 0.35/0.68 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.68 => ( ( groups4148127829035722712t_real
% 0.35/0.68 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_real ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7987_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_real,A2: real,B3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ S )
% 0.35/0.68 => ( ( ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat
% 0.35/0.68 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat
% 0.35/0.68 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7988_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_o,A2: $o,B3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ S )
% 0.35/0.68 => ( ( ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat
% 0.35/0.68 @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ A2 @ S )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat
% 0.35/0.68 @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7989_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_nat,A2: nat,B3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ S )
% 0.35/0.68 => ( ( ( member_nat @ A2 @ S )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_nat @ A2 @ S )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7990_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_int,A2: int,B3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ( ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat
% 0.35/0.68 @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ A2 @ S )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat
% 0.35/0.68 @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7991_sum_Odelta,axiom,
% 0.35/0.68 ! [S: set_complex,A2: complex,B3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat
% 0.35/0.68 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = ( B3 @ A2 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat
% 0.35/0.68 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 0.35/0.68 @ S )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.delta
% 0.35/0.68 thf(fact_7992_sum__abs,axiom,
% 0.35/0.68 ! [F: int > int,A3: set_int] :
% 0.35/0.68 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
% 0.35/0.68 @ ( groups4538972089207619220nt_int
% 0.35/0.68 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_abs
% 0.35/0.68 thf(fact_7993_sum__abs,axiom,
% 0.35/0.68 ! [F: nat > real,A3: set_nat] :
% 0.35/0.68 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
% 0.35/0.68 @ ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_abs
% 0.35/0.68 thf(fact_7994_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_real,X2: real,G3: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ~ ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_7995_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_o,X2: $o,G3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ~ ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_7996_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_int,X2: int,G3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ~ ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_7997_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ~ ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_7998_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ~ ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_7999_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_real,X2: real,G3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ~ ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_8000_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_o,X2: $o,G3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ~ ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_8001_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_nat,X2: nat,G3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ~ ( member_nat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_8002_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_int,X2: int,G3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ~ ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_8003_sum_Oinsert,axiom,
% 0.35/0.68 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ~ ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert
% 0.35/0.68 thf(fact_8004_sum__abs__ge__zero,axiom,
% 0.35/0.68 ! [F: int > int,A3: set_int] :
% 0.35/0.68 ( ord_less_eq_int @ zero_zero_int
% 0.35/0.68 @ ( groups4538972089207619220nt_int
% 0.35/0.68 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_abs_ge_zero
% 0.35/0.68 thf(fact_8005_sum__abs__ge__zero,axiom,
% 0.35/0.68 ! [F: nat > real,A3: set_nat] :
% 0.35/0.68 ( ord_less_eq_real @ zero_zero_real
% 0.35/0.68 @ ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_abs_ge_zero
% 0.35/0.68 thf(fact_8006_semiring__norm_I74_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 0.35/0.68 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(74)
% 0.35/0.68 thf(fact_8007_semiring__norm_I79_J,axiom,
% 0.35/0.68 ! [M: num,N: num] :
% 0.35/0.68 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.68 = ( ord_less_eq_num @ M @ N ) ) ).
% 0.35/0.68
% 0.35/0.68 % semiring_norm(79)
% 0.35/0.68 thf(fact_8008_powser__sums__zero__iff,axiom,
% 0.35/0.68 ! [A2: nat > complex,X2: complex] :
% 0.35/0.68 ( ( sums_complex
% 0.35/0.68 @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.35/0.68 @ X2 )
% 0.35/0.68 = ( ( A2 @ zero_zero_nat )
% 0.35/0.68 = X2 ) ) ).
% 0.35/0.68
% 0.35/0.68 % powser_sums_zero_iff
% 0.35/0.68 thf(fact_8009_powser__sums__zero__iff,axiom,
% 0.35/0.68 ! [A2: nat > real,X2: real] :
% 0.35/0.68 ( ( sums_real
% 0.35/0.68 @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.35/0.68 @ X2 )
% 0.35/0.68 = ( ( A2 @ zero_zero_nat )
% 0.35/0.68 = X2 ) ) ).
% 0.35/0.68
% 0.35/0.68 % powser_sums_zero_iff
% 0.35/0.68 thf(fact_8010_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.68 ! [N: nat,M: nat,G3: nat > rat] :
% 0.35/0.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = zero_zero_rat ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.cl_ivl_Suc
% 0.35/0.68 thf(fact_8011_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.68 ! [N: nat,M: nat,G3: nat > int] :
% 0.35/0.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = zero_zero_int ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.cl_ivl_Suc
% 0.35/0.68 thf(fact_8012_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.68 ! [N: nat,M: nat,G3: nat > nat] :
% 0.35/0.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = zero_zero_nat ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.cl_ivl_Suc
% 0.35/0.68 thf(fact_8013_sum_Ocl__ivl__Suc,axiom,
% 0.35/0.68 ! [N: nat,M: nat,G3: nat > real] :
% 0.35/0.68 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.68 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.cl_ivl_Suc
% 0.35/0.68 thf(fact_8014_sum__zero__power,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > complex] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( C @ zero_zero_nat ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_complex ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power
% 0.35/0.68 thf(fact_8015_sum__zero__power,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > rat] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( C @ zero_zero_nat ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power
% 0.35/0.68 thf(fact_8016_sum__zero__power,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > real] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( C @ zero_zero_nat ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power
% 0.35/0.68 thf(fact_8017_sum__zero__power_H,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > rat,D: nat > rat] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power'
% 0.35/0.68 thf(fact_8018_sum__zero__power_H,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > complex,D: nat > complex] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_complex ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power'
% 0.35/0.68 thf(fact_8019_sum__zero__power_H,axiom,
% 0.35/0.68 ! [A3: set_nat,C: nat > real,D: nat > real] :
% 0.35/0.68 ( ( ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.35/0.68 & ( ~ ( ( finite_finite_nat @ A3 )
% 0.35/0.68 & ( member_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_zero_power'
% 0.35/0.68 thf(fact_8020_sums__le,axiom,
% 0.35/0.68 ! [F: nat > nat,G3: nat > nat,S2: nat,T: nat] :
% 0.35/0.68 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.68 => ( ( sums_nat @ F @ S2 )
% 0.35/0.68 => ( ( sums_nat @ G3 @ T )
% 0.35/0.68 => ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_le
% 0.35/0.68 thf(fact_8021_sums__le,axiom,
% 0.35/0.68 ! [F: nat > int,G3: nat > int,S2: int,T: int] :
% 0.35/0.68 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.68 => ( ( sums_int @ F @ S2 )
% 0.35/0.68 => ( ( sums_int @ G3 @ T )
% 0.35/0.68 => ( ord_less_eq_int @ S2 @ T ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_le
% 0.35/0.68 thf(fact_8022_sums__0,axiom,
% 0.35/0.68 ! [F: nat > real] :
% 0.35/0.68 ( ! [N3: nat] :
% 0.35/0.68 ( ( F @ N3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 => ( sums_real @ F @ zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_0
% 0.35/0.68 thf(fact_8023_sums__0,axiom,
% 0.35/0.68 ! [F: nat > nat] :
% 0.35/0.68 ( ! [N3: nat] :
% 0.35/0.68 ( ( F @ N3 )
% 0.35/0.68 = zero_zero_nat )
% 0.35/0.68 => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_0
% 0.35/0.68 thf(fact_8024_sums__0,axiom,
% 0.35/0.68 ! [F: nat > int] :
% 0.35/0.68 ( ! [N3: nat] :
% 0.35/0.68 ( ( F @ N3 )
% 0.35/0.68 = zero_zero_int )
% 0.35/0.68 => ( sums_int @ F @ zero_zero_int ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_0
% 0.35/0.68 thf(fact_8025_sum_Oneutral,axiom,
% 0.35/0.68 ! [A3: set_nat,G3: nat > nat] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ( G3 @ X5 )
% 0.35/0.68 = zero_zero_nat ) )
% 0.35/0.68 => ( ( groups3542108847815614940at_nat @ G3 @ A3 )
% 0.35/0.68 = zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral
% 0.35/0.68 thf(fact_8026_sum_Oneutral,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > complex] :
% 0.35/0.68 ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ( G3 @ X5 )
% 0.35/0.68 = zero_zero_complex ) )
% 0.35/0.68 => ( ( groups7754918857620584856omplex @ G3 @ A3 )
% 0.35/0.68 = zero_zero_complex ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral
% 0.35/0.68 thf(fact_8027_sum_Oneutral,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > int] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ( G3 @ X5 )
% 0.35/0.68 = zero_zero_int ) )
% 0.35/0.68 => ( ( groups4538972089207619220nt_int @ G3 @ A3 )
% 0.35/0.68 = zero_zero_int ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral
% 0.35/0.68 thf(fact_8028_sum_Oneutral,axiom,
% 0.35/0.68 ! [A3: set_nat,G3: nat > real] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ( G3 @ X5 )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ G3 @ A3 )
% 0.35/0.68 = zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.neutral
% 0.35/0.68 thf(fact_8029_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: real > real,A3: set_real] :
% 0.35/0.68 ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 0.35/0.68 != zero_zero_real )
% 0.35/0.68 => ~ ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8030_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: $o > real,A3: set_o] :
% 0.35/0.68 ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 0.35/0.68 != zero_zero_real )
% 0.35/0.68 => ~ ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8031_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: int > real,A3: set_int] :
% 0.35/0.68 ( ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 0.35/0.68 != zero_zero_real )
% 0.35/0.68 => ~ ! [A: int] :
% 0.35/0.68 ( ( member_int @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8032_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: real > rat,A3: set_real] :
% 0.35/0.68 ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_rat )
% 0.35/0.68 => ~ ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8033_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: $o > rat,A3: set_o] :
% 0.35/0.68 ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_rat )
% 0.35/0.68 => ~ ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8034_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: nat > rat,A3: set_nat] :
% 0.35/0.68 ( ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_rat )
% 0.35/0.68 => ~ ! [A: nat] :
% 0.35/0.68 ( ( member_nat @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8035_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: int > rat,A3: set_int] :
% 0.35/0.68 ( ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_rat )
% 0.35/0.68 => ~ ! [A: int] :
% 0.35/0.68 ( ( member_int @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_rat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8036_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: real > nat,A3: set_real] :
% 0.35/0.68 ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_nat )
% 0.35/0.68 => ~ ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_nat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8037_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: $o > nat,A3: set_o] :
% 0.35/0.68 ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_nat )
% 0.35/0.68 => ~ ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_nat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8038_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.35/0.68 ! [G3: int > nat,A3: set_int] :
% 0.35/0.68 ( ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 0.35/0.68 != zero_zero_nat )
% 0.35/0.68 => ~ ! [A: int] :
% 0.35/0.68 ( ( member_int @ A @ A3 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_nat ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.not_neutral_contains_not_neutral
% 0.35/0.68 thf(fact_8039_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_real,F: real > complex,G3: real > real] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8040_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_o,F: $o > complex,G3: $o > real] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( groups8691415230153176458o_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8041_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_set_nat,F: set_nat > complex,G3: set_nat > real] :
% 0.35/0.68 ( ! [X5: set_nat] :
% 0.35/0.68 ( ( member_set_nat @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8042_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_int,F: int > complex,G3: int > real] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8043_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_nat,F: nat > complex,G3: nat > real] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8044_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_complex,F: complex > complex,G3: complex > real] :
% 0.35/0.68 ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8045_sum__norm__le,axiom,
% 0.35/0.68 ! [S: set_nat,F: nat > real,G3: nat > real] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ S )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_norm_le
% 0.35/0.68 thf(fact_8046_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_real,F: real > rat,G3: real > rat] :
% 0.35/0.68 ( ! [I2: real] :
% 0.35/0.68 ( ( member_real @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8047_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_o,F: $o > rat,G3: $o > rat] :
% 0.35/0.68 ( ! [I2: $o] :
% 0.35/0.68 ( ( member_o @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K4 ) @ ( groups7872700643590313910_o_rat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8048_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_nat,F: nat > rat,G3: nat > rat] :
% 0.35/0.68 ( ! [I2: nat] :
% 0.35/0.68 ( ( member_nat @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8049_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_int,F: int > rat,G3: int > rat] :
% 0.35/0.68 ( ! [I2: int] :
% 0.35/0.68 ( ( member_int @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8050_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_real,F: real > nat,G3: real > nat] :
% 0.35/0.68 ( ! [I2: real] :
% 0.35/0.68 ( ( member_real @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8051_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_o,F: $o > nat,G3: $o > nat] :
% 0.35/0.68 ( ! [I2: $o] :
% 0.35/0.68 ( ( member_o @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K4 ) @ ( groups8507830703676809646_o_nat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8052_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_int,F: int > nat,G3: int > nat] :
% 0.35/0.68 ( ! [I2: int] :
% 0.35/0.68 ( ( member_int @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8053_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_real,F: real > int,G3: real > int] :
% 0.35/0.68 ( ! [I2: real] :
% 0.35/0.68 ( ( member_real @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8054_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_o,F: $o > int,G3: $o > int] :
% 0.35/0.68 ( ! [I2: $o] :
% 0.35/0.68 ( ( member_o @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K4 ) @ ( groups8505340233167759370_o_int @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8055_sum__mono,axiom,
% 0.35/0.68 ! [K4: set_nat,F: nat > int,G3: nat > int] :
% 0.35/0.68 ( ! [I2: nat] :
% 0.35/0.68 ( ( member_nat @ I2 @ K4 )
% 0.35/0.68 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G3 @ K4 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono
% 0.35/0.68 thf(fact_8056_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_o,B2: set_nat,G3: $o > nat > nat,R: $o > nat > $o] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( finite_finite_nat @ B2 )
% 0.35/0.68 => ( ( groups8507830703676809646_o_nat
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( ( member_nat @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( groups8507830703676809646_o_nat
% 0.35/0.68 @ ^ [X: $o] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8057_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_real,B2: set_nat,G3: real > nat > nat,R: real > nat > $o] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( finite_finite_nat @ B2 )
% 0.35/0.68 => ( ( groups1935376822645274424al_nat
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( ( member_nat @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( groups1935376822645274424al_nat
% 0.35/0.68 @ ^ [X: real] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8058_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_int,B2: set_nat,G3: int > nat > nat,R: int > nat > $o] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( finite_finite_nat @ B2 )
% 0.35/0.68 => ( ( groups4541462559716669496nt_nat
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( ( member_nat @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( groups4541462559716669496nt_nat
% 0.35/0.68 @ ^ [X: int] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8059_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_complex,B2: set_nat,G3: complex > nat > nat,R: complex > nat > $o] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( finite_finite_nat @ B2 )
% 0.35/0.68 => ( ( groups5693394587270226106ex_nat
% 0.35/0.68 @ ^ [X: complex] :
% 0.35/0.68 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( ( member_nat @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( groups5693394587270226106ex_nat
% 0.35/0.68 @ ^ [X: complex] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8060_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,B2: set_nat,G3: extended_enat > nat > nat,R: extended_enat > nat > $o] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( finite_finite_nat @ B2 )
% 0.35/0.68 => ( ( groups2027974829824023292at_nat
% 0.35/0.68 @ ^ [X: extended_enat] :
% 0.35/0.68 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( ( member_nat @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat
% 0.35/0.68 @ ^ [Y: nat] :
% 0.35/0.68 ( groups2027974829824023292at_nat
% 0.35/0.68 @ ^ [X: extended_enat] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [X: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8061_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_o,B2: set_complex,G3: $o > complex > complex,R: $o > complex > $o] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68 => ( ( groups5328290441151304332omplex
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( groups7754918857620584856omplex @ ( G3 @ X )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( ( member_complex @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( groups5328290441151304332omplex
% 0.35/0.68 @ ^ [X: $o] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8062_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_real,B2: set_complex,G3: real > complex > complex,R: real > complex > $o] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68 => ( ( groups5754745047067104278omplex
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( groups7754918857620584856omplex @ ( G3 @ X )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( ( member_complex @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( groups5754745047067104278omplex
% 0.35/0.68 @ ^ [X: real] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8063_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_nat,B2: set_complex,G3: nat > complex > complex,R: nat > complex > $o] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68 => ( ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [X: nat] :
% 0.35/0.68 ( groups7754918857620584856omplex @ ( G3 @ X )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( ( member_complex @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( groups2073611262835488442omplex
% 0.35/0.68 @ ^ [X: nat] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [X: nat] :
% 0.35/0.68 ( ( member_nat @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8064_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_int,B2: set_complex,G3: int > complex > complex,R: int > complex > $o] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68 => ( ( groups3049146728041665814omplex
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( groups7754918857620584856omplex @ ( G3 @ X )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( ( member_complex @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( groups3049146728041665814omplex
% 0.35/0.68 @ ^ [X: int] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8065_sum_Oswap__restrict,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,B2: set_complex,G3: extended_enat > complex > complex,R: extended_enat > complex > $o] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.68 => ( ( groups6818542070133387226omplex
% 0.35/0.68 @ ^ [X: extended_enat] :
% 0.35/0.68 ( groups7754918857620584856omplex @ ( G3 @ X )
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( ( member_complex @ Y @ B2 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ A3 )
% 0.35/0.68 = ( groups7754918857620584856omplex
% 0.35/0.68 @ ^ [Y: complex] :
% 0.35/0.68 ( groups6818542070133387226omplex
% 0.35/0.68 @ ^ [X: extended_enat] : ( G3 @ X @ Y )
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [X: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.68 & ( R @ X @ Y ) ) ) )
% 0.35/0.68 @ B2 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.swap_restrict
% 0.35/0.68 thf(fact_8066_sums__If__finite__set,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > int] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( sums_int
% 0.35/0.68 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.68 @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite_set
% 0.35/0.68 thf(fact_8067_sums__If__finite__set,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > nat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( sums_nat
% 0.35/0.68 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.68 @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite_set
% 0.35/0.68 thf(fact_8068_sums__If__finite__set,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > real] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( sums_real
% 0.35/0.68 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.68 @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite_set
% 0.35/0.68 thf(fact_8069_sums__If__finite,axiom,
% 0.35/0.68 ! [P: nat > $o,F: nat > int] :
% 0.35/0.68 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.68 => ( sums_int
% 0.35/0.68 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.68 @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite
% 0.35/0.68 thf(fact_8070_sums__If__finite,axiom,
% 0.35/0.68 ! [P: nat > $o,F: nat > nat] :
% 0.35/0.68 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.68 => ( sums_nat
% 0.35/0.68 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.68 @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite
% 0.35/0.68 thf(fact_8071_sums__If__finite,axiom,
% 0.35/0.68 ! [P: nat > $o,F: nat > real] :
% 0.35/0.68 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.35/0.68 => ( sums_real
% 0.35/0.68 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.68 @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite
% 0.35/0.68 thf(fact_8072_sums__finite,axiom,
% 0.35/0.68 ! [N6: set_nat,F: nat > int] :
% 0.35/0.68 ( ( finite_finite_nat @ N6 )
% 0.35/0.68 => ( ! [N3: nat] :
% 0.35/0.68 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.68 => ( ( F @ N3 )
% 0.35/0.68 = zero_zero_int ) )
% 0.35/0.68 => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_finite
% 0.35/0.68 thf(fact_8073_sums__finite,axiom,
% 0.35/0.68 ! [N6: set_nat,F: nat > nat] :
% 0.35/0.68 ( ( finite_finite_nat @ N6 )
% 0.35/0.68 => ( ! [N3: nat] :
% 0.35/0.68 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.68 => ( ( F @ N3 )
% 0.35/0.68 = zero_zero_nat ) )
% 0.35/0.68 => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_finite
% 0.35/0.68 thf(fact_8074_sums__finite,axiom,
% 0.35/0.68 ! [N6: set_nat,F: nat > real] :
% 0.35/0.68 ( ( finite_finite_nat @ N6 )
% 0.35/0.68 => ( ! [N3: nat] :
% 0.35/0.68 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.68 => ( ( F @ N3 )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_finite
% 0.35/0.68 thf(fact_8075_norm__sum,axiom,
% 0.35/0.68 ! [F: nat > complex,A3: set_nat] :
% 0.35/0.68 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A3 ) )
% 0.35/0.68 @ ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % norm_sum
% 0.35/0.68 thf(fact_8076_norm__sum,axiom,
% 0.35/0.68 ! [F: complex > complex,A3: set_complex] :
% 0.35/0.68 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A3 ) )
% 0.35/0.68 @ ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % norm_sum
% 0.35/0.68 thf(fact_8077_norm__sum,axiom,
% 0.35/0.68 ! [F: nat > real,A3: set_nat] :
% 0.35/0.68 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
% 0.35/0.68 @ ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 0.35/0.68 @ A3 ) ) ).
% 0.35/0.68
% 0.35/0.68 % norm_sum
% 0.35/0.68 thf(fact_8078_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > real] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8079_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > real] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8080_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > real] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8081_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > rat] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8082_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > rat] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8083_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > rat] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8084_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > rat] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8085_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > nat] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8086_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > nat] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8087_sum__nonpos,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > nat] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 0.35/0.68 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonpos
% 0.35/0.68 thf(fact_8088_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > real] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8089_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > real] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8090_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > real] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8091_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > rat] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8092_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > rat] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8093_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > rat] :
% 0.35/0.68 ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8094_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > rat] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8095_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > nat] :
% 0.35/0.68 ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8096_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > nat] :
% 0.35/0.68 ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8097_sum__nonneg,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > nat] :
% 0.35/0.68 ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg
% 0.35/0.68 thf(fact_8098_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: real > rat,I5: set_real,G3: real > rat,I: real] :
% 0.35/0.68 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 0.35/0.68 = ( groups1300246762558778688al_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: real] :
% 0.35/0.68 ( ( member_real @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_real @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_real @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8099_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: $o > rat,I5: set_o,G3: $o > rat,I: $o] :
% 0.35/0.68 ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
% 0.35/0.68 = ( groups7872700643590313910_o_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: $o] :
% 0.35/0.68 ( ( member_o @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_o @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_o @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8100_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: nat > rat,I5: set_nat,G3: nat > rat,I: nat] :
% 0.35/0.68 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 0.35/0.68 = ( groups2906978787729119204at_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: nat] :
% 0.35/0.68 ( ( member_nat @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_nat @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_nat @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8101_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: int > rat,I5: set_int,G3: int > rat,I: int] :
% 0.35/0.68 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 0.35/0.68 = ( groups3906332499630173760nt_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: int] :
% 0.35/0.68 ( ( member_int @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_int @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_int @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8102_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: complex > rat,I5: set_complex,G3: complex > rat,I: complex] :
% 0.35/0.68 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 0.35/0.68 = ( groups5058264527183730370ex_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: complex] :
% 0.35/0.68 ( ( member_complex @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_complex @ I @ I5 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8103_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: extended_enat > rat,I5: set_Extended_enat,G3: extended_enat > rat,I: extended_enat] :
% 0.35/0.68 ( ( ( groups1392844769737527556at_rat @ F @ I5 )
% 0.35/0.68 = ( groups1392844769737527556at_rat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_Extended_enat @ I @ I5 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8104_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: real > nat,I5: set_real,G3: real > nat,I: real] :
% 0.35/0.68 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 0.35/0.68 = ( groups1935376822645274424al_nat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: real] :
% 0.35/0.68 ( ( member_real @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_real @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_real @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8105_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: $o > nat,I5: set_o,G3: $o > nat,I: $o] :
% 0.35/0.68 ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
% 0.35/0.68 = ( groups8507830703676809646_o_nat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: $o] :
% 0.35/0.68 ( ( member_o @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_o @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_o @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8106_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: int > nat,I5: set_int,G3: int > nat,I: int] :
% 0.35/0.68 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 0.35/0.68 = ( groups4541462559716669496nt_nat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: int] :
% 0.35/0.68 ( ( member_int @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_int @ I @ I5 )
% 0.35/0.68 => ( ( finite_finite_int @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8107_sum__mono__inv,axiom,
% 0.35/0.68 ! [F: complex > nat,I5: set_complex,G3: complex > nat,I: complex] :
% 0.35/0.68 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 0.35/0.68 = ( groups5693394587270226106ex_nat @ G3 @ I5 ) )
% 0.35/0.68 => ( ! [I2: complex] :
% 0.35/0.68 ( ( member_complex @ I2 @ I5 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 0.35/0.68 => ( ( member_complex @ I @ I5 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.68 => ( ( F @ I )
% 0.35/0.68 = ( G3 @ I ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_mono_inv
% 0.35/0.68 thf(fact_8108_sum__cong__Suc,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > nat,G3: nat > nat] :
% 0.35/0.68 ( ~ ( member_nat @ zero_zero_nat @ A3 )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ ( suc @ X5 ) @ A3 )
% 0.35/0.68 => ( ( F @ ( suc @ X5 ) )
% 0.35/0.68 = ( G3 @ ( suc @ X5 ) ) ) )
% 0.35/0.68 => ( ( groups3542108847815614940at_nat @ F @ A3 )
% 0.35/0.68 = ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_cong_Suc
% 0.35/0.68 thf(fact_8109_sum__cong__Suc,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > real,G3: nat > real] :
% 0.35/0.68 ( ~ ( member_nat @ zero_zero_nat @ A3 )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ ( suc @ X5 ) @ A3 )
% 0.35/0.68 => ( ( F @ ( suc @ X5 ) )
% 0.35/0.68 = ( G3 @ ( suc @ X5 ) ) ) )
% 0.35/0.68 => ( ( groups6591440286371151544t_real @ F @ A3 )
% 0.35/0.68 = ( groups6591440286371151544t_real @ G3 @ A3 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_cong_Suc
% 0.35/0.68 thf(fact_8110_sums__single,axiom,
% 0.35/0.68 ! [I: nat,F: nat > real] :
% 0.35/0.68 ( sums_real
% 0.35/0.68 @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 0.35/0.68 @ ( F @ I ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_single
% 0.35/0.68 thf(fact_8111_sums__single,axiom,
% 0.35/0.68 ! [I: nat,F: nat > nat] :
% 0.35/0.68 ( sums_nat
% 0.35/0.68 @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.35/0.68 @ ( F @ I ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_single
% 0.35/0.68 thf(fact_8112_sums__single,axiom,
% 0.35/0.68 ! [I: nat,F: nat > int] :
% 0.35/0.68 ( sums_int
% 0.35/0.68 @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 0.35/0.68 @ ( F @ I ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_single
% 0.35/0.68 thf(fact_8113_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_o,G3: $o > real,P: $o > $o] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups8691415230153176458o_real
% 0.35/0.68 @ ^ [X: $o] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8114_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_real,G3: real > real,P: real > $o] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups8097168146408367636l_real
% 0.35/0.68 @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8115_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > real,P: int > $o] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( groups8778361861064173332t_real @ G3
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups8778361861064173332t_real
% 0.35/0.68 @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8116_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > real,P: complex > $o] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( groups5808333547571424918x_real @ G3
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups5808333547571424918x_real
% 0.35/0.68 @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8117_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,G3: extended_enat > real,P: extended_enat > $o] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( groups4148127829035722712t_real @ G3
% 0.35/0.68 @ ( collec4429806609662206161d_enat
% 0.35/0.68 @ ^ [X: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups4148127829035722712t_real
% 0.35/0.68 @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8118_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_o,G3: $o > rat,P: $o > $o] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat @ G3
% 0.35/0.68 @ ( collect_o
% 0.35/0.68 @ ^ [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups7872700643590313910_o_rat
% 0.35/0.68 @ ^ [X: $o] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8119_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_real,G3: real > rat,P: real > $o] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat @ G3
% 0.35/0.68 @ ( collect_real
% 0.35/0.68 @ ^ [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups1300246762558778688al_rat
% 0.35/0.68 @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8120_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_nat,G3: nat > rat,P: nat > $o] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3
% 0.35/0.68 @ ( collect_nat
% 0.35/0.68 @ ^ [X: nat] :
% 0.35/0.68 ( ( member_nat @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups2906978787729119204at_rat
% 0.35/0.68 @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8121_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_int,G3: int > rat,P: int > $o] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat @ G3
% 0.35/0.68 @ ( collect_int
% 0.35/0.68 @ ^ [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups3906332499630173760nt_rat
% 0.35/0.68 @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8122_sum_Ointer__filter,axiom,
% 0.35/0.68 ! [A3: set_complex,G3: complex > rat,P: complex > $o] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat @ G3
% 0.35/0.68 @ ( collect_complex
% 0.35/0.68 @ ^ [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ A3 )
% 0.35/0.68 & ( P @ X ) ) ) )
% 0.35/0.68 = ( groups5058264527183730370ex_rat
% 0.35/0.68 @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.68 @ A3 ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.inter_filter
% 0.35/0.68 thf(fact_8123_sums__If__finite__set_H,axiom,
% 0.35/0.68 ! [G3: nat > real,S: real,A3: set_nat,S5: real,F: nat > real] :
% 0.35/0.68 ( ( sums_real @ G3 @ S )
% 0.35/0.68 => ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( S5
% 0.35/0.68 = ( plus_plus_real @ S
% 0.35/0.68 @ ( groups6591440286371151544t_real
% 0.35/0.68 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 0.35/0.68 @ A3 ) ) )
% 0.35/0.68 => ( sums_real
% 0.35/0.68 @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A3 ) @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 0.35/0.68 @ S5 ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sums_If_finite_set'
% 0.35/0.68 thf(fact_8124_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_int,T: set_int,G3: int > real,I: int > int,F: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ S2 )
% 0.35/0.68 => ( ( finite_finite_int @ T )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: int] :
% 0.35/0.68 ( ( member_int @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8125_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_int,T: set_complex,G3: complex > real,I: complex > int,F: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ S2 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: complex] :
% 0.35/0.68 ( ( member_complex @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8126_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_int,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > int,F: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ S2 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ T )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8127_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_complex,T: set_int,G3: int > real,I: int > complex,F: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.68 => ( ( finite_finite_int @ T )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: int] :
% 0.35/0.68 ( ( member_int @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8128_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_complex,T: set_complex,G3: complex > real,I: complex > complex,F: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: complex] :
% 0.35/0.68 ( ( member_complex @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8129_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_complex,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > complex,F: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ T )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8130_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_Extended_enat,T: set_int,G3: int > real,I: int > extended_enat,F: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ S2 )
% 0.35/0.68 => ( ( finite_finite_int @ T )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: int] :
% 0.35/0.68 ( ( member_int @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8131_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_Extended_enat,T: set_complex,G3: complex > real,I: complex > extended_enat,F: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ S2 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: complex] :
% 0.35/0.68 ( ( member_complex @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8132_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_Extended_enat,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > extended_enat,F: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ S2 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ T )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8133_sum__le__included,axiom,
% 0.35/0.68 ! [S2: set_nat,T: set_nat,G3: nat > rat,I: nat > nat,F: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ S2 )
% 0.35/0.68 => ( ( finite_finite_nat @ T )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ T )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ S2 )
% 0.35/0.68 => ? [Xa: nat] :
% 0.35/0.68 ( ( member_nat @ Xa @ T )
% 0.35/0.68 & ( ( I @ Xa )
% 0.35/0.68 = X5 )
% 0.35/0.68 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 0.35/0.68 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G3 @ T ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_le_included
% 0.35/0.68 thf(fact_8134_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups8097168146408367636l_real @ F @ A3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 = ( ! [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8135_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups8691415230153176458o_real @ F @ A3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 = ( ! [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8136_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups8778361861064173332t_real @ F @ A3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 = ( ! [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8137_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups5808333547571424918x_real @ F @ A3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 = ( ! [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8138_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups4148127829035722712t_real @ F @ A3 )
% 0.35/0.68 = zero_zero_real )
% 0.35/0.68 = ( ! [X: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_real ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8139_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups1300246762558778688al_rat @ F @ A3 )
% 0.35/0.68 = zero_zero_rat )
% 0.35/0.68 = ( ! [X: real] :
% 0.35/0.68 ( ( member_real @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8140_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups7872700643590313910_o_rat @ F @ A3 )
% 0.35/0.68 = zero_zero_rat )
% 0.35/0.68 = ( ! [X: $o] :
% 0.35/0.68 ( ( member_o @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8141_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups2906978787729119204at_rat @ F @ A3 )
% 0.35/0.68 = zero_zero_rat )
% 0.35/0.68 = ( ! [X: nat] :
% 0.35/0.68 ( ( member_nat @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8142_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups3906332499630173760nt_rat @ F @ A3 )
% 0.35/0.68 = zero_zero_rat )
% 0.35/0.68 = ( ! [X: int] :
% 0.35/0.68 ( ( member_int @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8143_sum__nonneg__eq__0__iff,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.68 => ( ( ( groups5058264527183730370ex_rat @ F @ A3 )
% 0.35/0.68 = zero_zero_rat )
% 0.35/0.68 = ( ! [X: complex] :
% 0.35/0.68 ( ( member_complex @ X @ A3 )
% 0.35/0.68 => ( ( F @ X )
% 0.35/0.68 = zero_zero_rat ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_nonneg_eq_0_iff
% 0.35/0.68 thf(fact_8144_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > real,G3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: int] :
% 0.35/0.68 ( ( member_int @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8145_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > real,G3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: complex] :
% 0.35/0.68 ( ( member_complex @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8146_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8147_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: nat] :
% 0.35/0.68 ( ( member_nat @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8148_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > rat,G3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: int] :
% 0.35/0.68 ( ( member_int @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8149_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: complex] :
% 0.35/0.68 ( ( member_complex @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8150_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8151_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > nat,G3: int > nat] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: int] :
% 0.35/0.68 ( ( member_int @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8152_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > nat,G3: complex > nat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: complex] :
% 0.35/0.68 ( ( member_complex @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8153_sum__strict__mono__ex1,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > nat,G3: extended_enat > nat] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ? [X4: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X4 @ A3 )
% 0.35/0.68 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 0.35/0.68 => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono_ex1
% 0.35/0.68 thf(fact_8154_sum_Orelated,axiom,
% 0.35/0.68 ! [R: real > real > $o,S: set_int,H2: int > real,G3: int > real] :
% 0.35/0.68 ( ( R @ zero_zero_real @ zero_zero_real )
% 0.35/0.68 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups8778361861064173332t_real @ H2 @ S ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8155_sum_Orelated,axiom,
% 0.35/0.68 ! [R: real > real > $o,S: set_complex,H2: complex > real,G3: complex > real] :
% 0.35/0.68 ( ( R @ zero_zero_real @ zero_zero_real )
% 0.35/0.68 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8156_sum_Orelated,axiom,
% 0.35/0.68 ! [R: real > real > $o,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.68 ( ( R @ zero_zero_real @ zero_zero_real )
% 0.35/0.68 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups4148127829035722712t_real @ H2 @ S ) @ ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8157_sum_Orelated,axiom,
% 0.35/0.68 ! [R: rat > rat > $o,S: set_nat,H2: nat > rat,G3: nat > rat] :
% 0.35/0.68 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.68 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite_finite_nat @ S )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8158_sum_Orelated,axiom,
% 0.35/0.68 ! [R: rat > rat > $o,S: set_int,H2: int > rat,G3: int > rat] :
% 0.35/0.68 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.68 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S ) @ ( groups3906332499630173760nt_rat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8159_sum_Orelated,axiom,
% 0.35/0.68 ! [R: rat > rat > $o,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 0.35/0.68 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.68 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8160_sum_Orelated,axiom,
% 0.35/0.68 ! [R: rat > rat > $o,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.68 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.35/0.68 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups1392844769737527556at_rat @ H2 @ S ) @ ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8161_sum_Orelated,axiom,
% 0.35/0.68 ! [R: nat > nat > $o,S: set_int,H2: int > nat,G3: int > nat] :
% 0.35/0.68 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.35/0.68 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite_finite_int @ S )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S ) @ ( groups4541462559716669496nt_nat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8162_sum_Orelated,axiom,
% 0.35/0.68 ! [R: nat > nat > $o,S: set_complex,H2: complex > nat,G3: complex > nat] :
% 0.35/0.68 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.35/0.68 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ S )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8163_sum_Orelated,axiom,
% 0.35/0.68 ! [R: nat > nat > $o,S: set_Extended_enat,H2: extended_enat > nat,G3: extended_enat > nat] :
% 0.35/0.68 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.35/0.68 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 0.35/0.68 ( ( ( R @ X1 @ X23 )
% 0.35/0.68 & ( R @ Y1 @ Y22 ) )
% 0.35/0.68 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.68 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( R @ ( groups2027974829824023292at_nat @ H2 @ S ) @ ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.related
% 0.35/0.68 thf(fact_8164_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > real,G3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_complex )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8165_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8166_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > real,G3: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_real )
% 0.35/0.68 => ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8167_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > real,G3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_o )
% 0.35/0.68 => ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8168_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_int,F: int > real,G3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_int )
% 0.35/0.68 => ( ! [X5: int] :
% 0.35/0.68 ( ( member_int @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8169_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_complex )
% 0.35/0.68 => ( ! [X5: complex] :
% 0.35/0.68 ( ( member_complex @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8170_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.68 => ( ! [X5: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8171_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_real,F: real > rat,G3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_real )
% 0.35/0.68 => ( ! [X5: real] :
% 0.35/0.68 ( ( member_real @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8172_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_o,F: $o > rat,G3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_o )
% 0.35/0.68 => ( ! [X5: $o] :
% 0.35/0.68 ( ( member_o @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8173_sum__strict__mono,axiom,
% 0.35/0.68 ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( A3 != bot_bot_set_nat )
% 0.35/0.68 => ( ! [X5: nat] :
% 0.35/0.68 ( ( member_nat @ X5 @ A3 )
% 0.35/0.68 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 0.35/0.68 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum_strict_mono
% 0.35/0.68 thf(fact_8174_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_real,X2: real,G3: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( groups8097168146408367636l_real @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8175_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_o,X2: $o,G3: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( groups8691415230153176458o_real @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8176_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_int,X2: int,G3: int > real] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( groups8778361861064173332t_real @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8177_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( groups5808333547571424918x_real @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8178_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 0.35/0.68 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.68 => ( ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.68 = ( groups4148127829035722712t_real @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8179_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_real,X2: real,G3: real > rat] :
% 0.35/0.68 ( ( finite_finite_real @ A3 )
% 0.35/0.68 => ( ( ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_real @ X2 @ A3 )
% 0.35/0.68 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8180_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_o,X2: $o,G3: $o > rat] :
% 0.35/0.68 ( ( finite_finite_o @ A3 )
% 0.35/0.68 => ( ( ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_o @ X2 @ A3 )
% 0.35/0.68 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8181_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_nat,X2: nat,G3: nat > rat] :
% 0.35/0.68 ( ( finite_finite_nat @ A3 )
% 0.35/0.68 => ( ( ( member_nat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 0.35/0.68 = ( groups2906978787729119204at_rat @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_nat @ X2 @ A3 )
% 0.35/0.68 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8182_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_int,X2: int,G3: int > rat] :
% 0.35/0.68 ( ( finite_finite_int @ A3 )
% 0.35/0.68 => ( ( ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_int @ X2 @ A3 )
% 0.35/0.68 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8183_sum_Oinsert__if,axiom,
% 0.35/0.68 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 0.35/0.68 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.68 => ( ( ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) )
% 0.35/0.68 & ( ~ ( member_complex @ X2 @ A3 )
% 0.35/0.68 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.68 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.insert_if
% 0.35/0.68 thf(fact_8184_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_real,T5: set_real,S: set_real,I: real > real,J: real > real,T2: set_real,G3: real > real,H2: real > real] :
% 0.35/0.68 ( ( finite_finite_real @ S5 )
% 0.35/0.68 => ( ( finite_finite_real @ T5 )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 0.35/0.68 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.68 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8185_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_real,T5: set_o,S: set_real,I: $o > real,J: real > $o,T2: set_o,G3: real > real,H2: $o > real] :
% 0.35/0.68 ( ( finite_finite_real @ S5 )
% 0.35/0.68 => ( ( finite_finite_o @ T5 )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.35/0.68 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.68 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8186_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_o,T5: set_real,S: set_o,I: real > $o,J: $o > real,T2: set_real,G3: $o > real,H2: real > real] :
% 0.35/0.68 ( ( finite_finite_o @ S5 )
% 0.35/0.68 => ( ( finite_finite_real @ T5 )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 0.35/0.68 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: real] :
% 0.35/0.68 ( ( member_real @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.68 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8187_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_o,T5: set_o,S: set_o,I: $o > $o,J: $o > $o,T2: set_o,G3: $o > real,H2: $o > real] :
% 0.35/0.68 ( ( finite_finite_o @ S5 )
% 0.35/0.68 => ( ( finite_finite_o @ T5 )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 0.35/0.68 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: $o] :
% 0.35/0.68 ( ( member_o @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.68 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8188_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_real,T5: set_int,S: set_real,I: int > real,J: real > int,T2: set_int,G3: real > real,H2: int > real] :
% 0.35/0.68 ( ( finite_finite_real @ S5 )
% 0.35/0.68 => ( ( finite_finite_int @ T5 )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.35/0.68 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.68 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8189_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_o,T5: set_int,S: set_o,I: int > $o,J: $o > int,T2: set_int,G3: $o > real,H2: int > real] :
% 0.35/0.68 ( ( finite_finite_o @ S5 )
% 0.35/0.68 => ( ( finite_finite_int @ T5 )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 0.35/0.68 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: int] :
% 0.35/0.68 ( ( member_int @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.68 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8190_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_real,T5: set_complex,S: set_real,I: complex > real,J: real > complex,T2: set_complex,G3: real > real,H2: complex > real] :
% 0.35/0.68 ( ( finite_finite_real @ S5 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ T5 )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 0.35/0.68 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.68 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8191_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_o,T5: set_complex,S: set_o,I: complex > $o,J: $o > complex,T2: set_complex,G3: $o > real,H2: complex > real] :
% 0.35/0.68 ( ( finite_finite_o @ S5 )
% 0.35/0.68 => ( ( finite3207457112153483333omplex @ T5 )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 0.35/0.68 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: complex] :
% 0.35/0.68 ( ( member_complex @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.68 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8192_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_real,T5: set_Extended_enat,S: set_real,I: extended_enat > real,J: real > extended_enat,T2: set_Extended_enat,G3: real > real,H2: extended_enat > real] :
% 0.35/0.68 ( ( finite_finite_real @ S5 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ T5 )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 0.35/0.68 => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 0.35/0.68 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: real] :
% 0.35/0.68 ( ( member_real @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.68 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.68
% 0.35/0.68 % sum.reindex_bij_witness_not_neutral
% 0.35/0.68 thf(fact_8193_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.35/0.68 ! [S5: set_o,T5: set_Extended_enat,S: set_o,I: extended_enat > $o,J: $o > extended_enat,T2: set_Extended_enat,G3: $o > real,H2: extended_enat > real] :
% 0.35/0.68 ( ( finite_finite_o @ S5 )
% 0.35/0.68 => ( ( finite4001608067531595151d_enat @ T5 )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( ( I @ ( J @ A ) )
% 0.35/0.68 = A ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 0.35/0.68 => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 0.35/0.68 => ( ( J @ ( I @ B ) )
% 0.35/0.68 = B ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 0.35/0.68 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S5 )
% 0.35/0.68 => ( ( G3 @ A )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [B: extended_enat] :
% 0.35/0.68 ( ( member_Extended_enat @ B @ T5 )
% 0.35/0.68 => ( ( H2 @ B )
% 0.35/0.68 = zero_zero_real ) )
% 0.35/0.68 => ( ! [A: $o] :
% 0.35/0.68 ( ( member_o @ A @ S )
% 0.35/0.68 => ( ( H2 @ ( J @ A ) )
% 0.35/0.68 = ( G3 @ A ) ) )
% 0.35/0.68 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.68 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.reindex_bij_witness_not_neutral
% 0.35/0.69 thf(fact_8194_sums__mult2__iff,axiom,
% 0.35/0.69 ! [C: complex,F: nat > complex,D: complex] :
% 0.35/0.69 ( ( C != zero_zero_complex )
% 0.35/0.69 => ( ( sums_complex
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 0.35/0.69 @ ( times_times_complex @ D @ C ) )
% 0.35/0.69 = ( sums_complex @ F @ D ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult2_iff
% 0.35/0.69 thf(fact_8195_sums__mult2__iff,axiom,
% 0.35/0.69 ! [C: real,F: nat > real,D: real] :
% 0.35/0.69 ( ( C != zero_zero_real )
% 0.35/0.69 => ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 0.35/0.69 @ ( times_times_real @ D @ C ) )
% 0.35/0.69 = ( sums_real @ F @ D ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult2_iff
% 0.35/0.69 thf(fact_8196_sums__mult__iff,axiom,
% 0.35/0.69 ! [C: complex,F: nat > complex,D: complex] :
% 0.35/0.69 ( ( C != zero_zero_complex )
% 0.35/0.69 => ( ( sums_complex
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.35/0.69 @ ( times_times_complex @ C @ D ) )
% 0.35/0.69 = ( sums_complex @ F @ D ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult_iff
% 0.35/0.69 thf(fact_8197_sums__mult__iff,axiom,
% 0.35/0.69 ! [C: real,F: nat > real,D: real] :
% 0.35/0.69 ( ( C != zero_zero_real )
% 0.35/0.69 => ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.35/0.69 @ ( times_times_real @ C @ D ) )
% 0.35/0.69 = ( sums_real @ F @ D ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult_iff
% 0.35/0.69 thf(fact_8198_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_real,F: real > real,B2: real,I: real] :
% 0.35/0.69 ( ( finite_finite_real @ S2 )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8097168146408367636l_real @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_real @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8199_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_o,F: $o > real,B2: real,I: $o] :
% 0.35/0.69 ( ( finite_finite_o @ S2 )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8691415230153176458o_real @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_o @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8200_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_int,F: int > real,B2: real,I: int] :
% 0.35/0.69 ( ( finite_finite_int @ S2 )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8778361861064173332t_real @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_int @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8201_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_complex,F: complex > real,B2: real,I: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups5808333547571424918x_real @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_complex @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8202_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_Extended_enat,F: extended_enat > real,B2: real,I: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ S2 )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups4148127829035722712t_real @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_Extended_enat @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8203_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_real,F: real > rat,B2: rat,I: real] :
% 0.35/0.69 ( ( finite_finite_real @ S2 )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_real @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8204_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_o,F: $o > rat,B2: rat,I: $o] :
% 0.35/0.69 ( ( finite_finite_o @ S2 )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_o @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8205_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_nat,F: nat > rat,B2: rat,I: nat] :
% 0.35/0.69 ( ( finite_finite_nat @ S2 )
% 0.35/0.69 => ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_nat @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8206_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_int,F: int > rat,B2: rat,I: int] :
% 0.35/0.69 ( ( finite_finite_int @ S2 )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_int @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8207_sum__nonneg__leq__bound,axiom,
% 0.35/0.69 ! [S2: set_complex,F: complex > rat,B2: rat,I: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
% 0.35/0.69 = B2 )
% 0.35/0.69 => ( ( member_complex @ I @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_leq_bound
% 0.35/0.69 thf(fact_8208_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_real,F: real > real,I: real] :
% 0.35/0.69 ( ( finite_finite_real @ S2 )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8097168146408367636l_real @ F @ S2 )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( member_real @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8209_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_o,F: $o > real,I: $o] :
% 0.35/0.69 ( ( finite_finite_o @ S2 )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8691415230153176458o_real @ F @ S2 )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( member_o @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8210_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_int,F: int > real,I: int] :
% 0.35/0.69 ( ( finite_finite_int @ S2 )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups8778361861064173332t_real @ F @ S2 )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( member_int @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8211_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_complex,F: complex > real,I: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups5808333547571424918x_real @ F @ S2 )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( member_complex @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8212_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ S2 )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups4148127829035722712t_real @ F @ S2 )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( member_Extended_enat @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8213_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_real,F: real > rat,I: real] :
% 0.35/0.69 ( ( finite_finite_real @ S2 )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
% 0.35/0.69 = zero_zero_rat )
% 0.35/0.69 => ( ( member_real @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8214_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_o,F: $o > rat,I: $o] :
% 0.35/0.69 ( ( finite_finite_o @ S2 )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
% 0.35/0.69 = zero_zero_rat )
% 0.35/0.69 => ( ( member_o @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8215_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_nat,F: nat > rat,I: nat] :
% 0.35/0.69 ( ( finite_finite_nat @ S2 )
% 0.35/0.69 => ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
% 0.35/0.69 = zero_zero_rat )
% 0.35/0.69 => ( ( member_nat @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8216_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_int,F: int > rat,I: int] :
% 0.35/0.69 ( ( finite_finite_int @ S2 )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
% 0.35/0.69 = zero_zero_rat )
% 0.35/0.69 => ( ( member_int @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8217_sum__nonneg__0,axiom,
% 0.35/0.69 ! [S2: set_complex,F: complex > rat,I: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ S2 )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ S2 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
% 0.35/0.69 = zero_zero_rat )
% 0.35/0.69 => ( ( member_complex @ I @ S2 )
% 0.35/0.69 => ( ( F @ I )
% 0.35/0.69 = zero_zero_rat ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_nonneg_0
% 0.35/0.69 thf(fact_8218_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > real,B2: set_real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.69 = ( groups8097168146408367636l_real
% 0.35/0.69 @ ^ [X: real] : ( if_real @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8219_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_o,G3: $o > real,B2: set_o] :
% 0.35/0.69 ( ( finite_finite_o @ A3 )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.69 = ( groups8691415230153176458o_real
% 0.35/0.69 @ ^ [X: $o] : ( if_real @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8220_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > real,B2: set_int] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( groups8778361861064173332t_real
% 0.35/0.69 @ ^ [X: int] : ( if_real @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8221_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > real,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( groups5808333547571424918x_real
% 0.35/0.69 @ ^ [X: complex] : ( if_real @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8222_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( groups4148127829035722712t_real
% 0.35/0.69 @ ^ [X: extended_enat] : ( if_real @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8223_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > rat,B2: set_real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
% 0.35/0.69 = ( groups1300246762558778688al_rat
% 0.35/0.69 @ ^ [X: real] : ( if_rat @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8224_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_o,G3: $o > rat,B2: set_o] :
% 0.35/0.69 ( ( finite_finite_o @ A3 )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
% 0.35/0.69 = ( groups7872700643590313910_o_rat
% 0.35/0.69 @ ^ [X: $o] : ( if_rat @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8225_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > rat,B2: set_int] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( groups3906332499630173760nt_rat
% 0.35/0.69 @ ^ [X: int] : ( if_rat @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8226_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( groups5058264527183730370ex_rat
% 0.35/0.69 @ ^ [X: complex] : ( if_rat @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8227_sum_Ointer__restrict,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( groups1392844769737527556at_rat
% 0.35/0.69 @ ^ [X: extended_enat] : ( if_rat @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 0.35/0.69 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.inter_restrict
% 0.35/0.69 thf(fact_8228_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3
% 0.35/0.69 @ ( minus_minus_set_real @ A3
% 0.35/0.69 @ ( collect_real
% 0.35/0.69 @ ^ [X: real] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_real ) ) ) )
% 0.35/0.69 = ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8229_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3
% 0.35/0.69 @ ( minus_minus_set_int @ A3
% 0.35/0.69 @ ( collect_int
% 0.35/0.69 @ ^ [X: int] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_real ) ) ) )
% 0.35/0.69 = ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8230_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3
% 0.35/0.69 @ ( minus_811609699411566653omplex @ A3
% 0.35/0.69 @ ( collect_complex
% 0.35/0.69 @ ^ [X: complex] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_real ) ) ) )
% 0.35/0.69 = ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8231_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3
% 0.35/0.69 @ ( minus_925952699566721837d_enat @ A3
% 0.35/0.69 @ ( collec4429806609662206161d_enat
% 0.35/0.69 @ ^ [X: extended_enat] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_real ) ) ) )
% 0.35/0.69 = ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8232_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3
% 0.35/0.69 @ ( minus_minus_set_real @ A3
% 0.35/0.69 @ ( collect_real
% 0.35/0.69 @ ^ [X: real] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_rat ) ) ) )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8233_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3
% 0.35/0.69 @ ( minus_minus_set_int @ A3
% 0.35/0.69 @ ( collect_int
% 0.35/0.69 @ ^ [X: int] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_rat ) ) ) )
% 0.35/0.69 = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8234_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3
% 0.35/0.69 @ ( minus_811609699411566653omplex @ A3
% 0.35/0.69 @ ( collect_complex
% 0.35/0.69 @ ^ [X: complex] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_rat ) ) ) )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8235_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3
% 0.35/0.69 @ ( minus_925952699566721837d_enat @ A3
% 0.35/0.69 @ ( collec4429806609662206161d_enat
% 0.35/0.69 @ ^ [X: extended_enat] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_rat ) ) ) )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8236_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1935376822645274424al_nat @ G3
% 0.35/0.69 @ ( minus_minus_set_real @ A3
% 0.35/0.69 @ ( collect_real
% 0.35/0.69 @ ^ [X: real] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_nat ) ) ) )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8237_sum_Osetdiff__irrelevant,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > nat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups4541462559716669496nt_nat @ G3
% 0.35/0.69 @ ( minus_minus_set_int @ A3
% 0.35/0.69 @ ( collect_int
% 0.35/0.69 @ ^ [X: int] :
% 0.35/0.69 ( ( G3 @ X )
% 0.35/0.69 = zero_zero_nat ) ) ) )
% 0.35/0.69 = ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.setdiff_irrelevant
% 0.35/0.69 thf(fact_8238_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.35/0.69 ! [G3: nat > nat,M: nat,N: nat] :
% 0.35/0.69 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.35/0.69 = ( groups3542108847815614940at_nat
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.shift_bounds_cl_Suc_ivl
% 0.35/0.69 thf(fact_8239_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.35/0.69 ! [G3: nat > real,M: nat,N: nat] :
% 0.35/0.69 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.35/0.69 = ( groups6591440286371151544t_real
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.shift_bounds_cl_Suc_ivl
% 0.35/0.69 thf(fact_8240_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.35/0.69 ! [G3: nat > nat,M: nat,K: nat,N: nat] :
% 0.35/0.69 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.35/0.69 = ( groups3542108847815614940at_nat
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.shift_bounds_cl_nat_ivl
% 0.35/0.69 thf(fact_8241_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.35/0.69 ! [G3: nat > real,M: nat,K: nat,N: nat] :
% 0.35/0.69 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 0.35/0.69 = ( groups6591440286371151544t_real
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.shift_bounds_cl_nat_ivl
% 0.35/0.69 thf(fact_8242_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_real,I: real,F: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ I5 )
% 0.35/0.69 => ( ( member_real @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8243_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_o,I: $o,F: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ I5 )
% 0.35/0.69 => ( ( member_o @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8244_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_int,I: int,F: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ I5 )
% 0.35/0.69 => ( ( member_int @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8245_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_complex,I: complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.69 => ( ( member_complex @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8246_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ I5 )
% 0.35/0.69 => ( ( member_Extended_enat @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8247_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_real,I: real,F: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ I5 )
% 0.35/0.69 => ( ( member_real @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8248_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_o,I: $o,F: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ I5 )
% 0.35/0.69 => ( ( member_o @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8249_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_nat,I: nat,F: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ I5 )
% 0.35/0.69 => ( ( member_nat @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8250_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_int,I: int,F: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ I5 )
% 0.35/0.69 => ( ( member_int @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8251_sum__pos2,axiom,
% 0.35/0.69 ! [I5: set_complex,I: complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.69 => ( ( member_complex @ I @ I5 )
% 0.35/0.69 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos2
% 0.35/0.69 thf(fact_8252_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_complex )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8253_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8254_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_real,F: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_real )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8255_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_o,F: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_o )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8256_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_int,F: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_int )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8257_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_complex )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8258_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8259_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_real,F: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_real )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8260_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_o,F: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_o )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8261_sum__pos,axiom,
% 0.35/0.69 ! [I5: set_nat,F: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ I5 )
% 0.35/0.69 => ( ( I5 != bot_bot_set_nat )
% 0.35/0.69 => ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ I5 )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_pos
% 0.35/0.69 thf(fact_8262_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_real,K4: real,F: real > real] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8263_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_o,K4: real,F: $o > real] :
% 0.35/0.69 ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8264_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_complex,K4: real,F: complex > real] :
% 0.35/0.69 ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8265_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_int,K4: real,F: int > real] :
% 0.35/0.69 ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8266_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_real,K4: rat,F: real > rat] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8267_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_o,K4: rat,F: $o > rat] :
% 0.35/0.69 ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8268_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_complex,K4: rat,F: complex > rat] :
% 0.35/0.69 ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8269_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_nat,K4: rat,F: nat > rat] :
% 0.35/0.69 ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8270_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_int,K4: rat,F: int > rat] :
% 0.35/0.69 ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8271_sum__bounded__below,axiom,
% 0.35/0.69 ! [A3: set_real,K4: nat,F: real > nat] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_nat @ K4 @ ( F @ I2 ) ) )
% 0.35/0.69 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_below
% 0.35/0.69 thf(fact_8272_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_real,F: real > real,K4: real] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8273_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_o,F: $o > real,K4: real] :
% 0.35/0.69 ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8274_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_complex,F: complex > real,K4: real] :
% 0.35/0.69 ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8275_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_int,F: int > real,K4: real] :
% 0.35/0.69 ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8276_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_real,F: real > rat,K4: rat] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8277_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_o,F: $o > rat,K4: rat] :
% 0.35/0.69 ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8278_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 0.35/0.69 ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8279_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 0.35/0.69 ( ! [I2: nat] :
% 0.35/0.69 ( ( member_nat @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8280_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_int,F: int > rat,K4: rat] :
% 0.35/0.69 ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8281_sum__bounded__above,axiom,
% 0.35/0.69 ! [A3: set_real,F: real > nat,K4: nat] :
% 0.35/0.69 ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ A3 )
% 0.35/0.69 => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.69 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_bounded_above
% 0.35/0.69 thf(fact_8282_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups8097168146408367636l_real @ G3 @ C2 )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8283_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups8691415230153176458o_real @ G3 @ C2 )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8284_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: complex] :
% 0.35/0.69 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups5808333547571424918x_real @ G3 @ C2 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8285_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups4148127829035722712t_real @ G3 @ C2 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8286_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups1300246762558778688al_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8287_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8288_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: complex] :
% 0.35/0.69 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8289_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups1392844769737527556at_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8290_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups1935376822645274424al_nat @ G3 @ C2 )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8291_sum_Osame__carrier,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ B2 ) )
% 0.35/0.69 = ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrier
% 0.35/0.69 thf(fact_8292_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups8097168146408367636l_real @ G3 @ C2 )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8293_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups8691415230153176458o_real @ G3 @ C2 )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8294_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: complex] :
% 0.35/0.69 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups5808333547571424918x_real @ G3 @ C2 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8295_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( ( groups4148127829035722712t_real @ G3 @ C2 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8296_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups1300246762558778688al_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8297_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8298_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: complex] :
% 0.35/0.69 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8299_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( ( groups1392844769737527556at_rat @ G3 @ C2 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8300_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: real] :
% 0.35/0.69 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( ( groups1935376822645274424al_nat @ G3 @ C2 )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8301_sum_Osame__carrierI,axiom,
% 0.35/0.69 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
% 0.35/0.69 ( ( finite_finite_o @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 0.35/0.69 => ( ! [A: $o] :
% 0.35/0.69 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 0.35/0.69 => ( ( G3 @ A )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 0.35/0.69 => ( ( H2 @ B )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ C2 ) )
% 0.35/0.69 => ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.same_carrierI
% 0.35/0.69 thf(fact_8302_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 0.35/0.69 = ( groups5808333547571424918x_real @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8303_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 0.35/0.69 = ( groups4148127829035722712t_real @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8304_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8305_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8306_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ S )
% 0.35/0.69 = ( groups5693394587270226106ex_nat @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8307_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ S )
% 0.35/0.69 = ( groups2027974829824023292at_nat @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8308_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ S )
% 0.35/0.69 = ( groups5690904116761175830ex_int @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8309_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ S )
% 0.35/0.69 = ( groups2025484359314973016at_int @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8310_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_nat,S: set_nat,G3: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: nat] :
% 0.35/0.69 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups2906978787729119204at_rat @ G3 @ S )
% 0.35/0.69 = ( groups2906978787729119204at_rat @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8311_sum_Omono__neutral__left,axiom,
% 0.35/0.69 ! [T2: set_nat,S: set_nat,G3: nat > int] :
% 0.35/0.69 ( ( finite_finite_nat @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: nat] :
% 0.35/0.69 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups3539618377306564664at_int @ G3 @ S )
% 0.35/0.69 = ( groups3539618377306564664at_int @ G3 @ T2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_left
% 0.35/0.69 thf(fact_8312_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ T2 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8313_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ T2 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8314_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8315_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8316_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ T2 )
% 0.35/0.69 = ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8317_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ T2 )
% 0.35/0.69 = ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8318_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ T2 )
% 0.35/0.69 = ( groups5690904116761175830ex_int @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8319_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ T2 )
% 0.35/0.69 = ( groups2025484359314973016at_int @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8320_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_nat,S: set_nat,G3: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: nat] :
% 0.35/0.69 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups2906978787729119204at_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8321_sum_Omono__neutral__right,axiom,
% 0.35/0.69 ! [T2: set_nat,S: set_nat,G3: nat > int] :
% 0.35/0.69 ( ( finite_finite_nat @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: nat] :
% 0.35/0.69 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups3539618377306564664at_int @ G3 @ T2 )
% 0.35/0.69 = ( groups3539618377306564664at_int @ G3 @ S ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_right
% 0.35/0.69 thf(fact_8322_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8323_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8324_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8325_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8326_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ S )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8327_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ G3 @ S )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8328_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8329_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8330_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,H2: real > nat,G3: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1935376822645274424al_nat @ G3 @ S )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8331_sum_Omono__neutral__cong__left,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,H2: $o > nat,G3: $o > nat] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8507830703676809646_o_nat @ G3 @ S )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_left
% 0.35/0.69 thf(fact_8332_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,G3: real > real,H2: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ T2 )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8333_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,G3: $o > real,H2: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ G3 @ T2 )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8334_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > real,H2: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ T2 )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8335_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ T2 )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8336_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,G3: real > rat,H2: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8337_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,G3: $o > rat,H2: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8338_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,G3: complex > rat,H2: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8339_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8340_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,G3: real > nat,H2: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ S @ T2 )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1935376822645274424al_nat @ G3 @ T2 )
% 0.35/0.69 = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8341_sum_Omono__neutral__cong__right,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,G3: $o > nat,H2: $o > nat] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ S @ T2 )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ S )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8507830703676809646_o_nat @ G3 @ T2 )
% 0.35/0.69 = ( groups8507830703676809646_o_nat @ H2 @ S ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong_right
% 0.35/0.69 thf(fact_8342_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8343_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8344_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8345_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8346_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8347_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8348_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8349_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > int] :
% 0.35/0.69 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8350_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_nat,A3: set_nat,G3: nat > rat] :
% 0.35/0.69 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8351_sum_Osubset__diff,axiom,
% 0.35/0.69 ! [B2: set_nat,A3: set_nat,G3: nat > int] :
% 0.35/0.69 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.69 => ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( groups3539618377306564664at_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups3539618377306564664at_int @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.subset_diff
% 0.35/0.69 thf(fact_8352_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8353_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8354_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8355_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8356_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8357_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8358_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_nat,B2: set_nat,F: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.69 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8359_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_nat,B2: set_nat,F: nat > int] :
% 0.35/0.69 ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.69 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8360_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,F: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8361_sum__diff,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,F: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff
% 0.35/0.69 thf(fact_8362_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( finite_finite_real @ S )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 0.35/0.69 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8363_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( finite_finite_o @ S )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 0.35/0.69 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8364_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_int,S: set_int,H2: int > real,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ T2 )
% 0.35/0.69 => ( ( finite_finite_int @ S )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: int] :
% 0.35/0.69 ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3 @ S )
% 0.35/0.69 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8365_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ S )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 0.35/0.69 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8366_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 0.35/0.69 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8367_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ T2 )
% 0.35/0.69 => ( ( finite_finite_real @ S )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [I2: real] :
% 0.35/0.69 ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: real] :
% 0.35/0.69 ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ S )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8368_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ T2 )
% 0.35/0.69 => ( ( finite_finite_o @ S )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [I2: $o] :
% 0.35/0.69 ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: $o] :
% 0.35/0.69 ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ G3 @ S )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8369_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_int,S: set_int,H2: int > rat,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ T2 )
% 0.35/0.69 => ( ( finite_finite_int @ S )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [I2: int] :
% 0.35/0.69 ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: int] :
% 0.35/0.69 ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3 @ S )
% 0.35/0.69 = ( groups3906332499630173760nt_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8370_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ S )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [I2: complex] :
% 0.35/0.69 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8371_sum_Omono__neutral__cong,axiom,
% 0.35/0.69 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ T2 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 0.35/0.69 => ( ( H2 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [I2: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ I2 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = ( H2 @ X5 ) ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.mono_neutral_cong
% 0.35/0.69 thf(fact_8372_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8373_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8374_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8375_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8376_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8377_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8378_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8379_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8380_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8381_sum_Ounion__inter,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 0.35/0.69 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter
% 0.35/0.69 thf(fact_8382_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > real,B2: set_int] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8383_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > real,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8384_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8385_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > rat,B2: set_int] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8386_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8387_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8388_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_int,G3: int > nat,B2: set_int] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8389_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > nat,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8390_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > nat,B2: set_Extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8391_sum_OInt__Diff,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > int,B2: set_complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.Int_Diff
% 0.35/0.69 thf(fact_8392_sums__mult__D,axiom,
% 0.35/0.69 ! [C: real,F: nat > real,A2: real] :
% 0.35/0.69 ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 0.35/0.69 @ A2 )
% 0.35/0.69 => ( ( C != zero_zero_real )
% 0.35/0.69 => ( sums_real @ F @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult_D
% 0.35/0.69 thf(fact_8393_sums__mult__D,axiom,
% 0.35/0.69 ! [C: complex,F: nat > complex,A2: complex] :
% 0.35/0.69 ( ( sums_complex
% 0.35/0.69 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 0.35/0.69 @ A2 )
% 0.35/0.69 => ( ( C != zero_zero_complex )
% 0.35/0.69 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A2 @ C ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_mult_D
% 0.35/0.69 thf(fact_8394_sums__Suc__imp,axiom,
% 0.35/0.69 ! [F: nat > real,S2: real] :
% 0.35/0.69 ( ( ( F @ zero_zero_nat )
% 0.35/0.69 = zero_zero_real )
% 0.35/0.69 => ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69 @ S2 )
% 0.35/0.69 => ( sums_real @ F @ S2 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_Suc_imp
% 0.35/0.69 thf(fact_8395_sums__Suc,axiom,
% 0.35/0.69 ! [F: nat > real,L: real] :
% 0.35/0.69 ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69 @ L )
% 0.35/0.69 => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_Suc
% 0.35/0.69 thf(fact_8396_sums__Suc,axiom,
% 0.35/0.69 ! [F: nat > nat,L: nat] :
% 0.35/0.69 ( ( sums_nat
% 0.35/0.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69 @ L )
% 0.35/0.69 => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_Suc
% 0.35/0.69 thf(fact_8397_sums__Suc,axiom,
% 0.35/0.69 ! [F: nat > int,L: int] :
% 0.35/0.69 ( ( sums_int
% 0.35/0.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69 @ L )
% 0.35/0.69 => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_Suc
% 0.35/0.69 thf(fact_8398_sums__Suc__iff,axiom,
% 0.35/0.69 ! [F: nat > real,S2: real] :
% 0.35/0.69 ( ( sums_real
% 0.35/0.69 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 0.35/0.69 @ S2 )
% 0.35/0.69 = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_Suc_iff
% 0.35/0.69 thf(fact_8399_sums__zero__iff__shift,axiom,
% 0.35/0.69 ! [N: nat,F: nat > real,S2: real] :
% 0.35/0.69 ( ! [I2: nat] :
% 0.35/0.69 ( ( ord_less_nat @ I2 @ N )
% 0.35/0.69 => ( ( F @ I2 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( sums_real
% 0.35/0.69 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 0.35/0.69 @ S2 )
% 0.35/0.69 = ( sums_real @ F @ S2 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sums_zero_iff_shift
% 0.35/0.69 thf(fact_8400_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_real,P: real > $o,H2: real > real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real
% 0.35/0.69 @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups8097168146408367636l_real @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8401_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_int,P: int > $o,H2: int > real,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real
% 0.35/0.69 @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups8778361861064173332t_real @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8402_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_complex,P: complex > $o,H2: complex > real,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real
% 0.35/0.69 @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8403_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > real,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real
% 0.35/0.69 @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8404_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_real,P: real > $o,H2: real > rat,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat
% 0.35/0.69 @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8405_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_int,P: int > $o,H2: int > rat,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat
% 0.35/0.69 @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8406_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_complex,P: complex > $o,H2: complex > rat,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat
% 0.35/0.69 @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8407_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > rat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat
% 0.35/0.69 @ ^ [X: extended_enat] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8408_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_real,P: real > $o,H2: real > nat,G3: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1935376822645274424al_nat
% 0.35/0.69 @ ^ [X: real] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups1935376822645274424al_nat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1935376822645274424al_nat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8409_sum_OIf__cases,axiom,
% 0.35/0.69 ! [A3: set_int,P: int > $o,H2: int > nat,G3: int > nat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( groups4541462559716669496nt_nat
% 0.35/0.69 @ ^ [X: int] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 0.35/0.69 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.If_cases
% 0.35/0.69 thf(fact_8410_sum__power__add,axiom,
% 0.35/0.69 ! [X2: complex,M: nat,I5: set_nat] :
% 0.35/0.69 ( ( groups2073611262835488442omplex
% 0.35/0.69 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.35/0.69 @ I5 )
% 0.35/0.69 = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_power_add
% 0.35/0.69 thf(fact_8411_sum__power__add,axiom,
% 0.35/0.69 ! [X2: rat,M: nat,I5: set_nat] :
% 0.35/0.69 ( ( groups2906978787729119204at_rat
% 0.35/0.69 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.35/0.69 @ I5 )
% 0.35/0.69 = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_power_add
% 0.35/0.69 thf(fact_8412_sum__power__add,axiom,
% 0.35/0.69 ! [X2: int,M: nat,I5: set_nat] :
% 0.35/0.69 ( ( groups3539618377306564664at_int
% 0.35/0.69 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.35/0.69 @ I5 )
% 0.35/0.69 = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_power_add
% 0.35/0.69 thf(fact_8413_sum__power__add,axiom,
% 0.35/0.69 ! [X2: real,M: nat,I5: set_nat] :
% 0.35/0.69 ( ( groups6591440286371151544t_real
% 0.35/0.69 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 0.35/0.69 @ I5 )
% 0.35/0.69 = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_power_add
% 0.35/0.69 thf(fact_8414_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_real,A3: set_real,F: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8415_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_o,A3: set_o,F: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8416_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8417_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8418_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_real,A3: set_real,F: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8419_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_o,A3: set_o,F: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: $o] :
% 0.35/0.69 ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8420_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_complex,A3: set_complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: complex] :
% 0.35/0.69 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8421_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8422_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_nat,A3: set_nat,F: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: nat] :
% 0.35/0.69 ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8423_sum__mono2,axiom,
% 0.35/0.69 ! [B2: set_real,A3: set_real,F: real > nat] :
% 0.35/0.69 ( ( finite_finite_real @ B2 )
% 0.35/0.69 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.69 => ( ! [B: real] :
% 0.35/0.69 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.69 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B ) ) )
% 0.35/0.69 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_mono2
% 0.35/0.69 thf(fact_8424_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ! [X5: int] :
% 0.35/0.69 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8425_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8426_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_real ) )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8427_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ! [X5: int] :
% 0.35/0.69 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8428_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8429_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_rat ) )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8430_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ! [X5: int] :
% 0.35/0.69 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8431_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8432_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ! [X5: extended_enat] :
% 0.35/0.69 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_nat ) )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8433_sum_Ounion__inter__neutral,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ! [X5: complex] :
% 0.35/0.69 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 0.35/0.69 => ( ( G3 @ X5 )
% 0.35/0.69 = zero_zero_int ) )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_inter_neutral
% 0.35/0.69 thf(fact_8434_sum_OatLeastAtMost__rev,axiom,
% 0.35/0.69 ! [G3: nat > nat,N: nat,M: nat] :
% 0.35/0.69 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.35/0.69 = ( groups3542108847815614940at_nat
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.atLeastAtMost_rev
% 0.35/0.69 thf(fact_8435_sum_OatLeastAtMost__rev,axiom,
% 0.35/0.69 ! [G3: nat > real,N: nat,M: nat] :
% 0.35/0.69 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 0.35/0.69 = ( groups6591440286371151544t_real
% 0.35/0.69 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 0.35/0.69 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.atLeastAtMost_rev
% 0.35/0.69 thf(fact_8436_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( member_complex @ X2 @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8437_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8438_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( member_complex @ X2 @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8439_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8440_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_complex,X2: complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( member_complex @ X2 @ A3 )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8441_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8442_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_complex,X2: complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( member_complex @ X2 @ A3 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8443_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( member_Extended_enat @ X2 @ A3 )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8444_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_real,X2: real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( member_real @ X2 @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8445_sum_Oremove,axiom,
% 0.35/0.69 ! [A3: set_real,X2: real,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( member_real @ X2 @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.remove
% 0.35/0.69 thf(fact_8446_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > real,X2: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8447_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > real,X2: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8448_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > rat,X2: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8449_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > rat,X2: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8450_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > nat,X2: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8451_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > nat,X2: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8452_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_complex,G3: complex > int,X2: complex] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8453_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,G3: extended_enat > int,X2: extended_enat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8454_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > real,X2: real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8455_sum_Oinsert__remove,axiom,
% 0.35/0.69 ! [A3: set_real,G3: real > rat,X2: real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.insert_remove
% 0.35/0.69 thf(fact_8456_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_complex,A2: complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( ( member_complex @ A2 @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_complex @ A2 @ A3 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 0.35/0.69 = ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8457_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( ( member_Extended_enat @ A2 @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_Extended_enat @ A2 @ A3 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.69 = ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8458_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_real,A2: real,F: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( ( member_real @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_real @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.69 = ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8459_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_o,A2: $o,F: $o > real] :
% 0.35/0.69 ( ( finite_finite_o @ A3 )
% 0.35/0.69 => ( ( ( member_o @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_o @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.69 = ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8460_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_int,A2: int,F: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( ( member_int @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.69 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_int @ A2 @ A3 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.69 = ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8461_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_complex,A2: complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( ( member_complex @ A2 @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_complex @ A2 @ A3 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 0.35/0.69 = ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8462_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( ( member_Extended_enat @ A2 @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_Extended_enat @ A2 @ A3 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.69 = ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8463_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_real,A2: real,F: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( ( member_real @ A2 @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_real @ A2 @ A3 )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.69 = ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8464_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_o,A2: $o,F: $o > rat] :
% 0.35/0.69 ( ( finite_finite_o @ A3 )
% 0.35/0.69 => ( ( ( member_o @ A2 @ A3 )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_o @ A2 @ A3 )
% 0.35/0.69 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.69 = ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8465_sum__diff1,axiom,
% 0.35/0.69 ! [A3: set_int,A2: int,F: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( ( member_int @ A2 @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.69 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.69 & ( ~ ( member_int @ A2 @ A3 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.69 = ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_diff1
% 0.35/0.69 thf(fact_8466_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,F: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8467_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8468_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8469_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,F: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8470_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8471_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8472_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_nat,B2: set_nat,F: nat > rat] :
% 0.35/0.69 ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( finite_finite_nat @ B2 )
% 0.35/0.69 => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8473_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8474_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( plus_plus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) @ ( groups2025484359314973016at_int @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8475_sum__Un,axiom,
% 0.35/0.69 ! [A3: set_nat,B2: set_nat,F: nat > int] :
% 0.35/0.69 ( ( finite_finite_nat @ A3 )
% 0.35/0.69 => ( ( finite_finite_nat @ B2 )
% 0.35/0.69 => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.69 = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum_Un
% 0.35/0.69 thf(fact_8476_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_complex )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8477_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 0.35/0.69 = bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8478_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_complex )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8479_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 0.35/0.69 = bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8480_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_complex )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8481_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 0.35/0.69 = bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8482_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_complex )
% 0.35/0.69 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8483_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > int] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 0.35/0.69 = bot_bo7653980558646680370d_enat )
% 0.35/0.69 => ( ( groups2025484359314973016at_int @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ A3 ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8484_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_real,B2: set_real,G3: real > real] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( finite_finite_real @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_real )
% 0.35/0.69 => ( ( groups8097168146408367636l_real @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( groups8097168146408367636l_real @ G3 @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8485_sum_Ounion__disjoint,axiom,
% 0.35/0.69 ! [A3: set_real,B2: set_real,G3: real > rat] :
% 0.35/0.69 ( ( finite_finite_real @ A3 )
% 0.35/0.69 => ( ( finite_finite_real @ B2 )
% 0.35/0.69 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 0.35/0.69 = bot_bot_set_real )
% 0.35/0.69 => ( ( groups1300246762558778688al_rat @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ G3 @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_disjoint
% 0.35/0.69 thf(fact_8486_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > real] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8487_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8488_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8489_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8490_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8491_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8492_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 0.35/0.69 ( ( finite_finite_int @ A3 )
% 0.35/0.69 => ( ( finite_finite_int @ B2 )
% 0.35/0.69 => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8493_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 0.35/0.69 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.69 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.69 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.69
% 0.35/0.69 % sum.union_diff2
% 0.35/0.69 thf(fact_8494_sum_Ounion__diff2,axiom,
% 0.35/0.69 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 0.35/0.69 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.69 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.69 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.69 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.union_diff2
% 0.35/0.70 thf(fact_8495_sum_Ounion__diff2,axiom,
% 0.35/0.70 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.70 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.union_diff2
% 0.35/0.70 thf(fact_8496_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_int,B2: set_int,F: int > real] :
% 0.35/0.70 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8497_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8498_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8499_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_int,B2: set_int,F: int > rat] :
% 0.35/0.70 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8500_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8501_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8502_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_int,B2: set_int,F: int > nat] :
% 0.35/0.70 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8503_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_complex,B2: set_complex,F: complex > nat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8504_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8505_sum__Un2,axiom,
% 0.35/0.70 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.70 = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Un2
% 0.35/0.70 thf(fact_8506_suminf__finite,axiom,
% 0.35/0.70 ! [N6: set_nat,F: nat > int] :
% 0.35/0.70 ( ( finite_finite_nat @ N6 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.70 => ( ( F @ N3 )
% 0.35/0.70 = zero_zero_int ) )
% 0.35/0.70 => ( ( suminf_int @ F )
% 0.35/0.70 = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % suminf_finite
% 0.35/0.70 thf(fact_8507_suminf__finite,axiom,
% 0.35/0.70 ! [N6: set_nat,F: nat > nat] :
% 0.35/0.70 ( ( finite_finite_nat @ N6 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.70 => ( ( F @ N3 )
% 0.35/0.70 = zero_zero_nat ) )
% 0.35/0.70 => ( ( suminf_nat @ F )
% 0.35/0.70 = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % suminf_finite
% 0.35/0.70 thf(fact_8508_suminf__finite,axiom,
% 0.35/0.70 ! [N6: set_nat,F: nat > real] :
% 0.35/0.70 ( ( finite_finite_nat @ N6 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ~ ( member_nat @ N3 @ N6 )
% 0.35/0.70 => ( ( F @ N3 )
% 0.35/0.70 = zero_zero_real ) )
% 0.35/0.70 => ( ( suminf_real @ F )
% 0.35/0.70 = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % suminf_finite
% 0.35/0.70 thf(fact_8509_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_complex,A2: complex,B3: complex > real,C: complex > real] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.70 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5808333547571424918x_real
% 0.35/0.70 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5808333547571424918x_real
% 0.35/0.70 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8510_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real,C: extended_enat > real] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.70 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups4148127829035722712t_real
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups4148127829035722712t_real
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8511_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_complex,A2: complex,B3: complex > rat,C: complex > rat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.70 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5058264527183730370ex_rat
% 0.35/0.70 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5058264527183730370ex_rat
% 0.35/0.70 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8512_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > rat,C: extended_enat > rat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.70 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups1392844769737527556at_rat
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups1392844769737527556at_rat
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8513_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_complex,A2: complex,B3: complex > nat,C: complex > nat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.70 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5693394587270226106ex_nat
% 0.35/0.70 @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5693394587270226106ex_nat
% 0.35/0.70 @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8514_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > nat,C: extended_enat > nat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.70 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups2027974829824023292at_nat
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups2027974829824023292at_nat
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8515_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_complex,A2: complex,B3: complex > int,C: complex > int] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ S )
% 0.35/0.70 => ( ( ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5690904116761175830ex_int
% 0.35/0.70 @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_complex @ A2 @ S )
% 0.35/0.70 => ( ( groups5690904116761175830ex_int
% 0.35/0.70 @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8516_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > int,C: extended_enat > int] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.70 => ( ( ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups2025484359314973016at_int
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_Extended_enat @ A2 @ S )
% 0.35/0.70 => ( ( groups2025484359314973016at_int
% 0.35/0.70 @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8517_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_real,A2: real,B3: real > real,C: real > real] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( ( member_real @ A2 @ S )
% 0.35/0.70 => ( ( groups8097168146408367636l_real
% 0.35/0.70 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.70 => ( ( groups8097168146408367636l_real
% 0.35/0.70 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8518_sum_Odelta__remove,axiom,
% 0.35/0.70 ! [S: set_real,A2: real,B3: real > rat,C: real > rat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( ( member_real @ A2 @ S )
% 0.35/0.70 => ( ( groups1300246762558778688al_rat
% 0.35/0.70 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
% 0.35/0.70 & ( ~ ( member_real @ A2 @ S )
% 0.35/0.70 => ( ( groups1300246762558778688al_rat
% 0.35/0.70 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.delta_remove
% 0.35/0.70 thf(fact_8519_powser__sums__if,axiom,
% 0.35/0.70 ! [M: nat,Z: complex] :
% 0.35/0.70 ( sums_complex
% 0.35/0.70 @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 0.35/0.70 @ ( power_power_complex @ Z @ M ) ) ).
% 0.35/0.70
% 0.35/0.70 % powser_sums_if
% 0.35/0.70 thf(fact_8520_powser__sums__if,axiom,
% 0.35/0.70 ! [M: nat,Z: real] :
% 0.35/0.70 ( sums_real
% 0.35/0.70 @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 0.35/0.70 @ ( power_power_real @ Z @ M ) ) ).
% 0.35/0.70
% 0.35/0.70 % powser_sums_if
% 0.35/0.70 thf(fact_8521_powser__sums__if,axiom,
% 0.35/0.70 ! [M: nat,Z: int] :
% 0.35/0.70 ( sums_int
% 0.35/0.70 @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 0.35/0.70 @ ( power_power_int @ Z @ M ) ) ).
% 0.35/0.70
% 0.35/0.70 % powser_sums_if
% 0.35/0.70 thf(fact_8522_powser__sums__zero,axiom,
% 0.35/0.70 ! [A2: nat > complex] :
% 0.35/0.70 ( sums_complex
% 0.35/0.70 @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 0.35/0.70 @ ( A2 @ zero_zero_nat ) ) ).
% 0.35/0.70
% 0.35/0.70 % powser_sums_zero
% 0.35/0.70 thf(fact_8523_powser__sums__zero,axiom,
% 0.35/0.70 ! [A2: nat > real] :
% 0.35/0.70 ( sums_real
% 0.35/0.70 @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 0.35/0.70 @ ( A2 @ zero_zero_nat ) ) ).
% 0.35/0.70
% 0.35/0.70 % powser_sums_zero
% 0.35/0.70 thf(fact_8524_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.70 ! [F: nat > rat,K: nat] :
% 0.35/0.70 ( ( ( F @ zero_zero_nat )
% 0.35/0.70 = zero_zero_rat )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.70 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_shift_lb_Suc0_0
% 0.35/0.70 thf(fact_8525_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.70 ! [F: nat > int,K: nat] :
% 0.35/0.70 ( ( ( F @ zero_zero_nat )
% 0.35/0.70 = zero_zero_int )
% 0.35/0.70 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.70 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_shift_lb_Suc0_0
% 0.35/0.70 thf(fact_8526_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.70 ! [F: nat > nat,K: nat] :
% 0.35/0.70 ( ( ( F @ zero_zero_nat )
% 0.35/0.70 = zero_zero_nat )
% 0.35/0.70 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.70 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_shift_lb_Suc0_0
% 0.35/0.70 thf(fact_8527_sum__shift__lb__Suc0__0,axiom,
% 0.35/0.70 ! [F: nat > real,K: nat] :
% 0.35/0.70 ( ( ( F @ zero_zero_nat )
% 0.35/0.70 = zero_zero_real )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.35/0.70 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_shift_lb_Suc0_0
% 0.35/0.70 thf(fact_8528_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.70 ! [G3: nat > rat,N: nat] :
% 0.35/0.70 ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast0_atMost_Suc
% 0.35/0.70 thf(fact_8529_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.70 ! [G3: nat > int,N: nat] :
% 0.35/0.70 ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast0_atMost_Suc
% 0.35/0.70 thf(fact_8530_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.70 ! [G3: nat > nat,N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast0_atMost_Suc
% 0.35/0.70 thf(fact_8531_sum_OatLeast0__atMost__Suc,axiom,
% 0.35/0.70 ! [G3: nat > real,N: nat] :
% 0.35/0.70 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast0_atMost_Suc
% 0.35/0.70 thf(fact_8532_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( plus_plus_rat @ ( G3 @ M ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast_Suc_atMost
% 0.35/0.70 thf(fact_8533_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( plus_plus_int @ ( G3 @ M ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast_Suc_atMost
% 0.35/0.70 thf(fact_8534_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( plus_plus_nat @ ( G3 @ M ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast_Suc_atMost
% 0.35/0.70 thf(fact_8535_sum_OatLeast__Suc__atMost,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( plus_plus_real @ ( G3 @ M ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.atLeast_Suc_atMost
% 0.35/0.70 thf(fact_8536_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_rat @ ( G3 @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.nat_ivl_Suc'
% 0.35/0.70 thf(fact_8537_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_int @ ( G3 @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.nat_ivl_Suc'
% 0.35/0.70 thf(fact_8538_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_nat @ ( G3 @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.nat_ivl_Suc'
% 0.35/0.70 thf(fact_8539_sum_Onat__ivl__Suc_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_real @ ( G3 @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.nat_ivl_Suc'
% 0.35/0.70 thf(fact_8540_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.70 ! [N: num,Q3: num] :
% 0.35/0.70 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 != zero_z3403309356797280102nteger ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(3)
% 0.35/0.70 thf(fact_8541_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.70 ! [N: num,Q3: num] :
% 0.35/0.70 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 != zero_zero_int ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(3)
% 0.35/0.70 thf(fact_8542_cong__exp__iff__simps_I3_J,axiom,
% 0.35/0.70 ! [N: num,Q3: num] :
% 0.35/0.70 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 != zero_zero_nat ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(3)
% 0.35/0.70 thf(fact_8543_numeral__3__eq__3,axiom,
% 0.35/0.70 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.35/0.70 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % numeral_3_eq_3
% 0.35/0.70 thf(fact_8544_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_real,A3: set_real,B3: real,F: real > real] :
% 0.35/0.70 ( ( finite_finite_real @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.70 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8545_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_o,A3: set_o,B3: $o,F: $o > real] :
% 0.35/0.70 ( ( finite_finite_o @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.35/0.70 => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8546_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > real] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.70 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.35/0.70 => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8547_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > real] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.70 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.35/0.70 => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8548_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_real,A3: set_real,B3: real,F: real > rat] :
% 0.35/0.70 ( ( finite_finite_real @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.70 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8549_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_o,A3: set_o,B3: $o,F: $o > rat] :
% 0.35/0.70 ( ( finite_finite_o @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 0.35/0.70 => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8550_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > rat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.70 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 0.35/0.70 => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8551_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > rat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.70 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 0.35/0.70 => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8552_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_nat,A3: set_nat,B3: nat,F: nat > rat] :
% 0.35/0.70 ( ( finite_finite_nat @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 0.35/0.70 => ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8553_sum__strict__mono2,axiom,
% 0.35/0.70 ! [B2: set_real,A3: set_real,B3: real,F: real > nat] :
% 0.35/0.70 ( ( finite_finite_real @ B2 )
% 0.35/0.70 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 0.35/0.70 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ B2 )
% 0.35/0.70 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_strict_mono2
% 0.35/0.70 thf(fact_8554_member__le__sum,axiom,
% 0.35/0.70 ! [I: complex,A3: set_complex,F: complex > real] :
% 0.35/0.70 ( ( member_complex @ I @ A3 )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8555_member__le__sum,axiom,
% 0.35/0.70 ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
% 0.35/0.70 ( ( member_Extended_enat @ I @ A3 )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I ) @ ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8556_member__le__sum,axiom,
% 0.35/0.70 ! [I: real,A3: set_real,F: real > real] :
% 0.35/0.70 ( ( member_real @ I @ A3 )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_real @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8557_member__le__sum,axiom,
% 0.35/0.70 ! [I: $o,A3: set_o,F: $o > real] :
% 0.35/0.70 ( ( member_o @ I @ A3 )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_o @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8558_member__le__sum,axiom,
% 0.35/0.70 ! [I: int,A3: set_int,F: int > real] :
% 0.35/0.70 ( ( member_int @ I @ A3 )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_int @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8559_member__le__sum,axiom,
% 0.35/0.70 ! [I: complex,A3: set_complex,F: complex > rat] :
% 0.35/0.70 ( ( member_complex @ I @ A3 )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8560_member__le__sum,axiom,
% 0.35/0.70 ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
% 0.35/0.70 ( ( member_Extended_enat @ I @ A3 )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8561_member__le__sum,axiom,
% 0.35/0.70 ! [I: real,A3: set_real,F: real > rat] :
% 0.35/0.70 ( ( member_real @ I @ A3 )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_real @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8562_member__le__sum,axiom,
% 0.35/0.70 ! [I: $o,A3: set_o,F: $o > rat] :
% 0.35/0.70 ( ( member_o @ I @ A3 )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_o @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8563_member__le__sum,axiom,
% 0.35/0.70 ! [I: int,A3: set_int,F: int > rat] :
% 0.35/0.70 ( ( member_int @ I @ A3 )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( finite_finite_int @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % member_le_sum
% 0.35/0.70 thf(fact_8564_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_real,F: real > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8565_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_o,F: $o > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8566_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: complex] :
% 0.35/0.70 ( ( member_complex @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8567_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8568_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_int,F: int > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 0.35/0.70 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8569_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_real,F: real > nat,K4: nat] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 0.35/0.70 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8570_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_o,F: $o > nat,K4: nat] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 0.35/0.70 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8571_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > nat,K4: nat] :
% 0.35/0.70 ( ! [I2: complex] :
% 0.35/0.70 ( ( member_complex @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 0.35/0.70 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8572_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_int,F: int > nat,K4: nat] :
% 0.35/0.70 ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 0.35/0.70 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8573_sum__bounded__above__strict,axiom,
% 0.35/0.70 ! [A3: set_real,F: real > int,K4: int] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
% 0.35/0.70 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 0.35/0.70 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_strict
% 0.35/0.70 thf(fact_8574_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > real,K4: real] :
% 0.35/0.70 ( ! [I2: complex] :
% 0.35/0.70 ( ( member_complex @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_complex )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8575_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,F: extended_enat > real,K4: real] :
% 0.35/0.70 ( ! [I2: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8576_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_real,F: real > real,K4: real] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_real @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_real )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8577_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_o,F: $o > real,K4: real] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_o @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_o )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8578_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_int,F: int > real,K4: real] :
% 0.35/0.70 ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_int @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_int )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8579_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: complex] :
% 0.35/0.70 ( ( member_complex @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_complex )
% 0.35/0.70 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8580_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,F: extended_enat > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.70 => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8581_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_real,F: real > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_real @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_real )
% 0.35/0.70 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8582_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_o,F: $o > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_o @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_o )
% 0.35/0.70 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8583_sum__bounded__above__divide,axiom,
% 0.35/0.70 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 0.35/0.70 ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ A3 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) ) ) )
% 0.35/0.70 => ( ( finite_finite_nat @ A3 )
% 0.35/0.70 => ( ( A3 != bot_bot_set_nat )
% 0.35/0.70 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_bounded_above_divide
% 0.35/0.70 thf(fact_8584_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_rat @ ( G3 @ M )
% 0.35/0.70 @ ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.Suc_reindex_ivl
% 0.35/0.70 thf(fact_8585_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_int @ ( G3 @ M )
% 0.35/0.70 @ ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.Suc_reindex_ivl
% 0.35/0.70 thf(fact_8586_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_nat @ ( G3 @ M )
% 0.35/0.70 @ ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.Suc_reindex_ivl
% 0.35/0.70 thf(fact_8587_sum_OSuc__reindex__ivl,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 0.35/0.70 = ( plus_plus_real @ ( G3 @ M )
% 0.35/0.70 @ ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.Suc_reindex_ivl
% 0.35/0.70 thf(fact_8588_sum__Suc__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Suc_diff
% 0.35/0.70 thf(fact_8589_sum__Suc__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Suc_diff
% 0.35/0.70 thf(fact_8590_sum__Suc__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 0.35/0.70 => ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_Suc_diff
% 0.35/0.70 thf(fact_8591_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_real,X2: real > code_integer,A2: real > code_integer,B3: code_integer,Delta: code_integer] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups7713935264441627589nteger @ X2 @ I5 )
% 0.35/0.70 = one_one_Code_integer )
% 0.35/0.70 => ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_le3102999989581377725nteger
% 0.35/0.70 @ ( abs_abs_Code_integer
% 0.35/0.70 @ ( minus_8373710615458151222nteger
% 0.35/0.70 @ ( groups7713935264441627589nteger
% 0.35/0.70 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8592_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_o,X2: $o > code_integer,A2: $o > code_integer,B3: code_integer,Delta: code_integer] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups4406642042086082107nteger @ X2 @ I5 )
% 0.35/0.70 = one_one_Code_integer )
% 0.35/0.70 => ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_le3102999989581377725nteger
% 0.35/0.70 @ ( abs_abs_Code_integer
% 0.35/0.70 @ ( minus_8373710615458151222nteger
% 0.35/0.70 @ ( groups4406642042086082107nteger
% 0.35/0.70 @ ^ [I4: $o] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8593_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_nat,X2: nat > code_integer,A2: nat > code_integer,B3: code_integer,Delta: code_integer] :
% 0.35/0.70 ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups7501900531339628137nteger @ X2 @ I5 )
% 0.35/0.70 = one_one_Code_integer )
% 0.35/0.70 => ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_le3102999989581377725nteger
% 0.35/0.70 @ ( abs_abs_Code_integer
% 0.35/0.70 @ ( minus_8373710615458151222nteger
% 0.35/0.70 @ ( groups7501900531339628137nteger
% 0.35/0.70 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8594_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_int,X2: int > code_integer,A2: int > code_integer,B3: code_integer,Delta: code_integer] :
% 0.35/0.70 ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups7873554091576472773nteger @ X2 @ I5 )
% 0.35/0.70 = one_one_Code_integer )
% 0.35/0.70 => ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ I5 )
% 0.35/0.70 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_le3102999989581377725nteger
% 0.35/0.70 @ ( abs_abs_Code_integer
% 0.35/0.70 @ ( minus_8373710615458151222nteger
% 0.35/0.70 @ ( groups7873554091576472773nteger
% 0.35/0.70 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8595_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_real,X2: real > real,A2: real > real,B3: real,Delta: real] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups8097168146408367636l_real @ X2 @ I5 )
% 0.35/0.70 = one_one_real )
% 0.35/0.70 => ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_real
% 0.35/0.70 @ ( abs_abs_real
% 0.35/0.70 @ ( minus_minus_real
% 0.35/0.70 @ ( groups8097168146408367636l_real
% 0.35/0.70 @ ^ [I4: real] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8596_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_o,X2: $o > real,A2: $o > real,B3: real,Delta: real] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups8691415230153176458o_real @ X2 @ I5 )
% 0.35/0.70 = one_one_real )
% 0.35/0.70 => ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_real
% 0.35/0.70 @ ( abs_abs_real
% 0.35/0.70 @ ( minus_minus_real
% 0.35/0.70 @ ( groups8691415230153176458o_real
% 0.35/0.70 @ ^ [I4: $o] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8597_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_int,X2: int > real,A2: int > real,B3: real,Delta: real] :
% 0.35/0.70 ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups8778361861064173332t_real @ X2 @ I5 )
% 0.35/0.70 = one_one_real )
% 0.35/0.70 => ( ! [I2: int] :
% 0.35/0.70 ( ( member_int @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_real
% 0.35/0.70 @ ( abs_abs_real
% 0.35/0.70 @ ( minus_minus_real
% 0.35/0.70 @ ( groups8778361861064173332t_real
% 0.35/0.70 @ ^ [I4: int] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8598_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_real,X2: real > rat,A2: real > rat,B3: rat,Delta: rat] :
% 0.35/0.70 ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups1300246762558778688al_rat @ X2 @ I5 )
% 0.35/0.70 = one_one_rat )
% 0.35/0.70 => ( ! [I2: real] :
% 0.35/0.70 ( ( member_real @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_rat
% 0.35/0.70 @ ( abs_abs_rat
% 0.35/0.70 @ ( minus_minus_rat
% 0.35/0.70 @ ( groups1300246762558778688al_rat
% 0.35/0.70 @ ^ [I4: real] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8599_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_o,X2: $o > rat,A2: $o > rat,B3: rat,Delta: rat] :
% 0.35/0.70 ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups7872700643590313910_o_rat @ X2 @ I5 )
% 0.35/0.70 = one_one_rat )
% 0.35/0.70 => ( ! [I2: $o] :
% 0.35/0.70 ( ( member_o @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_rat
% 0.35/0.70 @ ( abs_abs_rat
% 0.35/0.70 @ ( minus_minus_rat
% 0.35/0.70 @ ( groups7872700643590313910_o_rat
% 0.35/0.70 @ ^ [I4: $o] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8600_convex__sum__bound__le,axiom,
% 0.35/0.70 ! [I5: set_nat,X2: nat > rat,A2: nat > rat,B3: rat,Delta: rat] :
% 0.35/0.70 ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 0.35/0.70 => ( ( ( groups2906978787729119204at_rat @ X2 @ I5 )
% 0.35/0.70 = one_one_rat )
% 0.35/0.70 => ( ! [I2: nat] :
% 0.35/0.70 ( ( member_nat @ I2 @ I5 )
% 0.35/0.70 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 0.35/0.70 => ( ord_less_eq_rat
% 0.35/0.70 @ ( abs_abs_rat
% 0.35/0.70 @ ( minus_minus_rat
% 0.35/0.70 @ ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [I4: nat] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 0.35/0.70 @ I5 )
% 0.35/0.70 @ B3 ) )
% 0.35/0.70 @ Delta ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % convex_sum_bound_le
% 0.35/0.70 thf(fact_8601_num_Osize_I6_J,axiom,
% 0.35/0.70 ! [X32: num] :
% 0.35/0.70 ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.35/0.70 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % num.size(6)
% 0.35/0.70 thf(fact_8602_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_real,F: real > complex,K4: real] :
% 0.35/0.70 ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8603_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_o,F: $o > complex,K4: real] :
% 0.35/0.70 ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8604_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_list_nat,F: list_nat > complex,K4: real] :
% 0.35/0.70 ( ! [X5: list_nat] :
% 0.35/0.70 ( ( member_list_nat @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6529277132148336714omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8605_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_set_nat,F: set_nat > complex,K4: real] :
% 0.35/0.70 ( ! [X5: set_nat] :
% 0.35/0.70 ( ( member_set_nat @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8606_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_nat,F: nat > complex,K4: real] :
% 0.35/0.70 ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8607_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_int,F: int > complex,K4: real] :
% 0.35/0.70 ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8608_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_complex,F: complex > complex,K4: real] :
% 0.35/0.70 ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8609_sum__norm__bound,axiom,
% 0.35/0.70 ! [S: set_nat,F: nat > real,K4: real] :
% 0.35/0.70 ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ S )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ K4 ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_norm_bound
% 0.35/0.70 thf(fact_8610_sum_Oub__add__nat,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > rat,P6: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.35/0.70 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.ub_add_nat
% 0.35/0.70 thf(fact_8611_sum_Oub__add__nat,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > int,P6: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.70 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.35/0.70 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.ub_add_nat
% 0.35/0.70 thf(fact_8612_sum_Oub__add__nat,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > nat,P6: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.70 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.35/0.70 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.ub_add_nat
% 0.35/0.70 thf(fact_8613_sum_Oub__add__nat,axiom,
% 0.35/0.70 ! [M: nat,N: nat,G3: nat > real,P6: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 0.35/0.70 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.ub_add_nat
% 0.35/0.70 thf(fact_8614_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.70 ! [Q3: num,N: num] :
% 0.35/0.70 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.35/0.70 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(7)
% 0.35/0.70 thf(fact_8615_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.70 ! [Q3: num,N: num] :
% 0.35/0.70 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 0.35/0.70 = zero_zero_int ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(7)
% 0.35/0.70 thf(fact_8616_cong__exp__iff__simps_I7_J,axiom,
% 0.35/0.70 ! [Q3: num,N: num] :
% 0.35/0.70 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.35/0.70 = zero_zero_nat ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(7)
% 0.35/0.70 thf(fact_8617_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.70 ! [M: num,Q3: num] :
% 0.35/0.70 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 0.35/0.70 = zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(11)
% 0.35/0.70 thf(fact_8618_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.70 ! [M: num,Q3: num] :
% 0.35/0.70 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 0.35/0.70 = zero_zero_int ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(11)
% 0.35/0.70 thf(fact_8619_cong__exp__iff__simps_I11_J,axiom,
% 0.35/0.70 ! [M: num,Q3: num] :
% 0.35/0.70 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 0.35/0.70 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 0.35/0.70 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 0.35/0.70 = zero_zero_nat ) ) ).
% 0.35/0.70
% 0.35/0.70 % cong_exp_iff_simps(11)
% 0.35/0.70 thf(fact_8620_sum__le__suminf,axiom,
% 0.35/0.70 ! [F: nat > int,I5: set_nat] :
% 0.35/0.70 ( ( summable_int @ F )
% 0.35/0.70 => ( ( finite_finite_nat @ I5 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.35/0.70 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 0.35/0.70 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_le_suminf
% 0.35/0.70 thf(fact_8621_sum__le__suminf,axiom,
% 0.35/0.70 ! [F: nat > nat,I5: set_nat] :
% 0.35/0.70 ( ( summable_nat @ F )
% 0.35/0.70 => ( ( finite_finite_nat @ I5 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.35/0.70 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 0.35/0.70 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_le_suminf
% 0.35/0.70 thf(fact_8622_sum__le__suminf,axiom,
% 0.35/0.70 ! [F: nat > real,I5: set_nat] :
% 0.35/0.70 ( ( summable_real @ F )
% 0.35/0.70 => ( ( finite_finite_nat @ I5 )
% 0.35/0.70 => ( ! [N3: nat] :
% 0.35/0.70 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.35/0.70 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 0.35/0.70 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_le_suminf
% 0.35/0.70 thf(fact_8623_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_Pr1261947904930325089at_nat] :
% 0.35/0.70 ( ( ( finite711546835091564841at_nat @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ Z2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8624_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_complex] :
% 0.35/0.70 ( ( ( finite_card_complex @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: complex,Y: complex,Z2: complex] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_complex @ X @ ( insert_complex @ Y @ ( insert_complex @ Z2 @ bot_bot_set_complex ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8625_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_list_nat] :
% 0.35/0.70 ( ( ( finite_card_list_nat @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: list_nat,Y: list_nat,Z2: list_nat] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ ( insert_list_nat @ Z2 @ bot_bot_set_list_nat ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8626_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_set_nat] :
% 0.35/0.70 ( ( ( finite_card_set_nat @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: set_nat,Y: set_nat,Z2: set_nat] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ ( insert_set_nat @ Z2 @ bot_bot_set_set_nat ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8627_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_real] :
% 0.35/0.70 ( ( ( finite_card_real @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: real,Y: real,Z2: real] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_real @ X @ ( insert_real @ Y @ ( insert_real @ Z2 @ bot_bot_set_real ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8628_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_o] :
% 0.35/0.70 ( ( ( finite_card_o @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: $o,Y: $o,Z2: $o] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_o @ X @ ( insert_o @ Y @ ( insert_o @ Z2 @ bot_bot_set_o ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8629_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_nat] :
% 0.35/0.70 ( ( ( finite_card_nat @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: nat,Y: nat,Z2: nat] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_nat @ X @ ( insert_nat @ Y @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8630_card__3__iff,axiom,
% 0.35/0.70 ! [S: set_int] :
% 0.35/0.70 ( ( ( finite_card_int @ S )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.70 = ( ? [X: int,Y: int,Z2: int] :
% 0.35/0.70 ( ( S
% 0.35/0.70 = ( insert_int @ X @ ( insert_int @ Y @ ( insert_int @ Z2 @ bot_bot_set_int ) ) ) )
% 0.35/0.70 & ( X != Y )
% 0.35/0.70 & ( Y != Z2 )
% 0.35/0.70 & ( X != Z2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % card_3_iff
% 0.35/0.70 thf(fact_8631_exp__le,axiom,
% 0.35/0.70 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 0.35/0.70
% 0.35/0.70 % exp_le
% 0.35/0.70 thf(fact_8632_mod__exhaust__less__4,axiom,
% 0.35/0.70 ! [M: nat] :
% 0.35/0.70 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.70 = zero_zero_nat )
% 0.35/0.70 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.70 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.70 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % mod_exhaust_less_4
% 0.35/0.70 thf(fact_8633_sum__natinterval__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.70 ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = zero_zero_rat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_natinterval_diff
% 0.35/0.70 thf(fact_8634_sum__natinterval__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.70 ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = zero_zero_int ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_natinterval_diff
% 0.35/0.70 thf(fact_8635_sum__natinterval__diff,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.70 ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = zero_zero_real ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_natinterval_diff
% 0.35/0.70 thf(fact_8636_sum__telescope_H_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.70 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_telescope''
% 0.35/0.70 thf(fact_8637_sum__telescope_H_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.70 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_telescope''
% 0.35/0.70 thf(fact_8638_sum__telescope_H_H,axiom,
% 0.35/0.70 ! [M: nat,N: nat,F: nat > real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 0.35/0.70 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_telescope''
% 0.35/0.70 thf(fact_8639_summable__partial__sum__bound,axiom,
% 0.35/0.70 ! [F: nat > complex,E2: real] :
% 0.35/0.70 ( ( summable_complex @ F )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.70 => ~ ! [N9: nat] :
% 0.35/0.70 ~ ! [M3: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ N9 @ M3 )
% 0.35/0.70 => ! [N5: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % summable_partial_sum_bound
% 0.35/0.70 thf(fact_8640_summable__partial__sum__bound,axiom,
% 0.35/0.70 ! [F: nat > real,E2: real] :
% 0.35/0.70 ( ( summable_real @ F )
% 0.35/0.70 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.70 => ~ ! [N9: nat] :
% 0.35/0.70 ~ ! [M3: nat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ N9 @ M3 )
% 0.35/0.70 => ! [N5: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % summable_partial_sum_bound
% 0.35/0.70 thf(fact_8641_geometric__sums,axiom,
% 0.35/0.70 ! [C: real] :
% 0.35/0.70 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.35/0.70 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % geometric_sums
% 0.35/0.70 thf(fact_8642_geometric__sums,axiom,
% 0.35/0.70 ! [C: complex] :
% 0.35/0.70 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.35/0.70 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % geometric_sums
% 0.35/0.70 thf(fact_8643_mask__eq__sum__exp,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 0.35/0.70 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % mask_eq_sum_exp
% 0.35/0.70 thf(fact_8644_mask__eq__sum__exp,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
% 0.35/0.70 = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % mask_eq_sum_exp
% 0.35/0.70 thf(fact_8645_mask__eq__sum__exp,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 0.35/0.70 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % mask_eq_sum_exp
% 0.35/0.70 thf(fact_8646_sum__gp__multiplied,axiom,
% 0.35/0.70 ! [M: nat,N: nat,X2: complex] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.70 = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_multiplied
% 0.35/0.70 thf(fact_8647_sum__gp__multiplied,axiom,
% 0.35/0.70 ! [M: nat,N: nat,X2: rat] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.70 = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_multiplied
% 0.35/0.70 thf(fact_8648_sum__gp__multiplied,axiom,
% 0.35/0.70 ! [M: nat,N: nat,X2: int] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.70 = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_multiplied
% 0.35/0.70 thf(fact_8649_sum__gp__multiplied,axiom,
% 0.35/0.70 ! [M: nat,N: nat,X2: real] :
% 0.35/0.70 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.70 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 0.35/0.70 = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_multiplied
% 0.35/0.70 thf(fact_8650_sum_Oin__pairs,axiom,
% 0.35/0.70 ! [G3: nat > rat,M: nat,N: nat] :
% 0.35/0.70 ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.70 = ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_rat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.in_pairs
% 0.35/0.70 thf(fact_8651_sum_Oin__pairs,axiom,
% 0.35/0.70 ! [G3: nat > int,M: nat,N: nat] :
% 0.35/0.70 ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.70 = ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_int @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.in_pairs
% 0.35/0.70 thf(fact_8652_sum_Oin__pairs,axiom,
% 0.35/0.70 ! [G3: nat > nat,M: nat,N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.70 = ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.in_pairs
% 0.35/0.70 thf(fact_8653_sum_Oin__pairs,axiom,
% 0.35/0.70 ! [G3: nat > real,M: nat,N: nat] :
% 0.35/0.70 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.35/0.70 = ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_real @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum.in_pairs
% 0.35/0.70 thf(fact_8654_accp__subset__induct,axiom,
% 0.35/0.70 ! [D4: produc4953844613479565601on_nat > $o,R: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X2: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
% 0.35/0.70 ( ( ord_le8126618931240741628_nat_o @ D4 @ ( accp_P8646395344606611882on_nat @ R ) )
% 0.35/0.70 => ( ! [X5: produc4953844613479565601on_nat,Z4: produc4953844613479565601on_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ( R @ Z4 @ X5 )
% 0.35/0.70 => ( D4 @ Z4 ) ) )
% 0.35/0.70 => ( ( D4 @ X2 )
% 0.35/0.70 => ( ! [X5: produc4953844613479565601on_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ! [Z5: produc4953844613479565601on_nat] :
% 0.35/0.70 ( ( R @ Z5 @ X5 )
% 0.35/0.70 => ( P @ Z5 ) )
% 0.35/0.70 => ( P @ X5 ) ) )
% 0.35/0.70 => ( P @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % accp_subset_induct
% 0.35/0.70 thf(fact_8655_accp__subset__induct,axiom,
% 0.35/0.70 ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.35/0.70 ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
% 0.35/0.70 => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ( R @ Z4 @ X5 )
% 0.35/0.70 => ( D4 @ Z4 ) ) )
% 0.35/0.70 => ( ( D4 @ X2 )
% 0.35/0.70 => ( ! [X5: product_prod_nat_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ! [Z5: product_prod_nat_nat] :
% 0.35/0.70 ( ( R @ Z5 @ X5 )
% 0.35/0.70 => ( P @ Z5 ) )
% 0.35/0.70 => ( P @ X5 ) ) )
% 0.35/0.70 => ( P @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % accp_subset_induct
% 0.35/0.70 thf(fact_8656_accp__subset__induct,axiom,
% 0.35/0.70 ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
% 0.35/0.70 ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
% 0.35/0.70 => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ( R @ Z4 @ X5 )
% 0.35/0.70 => ( D4 @ Z4 ) ) )
% 0.35/0.70 => ( ( D4 @ X2 )
% 0.35/0.70 => ( ! [X5: product_prod_int_int] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ! [Z5: product_prod_int_int] :
% 0.35/0.70 ( ( R @ Z5 @ X5 )
% 0.35/0.70 => ( P @ Z5 ) )
% 0.35/0.70 => ( P @ X5 ) ) )
% 0.35/0.70 => ( P @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % accp_subset_induct
% 0.35/0.70 thf(fact_8657_accp__subset__induct,axiom,
% 0.35/0.70 ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X2: list_nat,P: list_nat > $o] :
% 0.35/0.70 ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
% 0.35/0.70 => ( ! [X5: list_nat,Z4: list_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ( R @ Z4 @ X5 )
% 0.35/0.70 => ( D4 @ Z4 ) ) )
% 0.35/0.70 => ( ( D4 @ X2 )
% 0.35/0.70 => ( ! [X5: list_nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ! [Z5: list_nat] :
% 0.35/0.70 ( ( R @ Z5 @ X5 )
% 0.35/0.70 => ( P @ Z5 ) )
% 0.35/0.70 => ( P @ X5 ) ) )
% 0.35/0.70 => ( P @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % accp_subset_induct
% 0.35/0.70 thf(fact_8658_accp__subset__induct,axiom,
% 0.35/0.70 ! [D4: nat > $o,R: nat > nat > $o,X2: nat,P: nat > $o] :
% 0.35/0.70 ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
% 0.35/0.70 => ( ! [X5: nat,Z4: nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ( R @ Z4 @ X5 )
% 0.35/0.70 => ( D4 @ Z4 ) ) )
% 0.35/0.70 => ( ( D4 @ X2 )
% 0.35/0.70 => ( ! [X5: nat] :
% 0.35/0.70 ( ( D4 @ X5 )
% 0.35/0.70 => ( ! [Z5: nat] :
% 0.35/0.70 ( ( R @ Z5 @ X5 )
% 0.35/0.70 => ( P @ Z5 ) )
% 0.35/0.70 => ( P @ X5 ) ) )
% 0.35/0.70 => ( P @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % accp_subset_induct
% 0.35/0.70 thf(fact_8659_mask__eq__sum__exp__nat,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % mask_eq_sum_exp_nat
% 0.35/0.70 thf(fact_8660_gauss__sum__nat,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [X: nat] : X
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum_nat
% 0.35/0.70 thf(fact_8661_gbinomial__sum__up__index,axiom,
% 0.35/0.70 ! [K: nat,N: nat] :
% 0.35/0.70 ( ( groups2073611262835488442omplex
% 0.35/0.70 @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gbinomial_sum_up_index
% 0.35/0.70 thf(fact_8662_gbinomial__sum__up__index,axiom,
% 0.35/0.70 ! [K: nat,N: nat] :
% 0.35/0.70 ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gbinomial_sum_up_index
% 0.35/0.70 thf(fact_8663_gbinomial__sum__up__index,axiom,
% 0.35/0.70 ! [K: nat,N: nat] :
% 0.35/0.70 ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gbinomial_sum_up_index
% 0.35/0.70 thf(fact_8664_double__arith__series,axiom,
% 0.35/0.70 ! [A2: complex,D: complex,N: nat] :
% 0.35/0.70 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups2073611262835488442omplex
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_complex @ A2 @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8665_double__arith__series,axiom,
% 0.35/0.70 ! [A2: rat,D: rat,N: nat] :
% 0.35/0.70 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_rat @ A2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8666_double__arith__series,axiom,
% 0.35/0.70 ! [A2: extended_enat,D: extended_enat,N: nat] :
% 0.35/0.70 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups7108830773950497114d_enat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A2 @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A2 ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8667_double__arith__series,axiom,
% 0.35/0.70 ! [A2: code_integer,D: code_integer,N: nat] :
% 0.35/0.70 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups7501900531339628137nteger
% 0.35/0.70 @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8668_double__arith__series,axiom,
% 0.35/0.70 ! [A2: int,D: int,N: nat] :
% 0.35/0.70 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8669_double__arith__series,axiom,
% 0.35/0.70 ! [A2: nat,D: nat,N: nat] :
% 0.35/0.70 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8670_double__arith__series,axiom,
% 0.35/0.70 ! [A2: real,D: real,N: nat] :
% 0.35/0.70 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.35/0.70 @ ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_real @ A2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_arith_series
% 0.35/0.70 thf(fact_8671_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8672_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8673_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8674_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8675_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8676_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8677_double__gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum
% 0.35/0.70 thf(fact_8678_arith__series__nat,axiom,
% 0.35/0.70 ! [A2: nat,D: nat,N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % arith_series_nat
% 0.35/0.70 thf(fact_8679_Sum__Icc__nat,axiom,
% 0.35/0.70 ! [M: nat,N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [X: nat] : X
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 0.35/0.70 = ( 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 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % Sum_Icc_nat
% 0.35/0.70 thf(fact_8680_arith__series,axiom,
% 0.35/0.70 ! [A2: code_integer,D: code_integer,N: nat] :
% 0.35/0.70 ( ( groups7501900531339628137nteger
% 0.35/0.70 @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % arith_series
% 0.35/0.70 thf(fact_8681_arith__series,axiom,
% 0.35/0.70 ! [A2: int,D: int,N: nat] :
% 0.35/0.70 ( ( groups3539618377306564664at_int
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( 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 ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % arith_series
% 0.35/0.70 thf(fact_8682_arith__series,axiom,
% 0.35/0.70 ! [A2: nat,D: nat,N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( 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 ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % arith_series
% 0.35/0.70 thf(fact_8683_gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum
% 0.35/0.70 thf(fact_8684_gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum
% 0.35/0.70 thf(fact_8685_gauss__sum,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 0.35/0.70 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum
% 0.35/0.70 thf(fact_8686_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8687_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8688_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8689_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8690_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8691_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8692_double__gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 0.35/0.70 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % double_gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8693_sum__gp__offset,axiom,
% 0.35/0.70 ! [X2: rat,M: nat,N: nat] :
% 0.35/0.70 ( ( ( X2 = one_one_rat )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 0.35/0.70 & ( ( X2 != one_one_rat )
% 0.35/0.70 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_offset
% 0.35/0.70 thf(fact_8694_sum__gp__offset,axiom,
% 0.35/0.70 ! [X2: complex,M: nat,N: nat] :
% 0.35/0.70 ( ( ( X2 = one_one_complex )
% 0.35/0.70 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 0.35/0.70 & ( ( X2 != one_one_complex )
% 0.35/0.70 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_offset
% 0.35/0.70 thf(fact_8695_sum__gp__offset,axiom,
% 0.35/0.70 ! [X2: real,M: nat,N: nat] :
% 0.35/0.70 ( ( ( X2 = one_one_real )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 0.35/0.70 & ( ( X2 != one_one_real )
% 0.35/0.70 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 0.35/0.70 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_gp_offset
% 0.35/0.70 thf(fact_8696_log__base__10__eq2,axiom,
% 0.35/0.70 ! [X2: real] :
% 0.35/0.70 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.70 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.70 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % log_base_10_eq2
% 0.35/0.70 thf(fact_8697_gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.70 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8698_gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.70 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8699_gauss__sum__from__Suc__0,axiom,
% 0.35/0.70 ! [N: nat] :
% 0.35/0.70 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.70 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gauss_sum_from_Suc_0
% 0.35/0.70 thf(fact_8700_gchoose__row__sum__weighted,axiom,
% 0.35/0.70 ! [R2: rat,M: nat] :
% 0.35/0.70 ( ( groups2906978787729119204at_rat
% 0.35/0.70 @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.70 = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gchoose_row_sum_weighted
% 0.35/0.70 thf(fact_8701_gchoose__row__sum__weighted,axiom,
% 0.35/0.70 ! [R2: complex,M: nat] :
% 0.35/0.70 ( ( groups2073611262835488442omplex
% 0.35/0.70 @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.70 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gchoose_row_sum_weighted
% 0.35/0.70 thf(fact_8702_gchoose__row__sum__weighted,axiom,
% 0.35/0.70 ! [R2: real,M: nat] :
% 0.35/0.70 ( ( groups6591440286371151544t_real
% 0.35/0.70 @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 0.35/0.70 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.35/0.70 = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % gchoose_row_sum_weighted
% 0.35/0.70 thf(fact_8703_and__int_Opinduct,axiom,
% 0.35/0.70 ! [A0: int,A13: int,P: int > int > $o] :
% 0.35/0.70 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
% 0.35/0.70 => ( ! [K2: int,L4: int] :
% 0.35/0.70 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 0.35/0.70 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.70 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.70 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.70 => ( P @ K2 @ L4 ) ) )
% 0.35/0.70 => ( P @ A0 @ A13 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % and_int.pinduct
% 0.35/0.70 thf(fact_8704_prod__decode__aux_Opelims,axiom,
% 0.35/0.70 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.35/0.70 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 0.35/0.70 = Y3 )
% 0.35/0.70 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.35/0.70 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.35/0.70 => ( Y3
% 0.35/0.70 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 0.35/0.70 => ( Y3
% 0.35/0.70 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
% 0.35/0.70 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % prod_decode_aux.pelims
% 0.35/0.70 thf(fact_8705_fold__atLeastAtMost__nat_Opinduct,axiom,
% 0.35/0.70 ! [A0: nat > nat > nat,A13: nat,A24: nat,A33: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
% 0.35/0.70 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A13 @ ( product_Pair_nat_nat @ A24 @ A33 ) ) ) )
% 0.35/0.70 => ( ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
% 0.35/0.70 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
% 0.35/0.70 => ( ( ~ ( ord_less_nat @ B @ A )
% 0.35/0.70 => ( P @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc ) ) )
% 0.35/0.70 => ( P @ F3 @ A @ B @ Acc ) ) )
% 0.35/0.70 => ( P @ A0 @ A13 @ A24 @ A33 ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % fold_atLeastAtMost_nat.pinduct
% 0.35/0.70 thf(fact_8706_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.35/0.70 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.35/0.70 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 0.35/0.70 = Y3 )
% 0.35/0.70 => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
% 0.35/0.70 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.35/0.70 => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.35/0.70 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
% 0.35/0.70 => ( ! [V2: product_prod_nat_nat] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.35/0.70 => ( ( Xb = none_P5556105721700978146at_nat )
% 0.35/0.70 => ( ( Y3 = none_P5556105721700978146at_nat )
% 0.35/0.70 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 0.35/0.70 => ~ ! [A: product_prod_nat_nat] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_P7363390416028606310at_nat @ A ) )
% 0.35/0.70 => ! [B: product_prod_nat_nat] :
% 0.35/0.70 ( ( Xb
% 0.35/0.70 = ( some_P7363390416028606310at_nat @ B ) )
% 0.35/0.70 => ( ( Y3
% 0.35/0.70 = ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) )
% 0.35/0.70 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % VEBT_internal.option_shift.pelims
% 0.35/0.70 thf(fact_8707_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.35/0.70 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 0.35/0.70 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 0.35/0.70 = Y3 )
% 0.35/0.70 => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
% 0.35/0.70 => ( ( ( Xa2 = none_num )
% 0.35/0.70 => ( ( Y3 = none_num )
% 0.35/0.70 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
% 0.35/0.70 => ( ! [V2: num] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_num @ V2 ) )
% 0.35/0.70 => ( ( Xb = none_num )
% 0.35/0.70 => ( ( Y3 = none_num )
% 0.35/0.70 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 0.35/0.70 => ~ ! [A: num] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_num @ A ) )
% 0.35/0.70 => ! [B: num] :
% 0.35/0.70 ( ( Xb
% 0.35/0.70 = ( some_num @ B ) )
% 0.35/0.70 => ( ( Y3
% 0.35/0.70 = ( some_num @ ( X2 @ A @ B ) ) )
% 0.35/0.70 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % VEBT_internal.option_shift.pelims
% 0.35/0.70 thf(fact_8708_VEBT__internal_Ooption__shift_Opelims,axiom,
% 0.35/0.70 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.35/0.70 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 0.35/0.70 = Y3 )
% 0.35/0.70 => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
% 0.35/0.70 => ( ( ( Xa2 = none_nat )
% 0.35/0.70 => ( ( Y3 = none_nat )
% 0.35/0.70 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
% 0.35/0.70 => ( ! [V2: nat] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_nat @ V2 ) )
% 0.35/0.70 => ( ( Xb = none_nat )
% 0.35/0.70 => ( ( Y3 = none_nat )
% 0.35/0.70 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 0.35/0.70 => ~ ! [A: nat] :
% 0.35/0.70 ( ( Xa2
% 0.35/0.70 = ( some_nat @ A ) )
% 0.35/0.70 => ! [B: nat] :
% 0.35/0.70 ( ( Xb
% 0.35/0.70 = ( some_nat @ B ) )
% 0.35/0.70 => ( ( Y3
% 0.35/0.70 = ( some_nat @ ( X2 @ A @ B ) ) )
% 0.35/0.70 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % VEBT_internal.option_shift.pelims
% 0.35/0.70 thf(fact_8709_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(8)
% 0.35/0.70 thf(fact_8710_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(8)
% 0.35/0.70 thf(fact_8711_divmod__algorithm__code_I8_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_num @ M @ N )
% 0.35/0.70 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(8)
% 0.35/0.70 thf(fact_8712_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(7)
% 0.35/0.70 thf(fact_8713_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(7)
% 0.35/0.70 thf(fact_8714_divmod__algorithm__code_I7_J,axiom,
% 0.35/0.70 ! [M: num,N: num] :
% 0.35/0.70 ( ( ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.35/0.70 & ( ~ ( ord_less_eq_num @ M @ N )
% 0.35/0.70 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 0.35/0.70 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(7)
% 0.35/0.70 thf(fact_8715_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.70 ! [M: num] :
% 0.35/0.70 ( ( unique5052692396658037445od_int @ M @ one )
% 0.35/0.70 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(2)
% 0.35/0.70 thf(fact_8716_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.70 ! [M: num] :
% 0.35/0.70 ( ( unique3479559517661332726nteger @ M @ one )
% 0.35/0.70 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(2)
% 0.35/0.70 thf(fact_8717_divmod__algorithm__code_I2_J,axiom,
% 0.35/0.70 ! [M: num] :
% 0.35/0.70 ( ( unique5055182867167087721od_nat @ M @ one )
% 0.35/0.70 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(2)
% 0.35/0.70 thf(fact_8718_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 0.35/0.70 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(3)
% 0.35/0.70 thf(fact_8719_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 0.35/0.70 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(3)
% 0.35/0.70 thf(fact_8720_divmod__algorithm__code_I3_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 0.35/0.70 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(3)
% 0.35/0.70 thf(fact_8721_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(4)
% 0.35/0.70 thf(fact_8722_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 0.35/0.70 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(4)
% 0.35/0.70 thf(fact_8723_divmod__algorithm__code_I4_J,axiom,
% 0.35/0.70 ! [N: num] :
% 0.35/0.70 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 0.35/0.70 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % divmod_algorithm_code(4)
% 0.35/0.70 thf(fact_8724_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_o,T: set_o,R: $o > $o > $o,K: $o > nat] :
% 0.35/0.70 ( ( finite_finite_o @ S2 )
% 0.35/0.70 => ( ( finite_finite_o @ T )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [J3: $o] :
% 0.35/0.70 ( ( member_o @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8725_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_o,T: set_real,R: $o > real > $o,K: real > nat] :
% 0.35/0.70 ( ( finite_finite_o @ S2 )
% 0.35/0.70 => ( ( finite_finite_real @ T )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [J3: real] :
% 0.35/0.70 ( ( member_real @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8726_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_real,T: set_o,R: real > $o > $o,K: $o > nat] :
% 0.35/0.70 ( ( finite_finite_real @ S2 )
% 0.35/0.70 => ( ( finite_finite_o @ T )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [J3: $o] :
% 0.35/0.70 ( ( member_o @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8727_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_real,T: set_real,R: real > real > $o,K: real > nat] :
% 0.35/0.70 ( ( finite_finite_real @ S2 )
% 0.35/0.70 => ( ( finite_finite_real @ T )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [J3: real] :
% 0.35/0.70 ( ( member_real @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8728_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_o,T: set_int,R: $o > int > $o,K: int > nat] :
% 0.35/0.70 ( ( finite_finite_o @ S2 )
% 0.35/0.70 => ( ( finite_finite_int @ T )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_int
% 0.35/0.70 @ ( collect_int
% 0.35/0.70 @ ^ [J3: int] :
% 0.35/0.70 ( ( member_int @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8729_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_real,T: set_int,R: real > int > $o,K: int > nat] :
% 0.35/0.70 ( ( finite_finite_real @ S2 )
% 0.35/0.70 => ( ( finite_finite_int @ T )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_int
% 0.35/0.70 @ ( collect_int
% 0.35/0.70 @ ^ [J3: int] :
% 0.35/0.70 ( ( member_int @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8730_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_o,T: set_complex,R: $o > complex > $o,K: complex > nat] :
% 0.35/0.70 ( ( finite_finite_o @ S2 )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_complex
% 0.35/0.70 @ ( collect_complex
% 0.35/0.70 @ ^ [J3: complex] :
% 0.35/0.70 ( ( member_complex @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8731_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_real,T: set_complex,R: real > complex > $o,K: complex > nat] :
% 0.35/0.70 ( ( finite_finite_real @ S2 )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ T )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_complex
% 0.35/0.70 @ ( collect_complex
% 0.35/0.70 @ ^ [J3: complex] :
% 0.35/0.70 ( ( member_complex @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8732_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_o,T: set_Extended_enat,R: $o > extended_enat > $o,K: extended_enat > nat] :
% 0.35/0.70 ( ( finite_finite_o @ S2 )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ T )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite121521170596916366d_enat
% 0.35/0.70 @ ( collec4429806609662206161d_enat
% 0.35/0.70 @ ^ [J3: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8733_sum__multicount__gen,axiom,
% 0.35/0.70 ! [S2: set_real,T: set_Extended_enat,R: real > extended_enat > $o,K: extended_enat > nat] :
% 0.35/0.70 ( ( finite_finite_real @ S2 )
% 0.35/0.70 => ( ( finite4001608067531595151d_enat @ T )
% 0.35/0.70 => ( ! [X5: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X5 @ T )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S2 )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = ( K @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite121521170596916366d_enat
% 0.35/0.70 @ ( collec4429806609662206161d_enat
% 0.35/0.70 @ ^ [J3: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ J3 @ T )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S2 )
% 0.35/0.70 = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount_gen
% 0.35/0.70 thf(fact_8734_sum__subtractf__nat,axiom,
% 0.35/0.70 ! [A3: set_real,G3: real > nat,F: real > nat] :
% 0.35/0.70 ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ A3 )
% 0.35/0.70 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 0.35/0.70 @ A3 )
% 0.35/0.70 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_subtractf_nat
% 0.35/0.70 thf(fact_8735_sum__subtractf__nat,axiom,
% 0.35/0.70 ! [A3: set_o,G3: $o > nat,F: $o > nat] :
% 0.35/0.70 ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ A3 )
% 0.35/0.70 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [X: $o] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 0.35/0.70 @ A3 )
% 0.35/0.70 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_subtractf_nat
% 0.35/0.70 thf(fact_8736_sum__subtractf__nat,axiom,
% 0.35/0.70 ! [A3: set_set_nat,G3: set_nat > nat,F: set_nat > nat] :
% 0.35/0.70 ( ! [X5: set_nat] :
% 0.35/0.70 ( ( member_set_nat @ X5 @ A3 )
% 0.35/0.70 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( groups8294997508430121362at_nat
% 0.35/0.70 @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 0.35/0.70 @ A3 )
% 0.35/0.70 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( groups8294997508430121362at_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_subtractf_nat
% 0.35/0.70 thf(fact_8737_sum__subtractf__nat,axiom,
% 0.35/0.70 ! [A3: set_int,G3: int > nat,F: int > nat] :
% 0.35/0.70 ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ A3 )
% 0.35/0.70 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( groups4541462559716669496nt_nat
% 0.35/0.70 @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 0.35/0.70 @ A3 )
% 0.35/0.70 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_subtractf_nat
% 0.35/0.70 thf(fact_8738_sum__subtractf__nat,axiom,
% 0.35/0.70 ! [A3: set_nat,G3: nat > nat,F: nat > nat] :
% 0.35/0.70 ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ A3 )
% 0.35/0.70 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 0.35/0.70 => ( ( groups3542108847815614940at_nat
% 0.35/0.70 @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 0.35/0.70 @ A3 )
% 0.35/0.70 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_subtractf_nat
% 0.35/0.70 thf(fact_8739_sum__eq__Suc0__iff,axiom,
% 0.35/0.70 ! [A3: set_int,F: int > nat] :
% 0.35/0.70 ( ( finite_finite_int @ A3 )
% 0.35/0.70 => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( ? [X: int] :
% 0.35/0.70 ( ( member_int @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 & ! [Y: int] :
% 0.35/0.70 ( ( member_int @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_Suc0_iff
% 0.35/0.70 thf(fact_8740_sum__eq__Suc0__iff,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > nat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( ? [X: complex] :
% 0.35/0.70 ( ( member_complex @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 & ! [Y: complex] :
% 0.35/0.70 ( ( member_complex @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_Suc0_iff
% 0.35/0.70 thf(fact_8741_sum__eq__Suc0__iff,axiom,
% 0.35/0.70 ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.70 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.70 => ( ( ( groups977919841031483927at_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( ? [X: product_prod_nat_nat] :
% 0.35/0.70 ( ( member8440522571783428010at_nat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 & ! [Y: product_prod_nat_nat] :
% 0.35/0.70 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_Suc0_iff
% 0.35/0.70 thf(fact_8742_sum__eq__Suc0__iff,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( ? [X: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 & ! [Y: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_Suc0_iff
% 0.35/0.70 thf(fact_8743_sum__eq__Suc0__iff,axiom,
% 0.35/0.70 ! [A3: set_nat,F: nat > nat] :
% 0.35/0.70 ( ( finite_finite_nat @ A3 )
% 0.35/0.70 => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 = ( ? [X: nat] :
% 0.35/0.70 ( ( member_nat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = ( suc @ zero_zero_nat ) )
% 0.35/0.70 & ! [Y: nat] :
% 0.35/0.70 ( ( member_nat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_Suc0_iff
% 0.35/0.70 thf(fact_8744_sum__SucD,axiom,
% 0.35/0.70 ! [F: nat > nat,A3: set_nat,N: nat] :
% 0.35/0.70 ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 0.35/0.70 = ( suc @ N ) )
% 0.35/0.70 => ? [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ A3 )
% 0.35/0.70 & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_SucD
% 0.35/0.70 thf(fact_8745_sum__eq__1__iff,axiom,
% 0.35/0.70 ! [A3: set_int,F: int > nat] :
% 0.35/0.70 ( ( finite_finite_int @ A3 )
% 0.35/0.70 => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 = ( ? [X: int] :
% 0.35/0.70 ( ( member_int @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 & ! [Y: int] :
% 0.35/0.70 ( ( member_int @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_1_iff
% 0.35/0.70 thf(fact_8746_sum__eq__1__iff,axiom,
% 0.35/0.70 ! [A3: set_complex,F: complex > nat] :
% 0.35/0.70 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.70 => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 = ( ? [X: complex] :
% 0.35/0.70 ( ( member_complex @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 & ! [Y: complex] :
% 0.35/0.70 ( ( member_complex @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_1_iff
% 0.35/0.70 thf(fact_8747_sum__eq__1__iff,axiom,
% 0.35/0.70 ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.70 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.70 => ( ( ( groups977919841031483927at_nat @ F @ A3 )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 = ( ? [X: product_prod_nat_nat] :
% 0.35/0.70 ( ( member8440522571783428010at_nat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 & ! [Y: product_prod_nat_nat] :
% 0.35/0.70 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_1_iff
% 0.35/0.70 thf(fact_8748_sum__eq__1__iff,axiom,
% 0.35/0.70 ! [A3: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.70 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.70 => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 = ( ? [X: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 & ! [Y: extended_enat] :
% 0.35/0.70 ( ( member_Extended_enat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_1_iff
% 0.35/0.70 thf(fact_8749_sum__eq__1__iff,axiom,
% 0.35/0.70 ! [A3: set_nat,F: nat > nat] :
% 0.35/0.70 ( ( finite_finite_nat @ A3 )
% 0.35/0.70 => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 = ( ? [X: nat] :
% 0.35/0.70 ( ( member_nat @ X @ A3 )
% 0.35/0.70 & ( ( F @ X )
% 0.35/0.70 = one_one_nat )
% 0.35/0.70 & ! [Y: nat] :
% 0.35/0.70 ( ( member_nat @ Y @ A3 )
% 0.35/0.70 => ( ( X != Y )
% 0.35/0.70 => ( ( F @ Y )
% 0.35/0.70 = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_eq_1_iff
% 0.35/0.70 thf(fact_8750_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_o,T2: set_o,R: $o > $o > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_o @ S )
% 0.35/0.70 => ( ( finite_finite_o @ T2 )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [J3: $o] :
% 0.35/0.70 ( ( member_o @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8751_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_o,T2: set_real,R: $o > real > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_o @ S )
% 0.35/0.70 => ( ( finite_finite_real @ T2 )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [J3: real] :
% 0.35/0.70 ( ( member_real @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8752_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_real,T2: set_o,R: real > $o > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( finite_finite_o @ T2 )
% 0.35/0.70 => ( ! [X5: $o] :
% 0.35/0.70 ( ( member_o @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [J3: $o] :
% 0.35/0.70 ( ( member_o @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8753_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_real,T2: set_real,R: real > real > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( finite_finite_real @ T2 )
% 0.35/0.70 => ( ! [X5: real] :
% 0.35/0.70 ( ( member_real @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [J3: real] :
% 0.35/0.70 ( ( member_real @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8754_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_o,T2: set_nat,R: $o > nat > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_o @ S )
% 0.35/0.70 => ( ( finite_finite_nat @ T2 )
% 0.35/0.70 => ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_nat
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [J3: nat] :
% 0.35/0.70 ( ( member_nat @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8755_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_real,T2: set_nat,R: real > nat > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( finite_finite_nat @ T2 )
% 0.35/0.70 => ( ! [X5: nat] :
% 0.35/0.70 ( ( member_nat @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_nat
% 0.35/0.70 @ ( collect_nat
% 0.35/0.70 @ ^ [J3: nat] :
% 0.35/0.70 ( ( member_nat @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8756_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_o,T2: set_int,R: $o > int > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_o @ S )
% 0.35/0.70 => ( ( finite_finite_int @ T2 )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_int
% 0.35/0.70 @ ( collect_int
% 0.35/0.70 @ ^ [J3: int] :
% 0.35/0.70 ( ( member_int @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8757_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_real,T2: set_int,R: real > int > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( finite_finite_int @ T2 )
% 0.35/0.70 => ( ! [X5: int] :
% 0.35/0.70 ( ( member_int @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_int
% 0.35/0.70 @ ( collect_int
% 0.35/0.70 @ ^ [J3: int] :
% 0.35/0.70 ( ( member_int @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8758_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_o,T2: set_complex,R: $o > complex > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_o @ S )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_o
% 0.35/0.70 @ ( collect_o
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( ( member_o @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups8507830703676809646_o_nat
% 0.35/0.70 @ ^ [I4: $o] :
% 0.35/0.70 ( finite_card_complex
% 0.35/0.70 @ ( collect_complex
% 0.35/0.70 @ ^ [J3: complex] :
% 0.35/0.70 ( ( member_complex @ J3 @ T2 )
% 0.35/0.70 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.70 @ S )
% 0.35/0.70 = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).
% 0.35/0.70
% 0.35/0.70 % sum_multicount
% 0.35/0.70 thf(fact_8759_sum__multicount,axiom,
% 0.35/0.70 ! [S: set_real,T2: set_complex,R: real > complex > $o,K: nat] :
% 0.35/0.70 ( ( finite_finite_real @ S )
% 0.35/0.70 => ( ( finite3207457112153483333omplex @ T2 )
% 0.35/0.70 => ( ! [X5: complex] :
% 0.35/0.70 ( ( member_complex @ X5 @ T2 )
% 0.35/0.70 => ( ( finite_card_real
% 0.35/0.70 @ ( collect_real
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( ( member_real @ I4 @ S )
% 0.35/0.70 & ( R @ I4 @ X5 ) ) ) )
% 0.35/0.70 = K ) )
% 0.35/0.70 => ( ( groups1935376822645274424al_nat
% 0.35/0.70 @ ^ [I4: real] :
% 0.35/0.70 ( finite_card_complex
% 0.35/0.70 @ ( collect_complex
% 0.35/0.70 @ ^ [J3: complex] :
% 0.35/0.70 ( ( member_complex @ J3 @ T2 )
% 0.35/0.71 & ( R @ I4 @ J3 ) ) ) )
% 0.35/0.71 @ S )
% 0.35/0.71 = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_multicount
% 0.35/0.71 thf(fact_8760_sum__diff__nat,axiom,
% 0.35/0.71 ! [B2: set_complex,A3: set_complex,F: complex > nat] :
% 0.35/0.71 ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.71 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 0.35/0.71 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_nat
% 0.35/0.71 thf(fact_8761_sum__diff__nat,axiom,
% 0.35/0.71 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.71 ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.71 => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 0.35/0.71 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_nat
% 0.35/0.71 thf(fact_8762_sum__diff__nat,axiom,
% 0.35/0.71 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.71 ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.71 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 0.35/0.71 => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_nat
% 0.35/0.71 thf(fact_8763_sum__diff__nat,axiom,
% 0.35/0.71 ! [B2: set_int,A3: set_int,F: int > nat] :
% 0.35/0.71 ( ( finite_finite_int @ B2 )
% 0.35/0.71 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 0.35/0.71 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_nat
% 0.35/0.71 thf(fact_8764_sum__diff__nat,axiom,
% 0.35/0.71 ! [B2: set_nat,A3: set_nat,F: nat > nat] :
% 0.35/0.71 ( ( finite_finite_nat @ B2 )
% 0.35/0.71 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 0.35/0.71 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_nat
% 0.35/0.71 thf(fact_8765_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.71 ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.35/0.71 = ( groups977919841031483927at_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8766_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: set_nat,A3: set_set_nat,F: set_nat > nat] :
% 0.35/0.71 ( ( ( member_set_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member_set_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 0.35/0.71 = ( groups8294997508430121362at_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8767_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: real,A3: set_real,F: real > nat] :
% 0.35/0.71 ( ( ( member_real @ A2 @ A3 )
% 0.35/0.71 => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member_real @ A2 @ A3 )
% 0.35/0.71 => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 0.35/0.71 = ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8768_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: $o,A3: set_o,F: $o > nat] :
% 0.35/0.71 ( ( ( member_o @ A2 @ A3 )
% 0.35/0.71 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member_o @ A2 @ A3 )
% 0.35/0.71 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 0.35/0.71 = ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8769_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: int,A3: set_int,F: int > nat] :
% 0.35/0.71 ( ( ( member_int @ A2 @ A3 )
% 0.35/0.71 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member_int @ A2 @ A3 )
% 0.35/0.71 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 0.35/0.71 = ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8770_sum__diff1__nat,axiom,
% 0.35/0.71 ! [A2: nat,A3: set_nat,F: nat > nat] :
% 0.35/0.71 ( ( ( member_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.35/0.71 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 0.35/0.71 & ( ~ ( member_nat @ A2 @ A3 )
% 0.35/0.71 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 0.35/0.71 = ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff1_nat
% 0.35/0.71 thf(fact_8771_sum__nth__roots,axiom,
% 0.35/0.71 ! [N: nat,C: complex] :
% 0.35/0.71 ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.71 => ( ( groups7754918857620584856omplex
% 0.35/0.71 @ ^ [X: complex] : X
% 0.35/0.71 @ ( collect_complex
% 0.35/0.71 @ ^ [Z2: complex] :
% 0.35/0.71 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.71 = C ) ) )
% 0.35/0.71 = zero_zero_complex ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_nth_roots
% 0.35/0.71 thf(fact_8772_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_int,B2: set_int,F: int > nat] :
% 0.35/0.71 ( ( finite_finite_int @ A3 )
% 0.35/0.71 => ( ( finite_finite_int @ B2 )
% 0.35/0.71 => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8773_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_complex,B2: set_complex,F: complex > nat] :
% 0.35/0.71 ( ( finite3207457112153483333omplex @ A3 )
% 0.35/0.71 => ( ( finite3207457112153483333omplex @ B2 )
% 0.35/0.71 => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8774_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
% 0.35/0.71 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.71 => ( ( finite4001608067531595151d_enat @ B2 )
% 0.35/0.71 => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8775_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.35/0.71 ( ( finite6177210948735845034at_nat @ A3 )
% 0.35/0.71 => ( ( finite6177210948735845034at_nat @ B2 )
% 0.35/0.71 => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8776_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
% 0.35/0.71 ( ( finite4392333629123659920at_nat @ A3 )
% 0.35/0.71 => ( ( finite4392333629123659920at_nat @ B2 )
% 0.35/0.71 => ( ( groups1900718384385340925at_nat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups1900718384385340925at_nat @ F @ A3 ) @ ( groups1900718384385340925at_nat @ F @ B2 ) ) @ ( groups1900718384385340925at_nat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8777_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 0.35/0.71 ( ( finite4343798906461161616at_nat @ A3 )
% 0.35/0.71 => ( ( finite4343798906461161616at_nat @ B2 )
% 0.35/0.71 => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A3 ) @ ( groups3860910324918113789at_nat @ F @ B2 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8778_sum__Un__nat,axiom,
% 0.35/0.71 ! [A3: set_nat,B2: set_nat,F: nat > nat] :
% 0.35/0.71 ( ( finite_finite_nat @ A3 )
% 0.35/0.71 => ( ( finite_finite_nat @ B2 )
% 0.35/0.71 => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 0.35/0.71 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_Un_nat
% 0.35/0.71 thf(fact_8779_sum__roots__unity,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ one_one_nat @ N )
% 0.35/0.71 => ( ( groups7754918857620584856omplex
% 0.35/0.71 @ ^ [X: complex] : X
% 0.35/0.71 @ ( collect_complex
% 0.35/0.71 @ ^ [Z2: complex] :
% 0.35/0.71 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.71 = one_one_complex ) ) )
% 0.35/0.71 = zero_zero_complex ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_roots_unity
% 0.35/0.71 thf(fact_8780_divmod__divmod__step,axiom,
% 0.35/0.71 ( unique5055182867167087721od_nat
% 0.35/0.71 = ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % divmod_divmod_step
% 0.35/0.71 thf(fact_8781_divmod__divmod__step,axiom,
% 0.35/0.71 ( unique5052692396658037445od_int
% 0.35/0.71 = ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % divmod_divmod_step
% 0.35/0.71 thf(fact_8782_divmod__divmod__step,axiom,
% 0.35/0.71 ( unique3479559517661332726nteger
% 0.35/0.71 = ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % divmod_divmod_step
% 0.35/0.71 thf(fact_8783_Sum__Icc__int,axiom,
% 0.35/0.71 ! [M: int,N: int] :
% 0.35/0.71 ( ( ord_less_eq_int @ M @ N )
% 0.35/0.71 => ( ( groups4538972089207619220nt_int
% 0.35/0.71 @ ^ [X: int] : X
% 0.35/0.71 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 0.35/0.71 = ( 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 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Sum_Icc_int
% 0.35/0.71 thf(fact_8784_fold__atLeastAtMost__nat_Opsimps,axiom,
% 0.35/0.71 ! [F: nat > nat > nat,A2: nat,B3: nat,Acc2: nat] :
% 0.35/0.71 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) )
% 0.35/0.71 => ( ( ( ord_less_nat @ B3 @ A2 )
% 0.35/0.71 => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
% 0.35/0.71 = Acc2 ) )
% 0.35/0.71 & ( ~ ( ord_less_nat @ B3 @ A2 )
% 0.35/0.71 => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
% 0.35/0.71 = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B3 @ ( F @ A2 @ Acc2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fold_atLeastAtMost_nat.psimps
% 0.35/0.71 thf(fact_8785_fold__atLeastAtMost__nat_Opelims,axiom,
% 0.35/0.71 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 0.35/0.71 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 0.35/0.71 = Y3 )
% 0.35/0.71 => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
% 0.35/0.71 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 0.35/0.71 => ( Y3 = Xc ) )
% 0.35/0.71 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 0.35/0.71 => ( Y3
% 0.35/0.71 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 0.35/0.71 => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fold_atLeastAtMost_nat.pelims
% 0.35/0.71 thf(fact_8786_upto_Opinduct,axiom,
% 0.35/0.71 ! [A0: int,A13: int,P: int > int > $o] :
% 0.35/0.71 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
% 0.35/0.71 => ( ! [I2: int,J2: int] :
% 0.35/0.71 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 0.35/0.71 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 0.35/0.71 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 0.35/0.71 => ( P @ I2 @ J2 ) ) )
% 0.35/0.71 => ( P @ A0 @ A13 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % upto.pinduct
% 0.35/0.71 thf(fact_8787_lemma__termdiff2,axiom,
% 0.35/0.71 ! [H2: rat,Z: rat,N: nat] :
% 0.35/0.71 ( ( H2 != zero_zero_rat )
% 0.35/0.71 => ( ( 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 ) ) ) ) )
% 0.35/0.71 = ( times_times_rat @ H2
% 0.35/0.71 @ ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [P5: nat] :
% 0.35/0.71 ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [Q6: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q6 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lemma_termdiff2
% 0.35/0.71 thf(fact_8788_lemma__termdiff2,axiom,
% 0.35/0.71 ! [H2: complex,Z: complex,N: nat] :
% 0.35/0.71 ( ( H2 != zero_zero_complex )
% 0.35/0.71 => ( ( 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 ) ) ) ) )
% 0.35/0.71 = ( times_times_complex @ H2
% 0.35/0.71 @ ( groups2073611262835488442omplex
% 0.35/0.71 @ ^ [P5: nat] :
% 0.35/0.71 ( groups2073611262835488442omplex
% 0.35/0.71 @ ^ [Q6: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q6 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lemma_termdiff2
% 0.35/0.71 thf(fact_8789_lemma__termdiff2,axiom,
% 0.35/0.71 ! [H2: real,Z: real,N: nat] :
% 0.35/0.71 ( ( H2 != zero_zero_real )
% 0.35/0.71 => ( ( 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 ) ) ) ) )
% 0.35/0.71 = ( times_times_real @ H2
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [P5: nat] :
% 0.35/0.71 ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [Q6: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q6 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lemma_termdiff2
% 0.35/0.71 thf(fact_8790_diffs__equiv,axiom,
% 0.35/0.71 ! [C: nat > complex,X2: complex] :
% 0.35/0.71 ( ( summable_complex
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 0.35/0.71 => ( sums_complex
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.71 @ ( suminf_complex
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diffs_equiv
% 0.35/0.71 thf(fact_8791_diffs__equiv,axiom,
% 0.35/0.71 ! [C: nat > real,X2: real] :
% 0.35/0.71 ( ( summable_real
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 0.35/0.71 => ( sums_real
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.71 @ ( suminf_real
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diffs_equiv
% 0.35/0.71 thf(fact_8792_lessThan__eq__iff,axiom,
% 0.35/0.71 ! [X2: nat,Y3: nat] :
% 0.35/0.71 ( ( ( set_ord_lessThan_nat @ X2 )
% 0.35/0.71 = ( set_ord_lessThan_nat @ Y3 ) )
% 0.35/0.71 = ( X2 = Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_eq_iff
% 0.35/0.71 thf(fact_8793_lessThan__eq__iff,axiom,
% 0.35/0.71 ! [X2: int,Y3: int] :
% 0.35/0.71 ( ( ( set_ord_lessThan_int @ X2 )
% 0.35/0.71 = ( set_ord_lessThan_int @ Y3 ) )
% 0.35/0.71 = ( X2 = Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_eq_iff
% 0.35/0.71 thf(fact_8794_lessThan__eq__iff,axiom,
% 0.35/0.71 ! [X2: real,Y3: real] :
% 0.35/0.71 ( ( ( set_or5984915006950818249n_real @ X2 )
% 0.35/0.71 = ( set_or5984915006950818249n_real @ Y3 ) )
% 0.35/0.71 = ( X2 = Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_eq_iff
% 0.35/0.71 thf(fact_8795_lessThan__iff,axiom,
% 0.35/0.71 ! [I: $o,K: $o] :
% 0.35/0.71 ( ( member_o @ I @ ( set_ord_lessThan_o @ K ) )
% 0.35/0.71 = ( ord_less_o @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8796_lessThan__iff,axiom,
% 0.35/0.71 ! [I: set_nat,K: set_nat] :
% 0.35/0.71 ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 0.35/0.71 = ( ord_less_set_nat @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8797_lessThan__iff,axiom,
% 0.35/0.71 ! [I: rat,K: rat] :
% 0.35/0.71 ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 0.35/0.71 = ( ord_less_rat @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8798_lessThan__iff,axiom,
% 0.35/0.71 ! [I: num,K: num] :
% 0.35/0.71 ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 0.35/0.71 = ( ord_less_num @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8799_lessThan__iff,axiom,
% 0.35/0.71 ! [I: nat,K: nat] :
% 0.35/0.71 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.35/0.71 = ( ord_less_nat @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8800_lessThan__iff,axiom,
% 0.35/0.71 ! [I: int,K: int] :
% 0.35/0.71 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.35/0.71 = ( ord_less_int @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8801_lessThan__iff,axiom,
% 0.35/0.71 ! [I: real,K: real] :
% 0.35/0.71 ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 0.35/0.71 = ( ord_less_real @ I @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_iff
% 0.35/0.71 thf(fact_8802_finite__lessThan,axiom,
% 0.35/0.71 ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % finite_lessThan
% 0.35/0.71 thf(fact_8803_card__lessThan,axiom,
% 0.35/0.71 ! [U: nat] :
% 0.35/0.71 ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
% 0.35/0.71 = U ) ).
% 0.35/0.71
% 0.35/0.71 % card_lessThan
% 0.35/0.71 thf(fact_8804_lessThan__subset__iff,axiom,
% 0.35/0.71 ! [X2: rat,Y3: rat] :
% 0.35/0.71 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 0.35/0.71 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_subset_iff
% 0.35/0.71 thf(fact_8805_lessThan__subset__iff,axiom,
% 0.35/0.71 ! [X2: num,Y3: num] :
% 0.35/0.71 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 0.35/0.71 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_subset_iff
% 0.35/0.71 thf(fact_8806_lessThan__subset__iff,axiom,
% 0.35/0.71 ! [X2: nat,Y3: nat] :
% 0.35/0.71 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.35/0.71 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_subset_iff
% 0.35/0.71 thf(fact_8807_lessThan__subset__iff,axiom,
% 0.35/0.71 ! [X2: int,Y3: int] :
% 0.35/0.71 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 0.35/0.71 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_subset_iff
% 0.35/0.71 thf(fact_8808_lessThan__subset__iff,axiom,
% 0.35/0.71 ! [X2: real,Y3: real] :
% 0.35/0.71 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 0.35/0.71 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_subset_iff
% 0.35/0.71 thf(fact_8809_lessThan__0,axiom,
% 0.35/0.71 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.35/0.71 = bot_bot_set_nat ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_0
% 0.35/0.71 thf(fact_8810_sum_OlessThan__Suc,axiom,
% 0.35/0.71 ! [G3: nat > rat,N: nat] :
% 0.35/0.71 ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc
% 0.35/0.71 thf(fact_8811_sum_OlessThan__Suc,axiom,
% 0.35/0.71 ! [G3: nat > int,N: nat] :
% 0.35/0.71 ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc
% 0.35/0.71 thf(fact_8812_sum_OlessThan__Suc,axiom,
% 0.35/0.71 ! [G3: nat > nat,N: nat] :
% 0.35/0.71 ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc
% 0.35/0.71 thf(fact_8813_sum_OlessThan__Suc,axiom,
% 0.35/0.71 ! [G3: nat > real,N: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc
% 0.35/0.71 thf(fact_8814_single__Diff__lessThan,axiom,
% 0.35/0.71 ! [K: $o] :
% 0.35/0.71 ( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
% 0.35/0.71 = ( insert_o @ K @ bot_bot_set_o ) ) ).
% 0.35/0.71
% 0.35/0.71 % single_Diff_lessThan
% 0.35/0.71 thf(fact_8815_single__Diff__lessThan,axiom,
% 0.35/0.71 ! [K: nat] :
% 0.35/0.71 ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 0.35/0.71 = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % single_Diff_lessThan
% 0.35/0.71 thf(fact_8816_single__Diff__lessThan,axiom,
% 0.35/0.71 ! [K: int] :
% 0.35/0.71 ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 0.35/0.71 = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 0.35/0.71
% 0.35/0.71 % single_Diff_lessThan
% 0.35/0.71 thf(fact_8817_single__Diff__lessThan,axiom,
% 0.35/0.71 ! [K: real] :
% 0.35/0.71 ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 0.35/0.71 = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 0.35/0.71
% 0.35/0.71 % single_Diff_lessThan
% 0.35/0.71 thf(fact_8818_lessThan__non__empty,axiom,
% 0.35/0.71 ! [X2: int] :
% 0.35/0.71 ( ( set_ord_lessThan_int @ X2 )
% 0.35/0.71 != bot_bot_set_int ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_non_empty
% 0.35/0.71 thf(fact_8819_lessThan__non__empty,axiom,
% 0.35/0.71 ! [X2: real] :
% 0.35/0.71 ( ( set_or5984915006950818249n_real @ X2 )
% 0.35/0.71 != bot_bot_set_real ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_non_empty
% 0.35/0.71 thf(fact_8820_infinite__Iio,axiom,
% 0.35/0.71 ! [A2: int] :
% 0.35/0.71 ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A2 ) ) ).
% 0.35/0.71
% 0.35/0.71 % infinite_Iio
% 0.35/0.71 thf(fact_8821_infinite__Iio,axiom,
% 0.35/0.71 ! [A2: real] :
% 0.35/0.71 ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A2 ) ) ).
% 0.35/0.71
% 0.35/0.71 % infinite_Iio
% 0.35/0.71 thf(fact_8822_lessThan__def,axiom,
% 0.35/0.71 ( set_or890127255671739683et_nat
% 0.35/0.71 = ( ^ [U2: set_nat] :
% 0.35/0.71 ( collect_set_nat
% 0.35/0.71 @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8823_lessThan__def,axiom,
% 0.35/0.71 ( set_ord_lessThan_rat
% 0.35/0.71 = ( ^ [U2: rat] :
% 0.35/0.71 ( collect_rat
% 0.35/0.71 @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8824_lessThan__def,axiom,
% 0.35/0.71 ( set_ord_lessThan_num
% 0.35/0.71 = ( ^ [U2: num] :
% 0.35/0.71 ( collect_num
% 0.35/0.71 @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8825_lessThan__def,axiom,
% 0.35/0.71 ( set_ord_lessThan_nat
% 0.35/0.71 = ( ^ [U2: nat] :
% 0.35/0.71 ( collect_nat
% 0.35/0.71 @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8826_lessThan__def,axiom,
% 0.35/0.71 ( set_ord_lessThan_int
% 0.35/0.71 = ( ^ [U2: int] :
% 0.35/0.71 ( collect_int
% 0.35/0.71 @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8827_lessThan__def,axiom,
% 0.35/0.71 ( set_or5984915006950818249n_real
% 0.35/0.71 = ( ^ [U2: real] :
% 0.35/0.71 ( collect_real
% 0.35/0.71 @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_def
% 0.35/0.71 thf(fact_8828_Iio__eq__empty__iff,axiom,
% 0.35/0.71 ! [N: $o] :
% 0.35/0.71 ( ( ( set_ord_lessThan_o @ N )
% 0.35/0.71 = bot_bot_set_o )
% 0.35/0.71 = ( N = bot_bot_o ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_eq_empty_iff
% 0.35/0.71 thf(fact_8829_Iio__eq__empty__iff,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( ( set_ord_lessThan_nat @ N )
% 0.35/0.71 = bot_bot_set_nat )
% 0.35/0.71 = ( N = bot_bot_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_eq_empty_iff
% 0.35/0.71 thf(fact_8830_lessThan__strict__subset__iff,axiom,
% 0.35/0.71 ! [M: rat,N: rat] :
% 0.35/0.71 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 0.35/0.71 = ( ord_less_rat @ M @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_strict_subset_iff
% 0.35/0.71 thf(fact_8831_lessThan__strict__subset__iff,axiom,
% 0.35/0.71 ! [M: num,N: num] :
% 0.35/0.71 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 0.35/0.71 = ( ord_less_num @ M @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_strict_subset_iff
% 0.35/0.71 thf(fact_8832_lessThan__strict__subset__iff,axiom,
% 0.35/0.71 ! [M: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_strict_subset_iff
% 0.35/0.71 thf(fact_8833_lessThan__strict__subset__iff,axiom,
% 0.35/0.71 ! [M: int,N: int] :
% 0.35/0.71 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 0.35/0.71 = ( ord_less_int @ M @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_strict_subset_iff
% 0.35/0.71 thf(fact_8834_lessThan__strict__subset__iff,axiom,
% 0.35/0.71 ! [M: real,N: real] :
% 0.35/0.71 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 0.35/0.71 = ( ord_less_real @ M @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_strict_subset_iff
% 0.35/0.71 thf(fact_8835_lessThan__Suc,axiom,
% 0.35/0.71 ! [K: nat] :
% 0.35/0.71 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.35/0.71 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_Suc
% 0.35/0.71 thf(fact_8836_lessThan__empty__iff,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( ( set_ord_lessThan_nat @ N )
% 0.35/0.71 = bot_bot_set_nat )
% 0.35/0.71 = ( N = zero_zero_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % lessThan_empty_iff
% 0.35/0.71 thf(fact_8837_finite__nat__bounded,axiom,
% 0.35/0.71 ! [S: set_nat] :
% 0.35/0.71 ( ( finite_finite_nat @ S )
% 0.35/0.71 => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % finite_nat_bounded
% 0.35/0.71 thf(fact_8838_finite__nat__iff__bounded,axiom,
% 0.35/0.71 ( finite_finite_nat
% 0.35/0.71 = ( ^ [S6: set_nat] :
% 0.35/0.71 ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % finite_nat_iff_bounded
% 0.35/0.71 thf(fact_8839_ivl__disj__int__one_I4_J,axiom,
% 0.35/0.71 ! [L: $o,U: $o] :
% 0.35/0.71 ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
% 0.35/0.71 = bot_bot_set_o ) ).
% 0.35/0.71
% 0.35/0.71 % ivl_disj_int_one(4)
% 0.35/0.71 thf(fact_8840_ivl__disj__int__one_I4_J,axiom,
% 0.35/0.71 ! [L: nat,U: nat] :
% 0.35/0.71 ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.35/0.71 = bot_bot_set_nat ) ).
% 0.35/0.71
% 0.35/0.71 % ivl_disj_int_one(4)
% 0.35/0.71 thf(fact_8841_ivl__disj__int__one_I4_J,axiom,
% 0.35/0.71 ! [L: int,U: int] :
% 0.35/0.71 ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.35/0.71 = bot_bot_set_int ) ).
% 0.35/0.71
% 0.35/0.71 % ivl_disj_int_one(4)
% 0.35/0.71 thf(fact_8842_ivl__disj__int__one_I4_J,axiom,
% 0.35/0.71 ! [L: real,U: real] :
% 0.35/0.71 ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ L ) @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.35/0.71 = bot_bot_set_real ) ).
% 0.35/0.71
% 0.35/0.71 % ivl_disj_int_one(4)
% 0.35/0.71 thf(fact_8843_sum_Onat__diff__reindex,axiom,
% 0.35/0.71 ! [G3: nat > nat,N: nat] :
% 0.35/0.71 ( ( groups3542108847815614940at_nat
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.nat_diff_reindex
% 0.35/0.71 thf(fact_8844_sum_Onat__diff__reindex,axiom,
% 0.35/0.71 ! [G3: nat > real,N: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.nat_diff_reindex
% 0.35/0.71 thf(fact_8845_sum__diff__distrib,axiom,
% 0.35/0.71 ! [Q: int > nat,P: int > nat,N: int] :
% 0.35/0.71 ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.35/0.71 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 0.35/0.71 = ( groups4541462559716669496nt_nat
% 0.35/0.71 @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.35/0.71 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_distrib
% 0.35/0.71 thf(fact_8846_sum__diff__distrib,axiom,
% 0.35/0.71 ! [Q: real > nat,P: real > nat,N: real] :
% 0.35/0.71 ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.35/0.71 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 0.35/0.71 = ( groups1935376822645274424al_nat
% 0.35/0.71 @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.35/0.71 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_distrib
% 0.35/0.71 thf(fact_8847_sum__diff__distrib,axiom,
% 0.35/0.71 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 0.35/0.71 ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 0.35/0.71 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 0.35/0.71 = ( groups3542108847815614940at_nat
% 0.35/0.71 @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_diff_distrib
% 0.35/0.71 thf(fact_8848_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: $o,K: $o] :
% 0.35/0.71 ( ( ( ord_less_o @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.71 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 0.35/0.71 & ( ~ ( ord_less_o @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 0.35/0.71 = bot_bot_set_o ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8849_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: rat,K: rat] :
% 0.35/0.71 ( ( ( ord_less_rat @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
% 0.35/0.71 = ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
% 0.35/0.71 & ( ~ ( ord_less_rat @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
% 0.35/0.71 = bot_bot_set_rat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8850_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: num,K: num] :
% 0.35/0.71 ( ( ( ord_less_num @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
% 0.35/0.71 = ( insert_num @ X2 @ bot_bot_set_num ) ) )
% 0.35/0.71 & ( ~ ( ord_less_num @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
% 0.35/0.71 = bot_bot_set_num ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8851_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: nat,K: nat] :
% 0.35/0.71 ( ( ( ord_less_nat @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.71 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.35/0.71 & ( ~ ( ord_less_nat @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 0.35/0.71 = bot_bot_set_nat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8852_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: int,K: int] :
% 0.35/0.71 ( ( ( ord_less_int @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.71 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.35/0.71 & ( ~ ( ord_less_int @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 0.35/0.71 = bot_bot_set_int ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8853_Iio__Int__singleton,axiom,
% 0.35/0.71 ! [X2: real,K: real] :
% 0.35/0.71 ( ( ( ord_less_real @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.71 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.35/0.71 & ( ~ ( ord_less_real @ X2 @ K )
% 0.35/0.71 => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 0.35/0.71 = bot_bot_set_real ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Iio_Int_singleton
% 0.35/0.71 thf(fact_8854_suminf__le__const,axiom,
% 0.35/0.71 ! [F: nat > int,X2: int] :
% 0.35/0.71 ( ( summable_int @ F )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % suminf_le_const
% 0.35/0.71 thf(fact_8855_suminf__le__const,axiom,
% 0.35/0.71 ! [F: nat > nat,X2: nat] :
% 0.35/0.71 ( ( summable_nat @ F )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % suminf_le_const
% 0.35/0.71 thf(fact_8856_suminf__le__const,axiom,
% 0.35/0.71 ! [F: nat > real,X2: real] :
% 0.35/0.71 ( ( summable_real @ F )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % suminf_le_const
% 0.35/0.71 thf(fact_8857_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.71 ! [G3: nat > rat,N: nat] :
% 0.35/0.71 ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_rat @ ( G3 @ zero_zero_nat )
% 0.35/0.71 @ ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc_shift
% 0.35/0.71 thf(fact_8858_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.71 ! [G3: nat > int,N: nat] :
% 0.35/0.71 ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_int @ ( G3 @ zero_zero_nat )
% 0.35/0.71 @ ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc_shift
% 0.35/0.71 thf(fact_8859_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.71 ! [G3: nat > nat,N: nat] :
% 0.35/0.71 ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_nat @ ( G3 @ zero_zero_nat )
% 0.35/0.71 @ ( groups3542108847815614940at_nat
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc_shift
% 0.35/0.71 thf(fact_8860_sum_OlessThan__Suc__shift,axiom,
% 0.35/0.71 ! [G3: nat > real,N: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 0.35/0.71 = ( plus_plus_real @ ( G3 @ zero_zero_nat )
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.lessThan_Suc_shift
% 0.35/0.71 thf(fact_8861_sum__lessThan__telescope_H,axiom,
% 0.35/0.71 ! [F: nat > rat,M: nat] :
% 0.35/0.71 ( ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope'
% 0.35/0.71 thf(fact_8862_sum__lessThan__telescope_H,axiom,
% 0.35/0.71 ! [F: nat > int,M: nat] :
% 0.35/0.71 ( ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope'
% 0.35/0.71 thf(fact_8863_sum__lessThan__telescope_H,axiom,
% 0.35/0.71 ! [F: nat > real,M: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope'
% 0.35/0.71 thf(fact_8864_sum__lessThan__telescope,axiom,
% 0.35/0.71 ! [F: nat > rat,M: nat] :
% 0.35/0.71 ( ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope
% 0.35/0.71 thf(fact_8865_sum__lessThan__telescope,axiom,
% 0.35/0.71 ! [F: nat > int,M: nat] :
% 0.35/0.71 ( ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope
% 0.35/0.71 thf(fact_8866_sum__lessThan__telescope,axiom,
% 0.35/0.71 ! [F: nat > real,M: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ M ) )
% 0.35/0.71 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_lessThan_telescope
% 0.35/0.71 thf(fact_8867_summableI__nonneg__bounded,axiom,
% 0.35/0.71 ! [F: nat > int,X2: int] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( summable_int @ F ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % summableI_nonneg_bounded
% 0.35/0.71 thf(fact_8868_summableI__nonneg__bounded,axiom,
% 0.35/0.71 ! [F: nat > nat,X2: nat] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( summable_nat @ F ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % summableI_nonneg_bounded
% 0.35/0.71 thf(fact_8869_summableI__nonneg__bounded,axiom,
% 0.35/0.71 ! [F: nat > real,X2: real] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.35/0.71 => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 0.35/0.71 => ( summable_real @ F ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % summableI_nonneg_bounded
% 0.35/0.71 thf(fact_8870_sum_OatLeast1__atMost__eq,axiom,
% 0.35/0.71 ! [G3: nat > nat,N: nat] :
% 0.35/0.71 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.71 = ( groups3542108847815614940at_nat
% 0.35/0.71 @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.atLeast1_atMost_eq
% 0.35/0.71 thf(fact_8871_sum_OatLeast1__atMost__eq,axiom,
% 0.35/0.71 ! [G3: nat > real,N: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.71 = ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum.atLeast1_atMost_eq
% 0.35/0.71 thf(fact_8872_fold__atLeastAtMost__nat_Osimps,axiom,
% 0.35/0.71 ( set_fo2584398358068434914at_nat
% 0.35/0.71 = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc3 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fold_atLeastAtMost_nat.simps
% 0.35/0.71 thf(fact_8873_fold__atLeastAtMost__nat_Oelims,axiom,
% 0.35/0.71 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 0.35/0.71 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 0.35/0.71 = Y3 )
% 0.35/0.71 => ( ( ( ord_less_nat @ Xb @ Xa2 )
% 0.35/0.71 => ( Y3 = Xc ) )
% 0.35/0.71 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 0.35/0.71 => ( Y3
% 0.35/0.71 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fold_atLeastAtMost_nat.elims
% 0.35/0.71 thf(fact_8874_finite__enumerate__initial__segment,axiom,
% 0.35/0.71 ! [S: set_Extended_enat,N: nat,S2: extended_enat] :
% 0.35/0.71 ( ( finite4001608067531595151d_enat @ S )
% 0.35/0.71 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) ) )
% 0.35/0.71 => ( ( infini7641415182203889163d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) @ N )
% 0.35/0.71 = ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % finite_enumerate_initial_segment
% 0.35/0.71 thf(fact_8875_finite__enumerate__initial__segment,axiom,
% 0.35/0.71 ! [S: set_nat,N: nat,S2: nat] :
% 0.35/0.71 ( ( finite_finite_nat @ S )
% 0.35/0.71 => ( ( ord_less_nat @ N @ ( finite_card_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) ) )
% 0.35/0.71 => ( ( infini8530281810654367211te_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) @ N )
% 0.35/0.71 = ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % finite_enumerate_initial_segment
% 0.35/0.71 thf(fact_8876_power__diff__1__eq,axiom,
% 0.35/0.71 ! [X2: complex,N: nat] :
% 0.35/0.71 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex )
% 0.35/0.71 = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_1_eq
% 0.35/0.71 thf(fact_8877_power__diff__1__eq,axiom,
% 0.35/0.71 ! [X2: rat,N: nat] :
% 0.35/0.71 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat )
% 0.35/0.71 = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_1_eq
% 0.35/0.71 thf(fact_8878_power__diff__1__eq,axiom,
% 0.35/0.71 ! [X2: int,N: nat] :
% 0.35/0.71 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
% 0.35/0.71 = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_1_eq
% 0.35/0.71 thf(fact_8879_power__diff__1__eq,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
% 0.35/0.71 = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_1_eq
% 0.35/0.71 thf(fact_8880_one__diff__power__eq,axiom,
% 0.35/0.71 ! [X2: complex,N: nat] :
% 0.35/0.71 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 0.35/0.71 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq
% 0.35/0.71 thf(fact_8881_one__diff__power__eq,axiom,
% 0.35/0.71 ! [X2: rat,N: nat] :
% 0.35/0.71 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 0.35/0.71 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq
% 0.35/0.71 thf(fact_8882_one__diff__power__eq,axiom,
% 0.35/0.71 ! [X2: int,N: nat] :
% 0.35/0.71 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 0.35/0.71 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq
% 0.35/0.71 thf(fact_8883_one__diff__power__eq,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 0.35/0.71 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq
% 0.35/0.71 thf(fact_8884_geometric__sum,axiom,
% 0.35/0.71 ! [X2: rat,N: nat] :
% 0.35/0.71 ( ( X2 != one_one_rat )
% 0.35/0.71 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % geometric_sum
% 0.35/0.71 thf(fact_8885_geometric__sum,axiom,
% 0.35/0.71 ! [X2: complex,N: nat] :
% 0.35/0.71 ( ( X2 != one_one_complex )
% 0.35/0.71 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % geometric_sum
% 0.35/0.71 thf(fact_8886_geometric__sum,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( X2 != one_one_real )
% 0.35/0.71 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % geometric_sum
% 0.35/0.71 thf(fact_8887_sum__less__suminf,axiom,
% 0.35/0.71 ! [F: nat > int,N: nat] :
% 0.35/0.71 ( ( summable_int @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf
% 0.35/0.71 thf(fact_8888_sum__less__suminf,axiom,
% 0.35/0.71 ! [F: nat > nat,N: nat] :
% 0.35/0.71 ( ( summable_nat @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf
% 0.35/0.71 thf(fact_8889_sum__less__suminf,axiom,
% 0.35/0.71 ! [F: nat > real,N: nat] :
% 0.35/0.71 ( ( summable_real @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf
% 0.35/0.71 thf(fact_8890_sum__gp__strict,axiom,
% 0.35/0.71 ! [X2: rat,N: nat] :
% 0.35/0.71 ( ( ( X2 = one_one_rat )
% 0.35/0.71 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( semiri681578069525770553at_rat @ N ) ) )
% 0.35/0.71 & ( ( X2 != one_one_rat )
% 0.35/0.71 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_gp_strict
% 0.35/0.71 thf(fact_8891_sum__gp__strict,axiom,
% 0.35/0.71 ! [X2: complex,N: nat] :
% 0.35/0.71 ( ( ( X2 = one_one_complex )
% 0.35/0.71 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( semiri8010041392384452111omplex @ N ) ) )
% 0.35/0.71 & ( ( X2 != one_one_complex )
% 0.35/0.71 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_gp_strict
% 0.35/0.71 thf(fact_8892_sum__gp__strict,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( ( X2 = one_one_real )
% 0.35/0.71 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.71 & ( ( X2 != one_one_real )
% 0.35/0.71 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_gp_strict
% 0.35/0.71 thf(fact_8893_diff__power__eq__sum,axiom,
% 0.35/0.71 ! [X2: complex,N: nat,Y3: complex] :
% 0.35/0.71 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 0.35/0.71 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 0.35/0.71 @ ( groups2073611262835488442omplex
% 0.35/0.71 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diff_power_eq_sum
% 0.35/0.71 thf(fact_8894_diff__power__eq__sum,axiom,
% 0.35/0.71 ! [X2: rat,N: nat,Y3: rat] :
% 0.35/0.71 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 0.35/0.71 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 0.35/0.71 @ ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diff_power_eq_sum
% 0.35/0.71 thf(fact_8895_diff__power__eq__sum,axiom,
% 0.35/0.71 ! [X2: int,N: nat,Y3: int] :
% 0.35/0.71 ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 0.35/0.71 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 0.35/0.71 @ ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X2 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diff_power_eq_sum
% 0.35/0.71 thf(fact_8896_diff__power__eq__sum,axiom,
% 0.35/0.71 ! [X2: real,N: nat,Y3: real] :
% 0.35/0.71 ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 0.35/0.71 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X2 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % diff_power_eq_sum
% 0.35/0.71 thf(fact_8897_power__diff__sumr2,axiom,
% 0.35/0.71 ! [X2: complex,N: nat,Y3: complex] :
% 0.35/0.71 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 0.35/0.71 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 0.35/0.71 @ ( groups2073611262835488442omplex
% 0.35/0.71 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X2 @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_sumr2
% 0.35/0.71 thf(fact_8898_power__diff__sumr2,axiom,
% 0.35/0.71 ! [X2: rat,N: nat,Y3: rat] :
% 0.35/0.71 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 0.35/0.71 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 0.35/0.71 @ ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X2 @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_sumr2
% 0.35/0.71 thf(fact_8899_power__diff__sumr2,axiom,
% 0.35/0.71 ! [X2: int,N: nat,Y3: int] :
% 0.35/0.71 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 0.35/0.71 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 0.35/0.71 @ ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X2 @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_sumr2
% 0.35/0.71 thf(fact_8900_power__diff__sumr2,axiom,
% 0.35/0.71 ! [X2: real,N: nat,Y3: real] :
% 0.35/0.71 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 0.35/0.71 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X2 @ I4 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % power_diff_sumr2
% 0.35/0.71 thf(fact_8901_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.71 ! [N: nat,F: nat > rat,K4: rat,K: nat] :
% 0.35/0.71 ( ! [P7: nat] :
% 0.35/0.71 ( ( ord_less_nat @ P7 @ N )
% 0.35/0.71 => ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
% 0.35/0.71 => ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
% 0.35/0.71 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_sum_nat_ivl_bounded2
% 0.35/0.71 thf(fact_8902_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.71 ! [N: nat,F: nat > int,K4: int,K: nat] :
% 0.35/0.71 ( ! [P7: nat] :
% 0.35/0.71 ( ( ord_less_nat @ P7 @ N )
% 0.35/0.71 => ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
% 0.35/0.71 => ( ( ord_less_eq_int @ zero_zero_int @ K4 )
% 0.35/0.71 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_sum_nat_ivl_bounded2
% 0.35/0.71 thf(fact_8903_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.71 ! [N: nat,F: nat > nat,K4: nat,K: nat] :
% 0.35/0.71 ( ! [P7: nat] :
% 0.35/0.71 ( ( ord_less_nat @ P7 @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_sum_nat_ivl_bounded2
% 0.35/0.71 thf(fact_8904_real__sum__nat__ivl__bounded2,axiom,
% 0.35/0.71 ! [N: nat,F: nat > real,K4: real,K: nat] :
% 0.35/0.71 ( ! [P7: nat] :
% 0.35/0.71 ( ( ord_less_nat @ P7 @ N )
% 0.35/0.71 => ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
% 0.35/0.71 => ( ( ord_less_eq_real @ zero_zero_real @ K4 )
% 0.35/0.71 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_sum_nat_ivl_bounded2
% 0.35/0.71 thf(fact_8905_sum__less__suminf2,axiom,
% 0.35/0.71 ! [F: nat > int,N: nat,I: nat] :
% 0.35/0.71 ( ( summable_int @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.71 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.35/0.71 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf2
% 0.35/0.71 thf(fact_8906_sum__less__suminf2,axiom,
% 0.35/0.71 ! [F: nat > nat,N: nat,I: nat] :
% 0.35/0.71 ( ( summable_nat @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.71 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.35/0.71 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf2
% 0.35/0.71 thf(fact_8907_sum__less__suminf2,axiom,
% 0.35/0.71 ! [F: nat > real,N: nat,I: nat] :
% 0.35/0.71 ( ( summable_real @ F )
% 0.35/0.71 => ( ! [M4: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ N @ M4 )
% 0.35/0.71 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ N @ I )
% 0.35/0.71 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.35/0.71 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_less_suminf2
% 0.35/0.71 thf(fact_8908_sum__atLeastAtMost__code,axiom,
% 0.35/0.71 ! [F: nat > rat,A2: nat,B3: nat] :
% 0.35/0.71 ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.71 = ( set_fo1949268297981939178at_rat
% 0.35/0.71 @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
% 0.35/0.71 @ A2
% 0.35/0.71 @ B3
% 0.35/0.71 @ zero_zero_rat ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_atLeastAtMost_code
% 0.35/0.71 thf(fact_8909_sum__atLeastAtMost__code,axiom,
% 0.35/0.71 ! [F: nat > int,A2: nat,B3: nat] :
% 0.35/0.71 ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.71 = ( set_fo2581907887559384638at_int
% 0.35/0.71 @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
% 0.35/0.71 @ A2
% 0.35/0.71 @ B3
% 0.35/0.71 @ zero_zero_int ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_atLeastAtMost_code
% 0.35/0.71 thf(fact_8910_sum__atLeastAtMost__code,axiom,
% 0.35/0.71 ! [F: nat > nat,A2: nat,B3: nat] :
% 0.35/0.71 ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.71 = ( set_fo2584398358068434914at_nat
% 0.35/0.71 @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
% 0.35/0.71 @ A2
% 0.35/0.71 @ B3
% 0.35/0.71 @ zero_zero_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_atLeastAtMost_code
% 0.35/0.71 thf(fact_8911_sum__atLeastAtMost__code,axiom,
% 0.35/0.71 ! [F: nat > real,A2: nat,B3: nat] :
% 0.35/0.71 ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.71 = ( set_fo3111899725591712190t_real
% 0.35/0.71 @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
% 0.35/0.71 @ A2
% 0.35/0.71 @ B3
% 0.35/0.71 @ zero_zero_real ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_atLeastAtMost_code
% 0.35/0.71 thf(fact_8912_one__diff__power__eq_H,axiom,
% 0.35/0.71 ! [X2: complex,N: nat] :
% 0.35/0.71 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 0.35/0.71 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
% 0.35/0.71 @ ( groups2073611262835488442omplex
% 0.35/0.71 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq'
% 0.35/0.71 thf(fact_8913_one__diff__power__eq_H,axiom,
% 0.35/0.71 ! [X2: rat,N: nat] :
% 0.35/0.71 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 0.35/0.71 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
% 0.35/0.71 @ ( groups2906978787729119204at_rat
% 0.35/0.71 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq'
% 0.35/0.71 thf(fact_8914_one__diff__power__eq_H,axiom,
% 0.35/0.71 ! [X2: int,N: nat] :
% 0.35/0.71 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 0.35/0.71 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
% 0.35/0.71 @ ( groups3539618377306564664at_int
% 0.35/0.71 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq'
% 0.35/0.71 thf(fact_8915_one__diff__power__eq_H,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 0.35/0.71 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % one_diff_power_eq'
% 0.35/0.71 thf(fact_8916_termdiff__converges,axiom,
% 0.35/0.71 ! [X2: real,K4: real,C: nat > real] :
% 0.35/0.71 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K4 )
% 0.35/0.71 => ( ! [X5: real] :
% 0.35/0.71 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K4 )
% 0.35/0.71 => ( summable_real
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 0.35/0.71 => ( summable_real
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % termdiff_converges
% 0.35/0.71 thf(fact_8917_termdiff__converges,axiom,
% 0.35/0.71 ! [X2: complex,K4: real,C: nat > complex] :
% 0.35/0.71 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K4 )
% 0.35/0.71 => ( ! [X5: complex] :
% 0.35/0.71 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K4 )
% 0.35/0.71 => ( summable_complex
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
% 0.35/0.71 => ( summable_complex
% 0.35/0.71 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % termdiff_converges
% 0.35/0.71 thf(fact_8918_sum__pos__lt__pair,axiom,
% 0.35/0.71 ! [F: nat > real,K: nat] :
% 0.35/0.71 ( ( summable_real @ F )
% 0.35/0.71 => ( ! [D6: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) @ one_one_nat ) ) ) ) )
% 0.35/0.71 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % sum_pos_lt_pair
% 0.35/0.71 thf(fact_8919_in__measure,axiom,
% 0.35/0.71 ! [X2: option_nat,Y3: option_nat,F: option_nat > nat] :
% 0.35/0.71 ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X2 @ Y3 ) @ ( measure_option_nat @ F ) )
% 0.35/0.71 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_measure
% 0.35/0.71 thf(fact_8920_in__measure,axiom,
% 0.35/0.71 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,F: set_Pr4329608150637261639at_nat > nat] :
% 0.35/0.71 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ ( measur4922264856574889999at_nat @ F ) )
% 0.35/0.71 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_measure
% 0.35/0.71 thf(fact_8921_in__measure,axiom,
% 0.35/0.71 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
% 0.35/0.71 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( measur1827424007717751593at_nat @ F ) )
% 0.35/0.71 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_measure
% 0.35/0.71 thf(fact_8922_in__measure,axiom,
% 0.35/0.71 ! [X2: nat,Y3: nat,F: nat > nat] :
% 0.35/0.71 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measure_nat @ F ) )
% 0.35/0.71 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_measure
% 0.35/0.71 thf(fact_8923_in__measure,axiom,
% 0.35/0.71 ! [X2: int,Y3: int,F: int > nat] :
% 0.35/0.71 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measure_int @ F ) )
% 0.35/0.71 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_measure
% 0.35/0.71 thf(fact_8924_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_nat,B2: set_nat] :
% 0.35/0.71 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B2 ) @ finite_psubset_nat )
% 0.35/0.71 = ( ( ord_less_set_nat @ A3 @ B2 )
% 0.35/0.71 & ( finite_finite_nat @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8925_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_int,B2: set_int] :
% 0.35/0.71 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A3 @ B2 ) @ finite_psubset_int )
% 0.35/0.71 = ( ( ord_less_set_int @ A3 @ B2 )
% 0.35/0.71 & ( finite_finite_int @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8926_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_complex,B2: set_complex] :
% 0.35/0.71 ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A3 @ B2 ) @ finite8643634255014194347omplex )
% 0.35/0.71 = ( ( ord_less_set_complex @ A3 @ B2 )
% 0.35/0.71 & ( finite3207457112153483333omplex @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8927_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 0.35/0.71 ( ( member4453595087596390480d_enat @ ( produc6639060556116774935d_enat @ A3 @ B2 ) @ finite4251489430341359785d_enat )
% 0.35/0.71 = ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
% 0.35/0.71 & ( finite4001608067531595151d_enat @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8928_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.35/0.71 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A3 @ B2 ) @ finite4695646753290404266at_nat )
% 0.35/0.71 = ( ( ord_le2604355607129572851at_nat @ A3 @ B2 )
% 0.35/0.71 & ( finite4343798906461161616at_nat @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8929_in__finite__psubset,axiom,
% 0.35/0.71 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.35/0.71 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B2 ) @ finite469560695537375940at_nat )
% 0.35/0.71 = ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
% 0.35/0.71 & ( finite6177210948735845034at_nat @ B2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % in_finite_psubset
% 0.35/0.71 thf(fact_8930_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > real] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8931_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > set_int] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_set_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8932_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > rat] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8933_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > num] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_num @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8934_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > nat] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8935_monoI1,axiom,
% 0.35/0.71 ! [X6: nat > int] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.71 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI1
% 0.35/0.71 thf(fact_8936_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > real] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8937_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > set_int] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8938_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > rat] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8939_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > num] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8940_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > nat] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8941_monoI2,axiom,
% 0.35/0.71 ! [X6: nat > int] :
% 0.35/0.71 ( ! [M4: nat,N3: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.71 => ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 0.35/0.71 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoI2
% 0.35/0.71 thf(fact_8942_monoseq__def,axiom,
% 0.35/0.71 ( topolo6980174941875973593q_real
% 0.35/0.71 = ( ^ [X8: nat > real] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8943_monoseq__def,axiom,
% 0.35/0.71 ( topolo3100542954746470799et_int
% 0.35/0.71 = ( ^ [X8: nat > set_int] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_set_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8944_monoseq__def,axiom,
% 0.35/0.71 ( topolo4267028734544971653eq_rat
% 0.35/0.71 = ( ^ [X8: nat > rat] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8945_monoseq__def,axiom,
% 0.35/0.71 ( topolo1459490580787246023eq_num
% 0.35/0.71 = ( ^ [X8: nat > num] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_num @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8946_monoseq__def,axiom,
% 0.35/0.71 ( topolo4902158794631467389eq_nat
% 0.35/0.71 = ( ^ [X8: nat > nat] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8947_monoseq__def,axiom,
% 0.35/0.71 ( topolo4899668324122417113eq_int
% 0.35/0.71 = ( ^ [X8: nat > int] :
% 0.35/0.71 ( ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 0.35/0.71 | ! [M2: nat,N2: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.71 => ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_def
% 0.35/0.71 thf(fact_8948_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo6980174941875973593q_real
% 0.35/0.71 = ( ^ [X8: nat > real] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8949_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo3100542954746470799et_int
% 0.35/0.71 = ( ^ [X8: nat > set_int] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8950_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo4267028734544971653eq_rat
% 0.35/0.71 = ( ^ [X8: nat > rat] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8951_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo1459490580787246023eq_num
% 0.35/0.71 = ( ^ [X8: nat > num] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8952_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo4902158794631467389eq_nat
% 0.35/0.71 = ( ^ [X8: nat > nat] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8953_monoseq__Suc,axiom,
% 0.35/0.71 ( topolo4899668324122417113eq_int
% 0.35/0.71 = ( ^ [X8: nat > int] :
% 0.35/0.71 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 0.35/0.71 | ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % monoseq_Suc
% 0.35/0.71 thf(fact_8954_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > real] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8955_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > set_int] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8956_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > rat] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8957_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > num] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8958_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > nat] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8959_mono__SucI2,axiom,
% 0.35/0.71 ! [X6: nat > int] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 0.35/0.71 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI2
% 0.35/0.71 thf(fact_8960_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > real] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8961_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > set_int] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8962_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > rat] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8963_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > num] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8964_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > nat] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8965_mono__SucI1,axiom,
% 0.35/0.71 ! [X6: nat > int] :
% 0.35/0.71 ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 0.35/0.71 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 0.35/0.71
% 0.35/0.71 % mono_SucI1
% 0.35/0.71 thf(fact_8966_Maclaurin__exp__lt,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ( ( X2 != zero_zero_real )
% 0.35/0.71 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71 => ? [T6: real] :
% 0.35/0.71 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.35/0.71 & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.71 & ( ( exp_real @ X2 )
% 0.35/0.71 = ( plus_plus_real
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Maclaurin_exp_lt
% 0.35/0.71 thf(fact_8967_floor__log__nat__eq__powr__iff,axiom,
% 0.35/0.71 ! [B3: nat,K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.71 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.71 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.71 = ( semiri1314217659103216013at_int @ N ) )
% 0.35/0.71 = ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 0.35/0.71 & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_log_nat_eq_powr_iff
% 0.35/0.71 thf(fact_8968_of__nat__code,axiom,
% 0.35/0.71 ( semiri681578069525770553at_rat
% 0.35/0.71 = ( ^ [N2: nat] :
% 0.35/0.71 ( semiri7787848453975740701ux_rat
% 0.35/0.71 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 0.35/0.71 @ N2
% 0.35/0.71 @ zero_zero_rat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % of_nat_code
% 0.35/0.71 thf(fact_8969_of__nat__code,axiom,
% 0.35/0.71 ( semiri1316708129612266289at_nat
% 0.35/0.71 = ( ^ [N2: nat] :
% 0.35/0.71 ( semiri8422978514062236437ux_nat
% 0.35/0.71 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 0.35/0.71 @ N2
% 0.35/0.71 @ zero_zero_nat ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % of_nat_code
% 0.35/0.71 thf(fact_8970_of__nat__code,axiom,
% 0.35/0.71 ( semiri1314217659103216013at_int
% 0.35/0.71 = ( ^ [N2: nat] :
% 0.35/0.71 ( semiri8420488043553186161ux_int
% 0.35/0.71 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 0.35/0.71 @ N2
% 0.35/0.71 @ zero_zero_int ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % of_nat_code
% 0.35/0.71 thf(fact_8971_of__nat__code,axiom,
% 0.35/0.71 ( semiri5074537144036343181t_real
% 0.35/0.71 = ( ^ [N2: nat] :
% 0.35/0.71 ( semiri7260567687927622513x_real
% 0.35/0.71 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 0.35/0.71 @ N2
% 0.35/0.71 @ zero_zero_real ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % of_nat_code
% 0.35/0.71 thf(fact_8972_central__binomial__lower__bound,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71 => ( 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 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % central_binomial_lower_bound
% 0.35/0.71 thf(fact_8973_binomial__0__Suc,axiom,
% 0.35/0.71 ! [K: nat] :
% 0.35/0.71 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.35/0.71 = zero_zero_nat ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_0_Suc
% 0.35/0.71 thf(fact_8974_binomial__1,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.71 = N ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_1
% 0.35/0.71 thf(fact_8975_binomial__eq__0__iff,axiom,
% 0.35/0.71 ! [N: nat,K: nat] :
% 0.35/0.71 ( ( ( binomial @ N @ K )
% 0.35/0.71 = zero_zero_nat )
% 0.35/0.71 = ( ord_less_nat @ N @ K ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_eq_0_iff
% 0.35/0.71 thf(fact_8976_binomial__n__0,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( binomial @ N @ zero_zero_nat )
% 0.35/0.71 = one_one_nat ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_n_0
% 0.35/0.71 thf(fact_8977_zero__less__binomial__iff,axiom,
% 0.35/0.71 ! [N: nat,K: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 0.35/0.71 = ( ord_less_eq_nat @ K @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % zero_less_binomial_iff
% 0.35/0.71 thf(fact_8978_fact__mono__nat,axiom,
% 0.35/0.71 ! [M: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_mono_nat
% 0.35/0.71 thf(fact_8979_fact__ge__self,axiom,
% 0.35/0.71 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_ge_self
% 0.35/0.71 thf(fact_8980_binomial__fact__lemma,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.71 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 0.35/0.71 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_fact_lemma
% 0.35/0.71 thf(fact_8981_binomial__altdef__nat,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.71 => ( ( binomial @ N @ K )
% 0.35/0.71 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_altdef_nat
% 0.35/0.71 thf(fact_8982_binomial__eq__0,axiom,
% 0.35/0.71 ! [N: nat,K: nat] :
% 0.35/0.71 ( ( ord_less_nat @ N @ K )
% 0.35/0.71 => ( ( binomial @ N @ K )
% 0.35/0.71 = zero_zero_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_eq_0
% 0.35/0.71 thf(fact_8983_fact__less__mono__nat,axiom,
% 0.35/0.71 ! [M: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.71 => ( ( ord_less_nat @ M @ N )
% 0.35/0.71 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_less_mono_nat
% 0.35/0.71 thf(fact_8984_binomial__symmetric,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.71 => ( ( binomial @ N @ K )
% 0.35/0.71 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_symmetric
% 0.35/0.71 thf(fact_8985_binomial__le__pow,axiom,
% 0.35/0.71 ! [R2: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ R2 @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_le_pow
% 0.35/0.71 thf(fact_8986_zero__less__binomial,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ N )
% 0.35/0.71 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % zero_less_binomial
% 0.35/0.71 thf(fact_8987_fact__ge__Suc__0__nat,axiom,
% 0.35/0.71 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_ge_Suc_0_nat
% 0.35/0.71 thf(fact_8988_choose__mult,axiom,
% 0.35/0.71 ! [K: nat,M: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ M )
% 0.35/0.71 => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.71 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 0.35/0.71 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % choose_mult
% 0.35/0.71 thf(fact_8989_fact__diff__Suc,axiom,
% 0.35/0.71 ! [N: nat,M: nat] :
% 0.35/0.71 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 0.35/0.71 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 0.35/0.71 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_diff_Suc
% 0.35/0.71 thf(fact_8990_fact__div__fact__le__pow,axiom,
% 0.35/0.71 ! [R2: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ R2 @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % fact_div_fact_le_pow
% 0.35/0.71 thf(fact_8991_real__of__int__floor__add__one__gt,axiom,
% 0.35/0.71 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_of_int_floor_add_one_gt
% 0.35/0.71 thf(fact_8992_floor__eq,axiom,
% 0.35/0.71 ! [N: int,X2: real] :
% 0.35/0.71 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.35/0.71 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.71 => ( ( archim6058952711729229775r_real @ X2 )
% 0.35/0.71 = N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_eq
% 0.35/0.71 thf(fact_8993_real__of__int__floor__add__one__ge,axiom,
% 0.35/0.71 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_of_int_floor_add_one_ge
% 0.35/0.71 thf(fact_8994_real__of__int__floor__gt__diff__one,axiom,
% 0.35/0.71 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_of_int_floor_gt_diff_one
% 0.35/0.71 thf(fact_8995_real__of__int__floor__ge__diff__one,axiom,
% 0.35/0.71 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % real_of_int_floor_ge_diff_one
% 0.35/0.71 thf(fact_8996_binomial__code,axiom,
% 0.35/0.71 ( binomial
% 0.35/0.71 = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_code
% 0.35/0.71 thf(fact_8997_binomial__maximum_H,axiom,
% 0.35/0.71 ! [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 ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_maximum'
% 0.35/0.71 thf(fact_8998_binomial__mono,axiom,
% 0.35/0.71 ! [K: nat,K5: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ K5 )
% 0.35/0.71 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_mono
% 0.35/0.71 thf(fact_8999_binomial__maximum,axiom,
% 0.35/0.71 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_maximum
% 0.35/0.71 thf(fact_9000_binomial__antimono,axiom,
% 0.35/0.71 ! [K: nat,K5: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ K @ K5 )
% 0.35/0.71 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.35/0.71 => ( ( ord_less_eq_nat @ K5 @ N )
% 0.35/0.71 => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_antimono
% 0.35/0.71 thf(fact_9001_binomial__le__pow2,axiom,
% 0.35/0.71 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_le_pow2
% 0.35/0.71 thf(fact_9002_choose__reduce__nat,axiom,
% 0.35/0.71 ! [N: nat,K: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.71 => ( ( binomial @ N @ K )
% 0.35/0.71 = ( 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 ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % choose_reduce_nat
% 0.35/0.71 thf(fact_9003_times__binomial__minus1__eq,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.71 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 0.35/0.71 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % times_binomial_minus1_eq
% 0.35/0.71 thf(fact_9004_floor__eq2,axiom,
% 0.35/0.71 ! [N: int,X2: real] :
% 0.35/0.71 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 0.35/0.71 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 0.35/0.71 => ( ( archim6058952711729229775r_real @ X2 )
% 0.35/0.71 = N ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_eq2
% 0.35/0.71 thf(fact_9005_floor__divide__real__eq__div,axiom,
% 0.35/0.71 ! [B3: int,A2: real] :
% 0.35/0.71 ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 0.35/0.71 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B3 ) ) )
% 0.35/0.71 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B3 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_divide_real_eq_div
% 0.35/0.71 thf(fact_9006_binomial__less__binomial__Suc,axiom,
% 0.35/0.71 ! [K: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.71 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_less_binomial_Suc
% 0.35/0.71 thf(fact_9007_binomial__strict__antimono,axiom,
% 0.35/0.71 ! [K: nat,K5: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ K @ K5 )
% 0.35/0.71 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ K5 @ N )
% 0.35/0.71 => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_strict_antimono
% 0.35/0.71 thf(fact_9008_binomial__strict__mono,axiom,
% 0.35/0.71 ! [K: nat,K5: nat,N: nat] :
% 0.35/0.71 ( ( ord_less_nat @ K @ K5 )
% 0.35/0.71 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 0.35/0.71 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_strict_mono
% 0.35/0.71 thf(fact_9009_binomial__addition__formula,axiom,
% 0.35/0.71 ! [N: nat,K: nat] :
% 0.35/0.71 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.71 => ( ( binomial @ N @ ( suc @ K ) )
% 0.35/0.71 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % binomial_addition_formula
% 0.35/0.71 thf(fact_9010_square__fact__le__2__fact,axiom,
% 0.35/0.71 ! [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 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % square_fact_le_2_fact
% 0.35/0.71 thf(fact_9011_Maclaurin__lemma,axiom,
% 0.35/0.71 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 0.35/0.71 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.71 => ? [B8: real] :
% 0.35/0.71 ( ( F @ H2 )
% 0.35/0.71 = ( plus_plus_real
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Maclaurin_lemma
% 0.35/0.71 thf(fact_9012_floor__log__eq__powr__iff,axiom,
% 0.35/0.71 ! [X2: real,B3: real,K: int] :
% 0.35/0.71 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.71 => ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.71 => ( ( ( archim6058952711729229775r_real @ ( log @ B3 @ X2 ) )
% 0.35/0.71 = K )
% 0.35/0.71 = ( ( ord_less_eq_real @ ( powr_real @ B3 @ ( ring_1_of_int_real @ K ) ) @ X2 )
% 0.35/0.71 & ( ord_less_real @ X2 @ ( powr_real @ B3 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_log_eq_powr_iff
% 0.35/0.71 thf(fact_9013_Maclaurin__exp__le,axiom,
% 0.35/0.71 ! [X2: real,N: nat] :
% 0.35/0.71 ? [T6: real] :
% 0.35/0.71 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.71 & ( ( exp_real @ X2 )
% 0.35/0.71 = ( plus_plus_real
% 0.35/0.71 @ ( groups6591440286371151544t_real
% 0.35/0.71 @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 0.35/0.71 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.71 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % Maclaurin_exp_le
% 0.35/0.71 thf(fact_9014_floor__log2__div2,axiom,
% 0.35/0.71 ! [N: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.71 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.71 = ( 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 ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_log2_div2
% 0.35/0.71 thf(fact_9015_floor__log__nat__eq__if,axiom,
% 0.35/0.71 ! [B3: nat,N: nat,K: nat] :
% 0.35/0.71 ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 0.35/0.71 => ( ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 0.35/0.71 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 0.35/0.71 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.35/0.71 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % floor_log_nat_eq_if
% 0.35/0.71 thf(fact_9016_powr__int,axiom,
% 0.35/0.71 ! [X2: real,I: int] :
% 0.35/0.71 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.71 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.71 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
% 0.35/0.71 = ( power_power_real @ X2 @ ( nat2 @ I ) ) ) )
% 0.35/0.71 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.35/0.71 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
% 0.35/0.71 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.35/0.71
% 0.35/0.71 % powr_int
% 0.35/0.71 thf(fact_9017_nat__dvd__1__iff__1,axiom,
% 0.35/0.71 ! [M: nat] :
% 0.35/0.71 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.35/0.71 = ( M = one_one_nat ) ) ).
% 0.35/0.71
% 0.35/0.71 % nat_dvd_1_iff_1
% 0.35/0.71 thf(fact_9018_dvd__1__left,axiom,
% 0.35/0.71 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.35/0.71
% 0.35/0.71 % dvd_1_left
% 0.35/0.71 thf(fact_9019_dvd__1__iff__1,axiom,
% 0.35/0.71 ! [M: nat] :
% 0.35/0.71 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.72 = ( M
% 0.35/0.72 = ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_1_iff_1
% 0.35/0.72 thf(fact_9020_nat__mult__dvd__cancel__disj,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.72 = ( ( K = zero_zero_nat )
% 0.35/0.72 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mult_dvd_cancel_disj
% 0.35/0.72 thf(fact_9021_finite__atMost,axiom,
% 0.35/0.72 ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 0.35/0.72
% 0.35/0.72 % finite_atMost
% 0.35/0.72 thf(fact_9022_card__atMost,axiom,
% 0.35/0.72 ! [U: nat] :
% 0.35/0.72 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.35/0.72 = ( suc @ U ) ) ).
% 0.35/0.72
% 0.35/0.72 % card_atMost
% 0.35/0.72 thf(fact_9023_nat__1,axiom,
% 0.35/0.72 ( ( nat2 @ one_one_int )
% 0.35/0.72 = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_1
% 0.35/0.72 thf(fact_9024_nat__0__iff,axiom,
% 0.35/0.72 ! [I: int] :
% 0.35/0.72 ( ( ( nat2 @ I )
% 0.35/0.72 = zero_zero_nat )
% 0.35/0.72 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_0_iff
% 0.35/0.72 thf(fact_9025_nat__le__0,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.35/0.72 => ( ( nat2 @ Z )
% 0.35/0.72 = zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_le_0
% 0.35/0.72 thf(fact_9026_zless__nat__conj,axiom,
% 0.35/0.72 ! [W2: int,Z: int] :
% 0.35/0.72 ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.72 & ( ord_less_int @ W2 @ Z ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % zless_nat_conj
% 0.35/0.72 thf(fact_9027_nat__neg__numeral,axiom,
% 0.35/0.72 ! [K: num] :
% 0.35/0.72 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.35/0.72 = zero_zero_nat ) ).
% 0.35/0.72
% 0.35/0.72 % nat_neg_numeral
% 0.35/0.72 thf(fact_9028_nat__zminus__int,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.72 = zero_zero_nat ) ).
% 0.35/0.72
% 0.35/0.72 % nat_zminus_int
% 0.35/0.72 thf(fact_9029_int__nat__eq,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.35/0.72 = Z ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.35/0.72 = zero_zero_int ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % int_nat_eq
% 0.35/0.72 thf(fact_9030_atMost__0,axiom,
% 0.35/0.72 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.35/0.72 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % atMost_0
% 0.35/0.72 thf(fact_9031_zero__less__nat__eq,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 0.35/0.72
% 0.35/0.72 % zero_less_nat_eq
% 0.35/0.72 thf(fact_9032_card__atLeastAtMost__int,axiom,
% 0.35/0.72 ! [L: int,U: int] :
% 0.35/0.72 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.35/0.72 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % card_atLeastAtMost_int
% 0.35/0.72 thf(fact_9033_nat__ceiling__le__eq,axiom,
% 0.35/0.72 ! [X2: real,A2: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A2 )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_ceiling_le_eq
% 0.35/0.72 thf(fact_9034_odd__Suc__minus__one,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 0.35/0.72 = N ) ) ).
% 0.35/0.72
% 0.35/0.72 % odd_Suc_minus_one
% 0.35/0.72 thf(fact_9035_even__diff__nat,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.72 = ( ( ord_less_nat @ M @ N )
% 0.35/0.72 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % even_diff_nat
% 0.35/0.72 thf(fact_9036_one__less__nat__eq,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 0.35/0.72
% 0.35/0.72 % one_less_nat_eq
% 0.35/0.72 thf(fact_9037_nat__less__numeral__power__cancel__iff,axiom,
% 0.35/0.72 ! [A2: int,X2: num,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.35/0.72 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_less_numeral_power_cancel_iff
% 0.35/0.72 thf(fact_9038_numeral__power__less__nat__cancel__iff,axiom,
% 0.35/0.72 ! [X2: num,N: nat,A2: int] :
% 0.35/0.72 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
% 0.35/0.72 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % numeral_power_less_nat_cancel_iff
% 0.35/0.72 thf(fact_9039_nat__le__numeral__power__cancel__iff,axiom,
% 0.35/0.72 ! [A2: int,X2: num,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 0.35/0.72 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_le_numeral_power_cancel_iff
% 0.35/0.72 thf(fact_9040_numeral__power__le__nat__cancel__iff,axiom,
% 0.35/0.72 ! [X2: num,N: nat,A2: int] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
% 0.35/0.72 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % numeral_power_le_nat_cancel_iff
% 0.35/0.72 thf(fact_9041_gcd__nat_Oextremum,axiom,
% 0.35/0.72 ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% 0.35/0.72
% 0.35/0.72 % gcd_nat.extremum
% 0.35/0.72 thf(fact_9042_gcd__nat_Oextremum__strict,axiom,
% 0.35/0.72 ! [A2: nat] :
% 0.35/0.72 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 0.35/0.72 & ( zero_zero_nat != A2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % gcd_nat.extremum_strict
% 0.35/0.72 thf(fact_9043_gcd__nat_Oextremum__unique,axiom,
% 0.35/0.72 ! [A2: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 0.35/0.72 = ( A2 = zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % gcd_nat.extremum_unique
% 0.35/0.72 thf(fact_9044_gcd__nat_Onot__eq__extremum,axiom,
% 0.35/0.72 ! [A2: nat] :
% 0.35/0.72 ( ( A2 != zero_zero_nat )
% 0.35/0.72 = ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
% 0.35/0.72 & ( A2 != zero_zero_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % gcd_nat.not_eq_extremum
% 0.35/0.72 thf(fact_9045_gcd__nat_Oextremum__uniqueI,axiom,
% 0.35/0.72 ! [A2: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 0.35/0.72 => ( A2 = zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % gcd_nat.extremum_uniqueI
% 0.35/0.72 thf(fact_9046_dvd__diff__nat,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ K @ M )
% 0.35/0.72 => ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.72 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_diff_nat
% 0.35/0.72 thf(fact_9047_nat__dvd__iff,axiom,
% 0.35/0.72 ! [Z: int,M: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 0.35/0.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_dvd_iff
% 0.35/0.72 thf(fact_9048_even__nat__iff,axiom,
% 0.35/0.72 ! [K: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 0.35/0.72 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % even_nat_iff
% 0.35/0.72 thf(fact_9049_nat__zero__as__int,axiom,
% 0.35/0.72 ( zero_zero_nat
% 0.35/0.72 = ( nat2 @ zero_zero_int ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_zero_as_int
% 0.35/0.72 thf(fact_9050_nat__mono,axiom,
% 0.35/0.72 ! [X2: int,Y3: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ X2 @ Y3 )
% 0.35/0.72 => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mono
% 0.35/0.72 thf(fact_9051_atMost__atLeast0,axiom,
% 0.35/0.72 ( set_ord_atMost_nat
% 0.35/0.72 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % atMost_atLeast0
% 0.35/0.72 thf(fact_9052_eq__nat__nat__iff,axiom,
% 0.35/0.72 ! [Z: int,Z7: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.35/0.72 => ( ( ( nat2 @ Z )
% 0.35/0.72 = ( nat2 @ Z7 ) )
% 0.35/0.72 = ( Z = Z7 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % eq_nat_nat_iff
% 0.35/0.72 thf(fact_9053_all__nat,axiom,
% 0.35/0.72 ( ( ^ [P2: nat > $o] :
% 0.35/0.72 ! [X3: nat] : ( P2 @ X3 ) )
% 0.35/0.72 = ( ^ [P3: nat > $o] :
% 0.35/0.72 ! [X: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.72 => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % all_nat
% 0.35/0.72 thf(fact_9054_ex__nat,axiom,
% 0.35/0.72 ( ( ^ [P2: nat > $o] :
% 0.35/0.72 ? [X3: nat] : ( P2 @ X3 ) )
% 0.35/0.72 = ( ^ [P3: nat > $o] :
% 0.35/0.72 ? [X: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.35/0.72 & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % ex_nat
% 0.35/0.72 thf(fact_9055_lessThan__Suc__atMost,axiom,
% 0.35/0.72 ! [K: nat] :
% 0.35/0.72 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.35/0.72 = ( set_ord_atMost_nat @ K ) ) ).
% 0.35/0.72
% 0.35/0.72 % lessThan_Suc_atMost
% 0.35/0.72 thf(fact_9056_dvd__pos__nat,axiom,
% 0.35/0.72 ! [N: nat,M: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72 => ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.72 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_pos_nat
% 0.35/0.72 thf(fact_9057_nat__dvd__not__less,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.72 => ( ( ord_less_nat @ M @ N )
% 0.35/0.72 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_dvd_not_less
% 0.35/0.72 thf(fact_9058_dvd__minus__self,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 0.35/0.72 = ( ( ord_less_nat @ N @ M )
% 0.35/0.72 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_minus_self
% 0.35/0.72 thf(fact_9059_atMost__Suc,axiom,
% 0.35/0.72 ! [K: nat] :
% 0.35/0.72 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.35/0.72 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % atMost_Suc
% 0.35/0.72 thf(fact_9060_zdvd__antisym__nonneg,axiom,
% 0.35/0.72 ! [M: int,N: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.35/0.72 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.35/0.72 => ( ( dvd_dvd_int @ M @ N )
% 0.35/0.72 => ( ( dvd_dvd_int @ N @ M )
% 0.35/0.72 => ( M = N ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % zdvd_antisym_nonneg
% 0.35/0.72 thf(fact_9061_less__eq__dvd__minus,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.72 => ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.72 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % less_eq_dvd_minus
% 0.35/0.72 thf(fact_9062_dvd__diffD1,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.72 => ( ( dvd_dvd_nat @ K @ M )
% 0.35/0.72 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_diffD1
% 0.35/0.72 thf(fact_9063_dvd__diffD,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 0.35/0.72 => ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.72 => ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_diffD
% 0.35/0.72 thf(fact_9064_zdvd__not__zless,axiom,
% 0.35/0.72 ! [M: int,N: int] :
% 0.35/0.72 ( ( ord_less_int @ zero_zero_int @ M )
% 0.35/0.72 => ( ( ord_less_int @ M @ N )
% 0.35/0.72 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % zdvd_not_zless
% 0.35/0.72 thf(fact_9065_dvd__fact,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.35/0.72 => ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.72 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_fact
% 0.35/0.72 thf(fact_9066_finite__nat__iff__bounded__le,axiom,
% 0.35/0.72 ( finite_finite_nat
% 0.35/0.72 = ( ^ [S6: set_nat] :
% 0.35/0.72 ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % finite_nat_iff_bounded_le
% 0.35/0.72 thf(fact_9067_nat__mono__iff,axiom,
% 0.35/0.72 ! [Z: int,W2: int] :
% 0.35/0.72 ( ( ord_less_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_int @ W2 @ Z ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mono_iff
% 0.35/0.72 thf(fact_9068_zless__nat__eq__int__zless,axiom,
% 0.35/0.72 ! [M: nat,Z: int] :
% 0.35/0.72 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 0.35/0.72
% 0.35/0.72 % zless_nat_eq_int_zless
% 0.35/0.72 thf(fact_9069_nat__le__iff,axiom,
% 0.35/0.72 ! [X2: int,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
% 0.35/0.72 = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_le_iff
% 0.35/0.72 thf(fact_9070_nat__0__le,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 0.35/0.72 = Z ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_0_le
% 0.35/0.72 thf(fact_9071_int__eq__iff,axiom,
% 0.35/0.72 ! [M: nat,Z: int] :
% 0.35/0.72 ( ( ( semiri1314217659103216013at_int @ M )
% 0.35/0.72 = Z )
% 0.35/0.72 = ( ( M
% 0.35/0.72 = ( nat2 @ Z ) )
% 0.35/0.72 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % int_eq_iff
% 0.35/0.72 thf(fact_9072_dvd__imp__le,axiom,
% 0.35/0.72 ! [K: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ K @ N )
% 0.35/0.72 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_imp_le
% 0.35/0.72 thf(fact_9073_nat__mult__dvd__cancel1,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.72 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mult_dvd_cancel1
% 0.35/0.72 thf(fact_9074_dvd__mult__cancel,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 0.35/0.72 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.35/0.72 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_mult_cancel
% 0.35/0.72 thf(fact_9075_bezout__add__strong__nat,axiom,
% 0.35/0.72 ! [A2: nat,B3: nat] :
% 0.35/0.72 ( ( A2 != zero_zero_nat )
% 0.35/0.72 => ? [D6: nat,X5: nat,Y4: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ D6 @ A2 )
% 0.35/0.72 & ( dvd_dvd_nat @ D6 @ B3 )
% 0.35/0.72 & ( ( times_times_nat @ A2 @ X5 )
% 0.35/0.72 = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ D6 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % bezout_add_strong_nat
% 0.35/0.72 thf(fact_9076_zdvd__imp__le,axiom,
% 0.35/0.72 ! [Z: int,N: int] :
% 0.35/0.72 ( ( dvd_dvd_int @ Z @ N )
% 0.35/0.72 => ( ( ord_less_int @ zero_zero_int @ N )
% 0.35/0.72 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % zdvd_imp_le
% 0.35/0.72 thf(fact_9077_mod__greater__zero__iff__not__dvd,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 0.35/0.72 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % mod_greater_zero_iff_not_dvd
% 0.35/0.72 thf(fact_9078_dvd__imp__le__int,axiom,
% 0.35/0.72 ! [I: int,D: int] :
% 0.35/0.72 ( ( I != zero_zero_int )
% 0.35/0.72 => ( ( dvd_dvd_int @ D @ I )
% 0.35/0.72 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_imp_le_int
% 0.35/0.72 thf(fact_9079_mod__eq__dvd__iff__nat,axiom,
% 0.35/0.72 ! [N: nat,M: nat,Q3: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.72 => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 0.35/0.72 = ( modulo_modulo_nat @ N @ Q3 ) )
% 0.35/0.72 = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % mod_eq_dvd_iff_nat
% 0.35/0.72 thf(fact_9080_real__nat__ceiling__ge,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_nat_ceiling_ge
% 0.35/0.72 thf(fact_9081_finite__divisors__nat,axiom,
% 0.35/0.72 ! [M: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.72 => ( finite_finite_nat
% 0.35/0.72 @ ( collect_nat
% 0.35/0.72 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % finite_divisors_nat
% 0.35/0.72 thf(fact_9082_nat__less__eq__zless,axiom,
% 0.35/0.72 ! [W2: int,Z: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_int @ W2 @ Z ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_less_eq_zless
% 0.35/0.72 thf(fact_9083_nat__le__eq__zle,axiom,
% 0.35/0.72 ! [W2: int,Z: int] :
% 0.35/0.72 ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 0.35/0.72 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 0.35/0.72 => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 0.35/0.72 = ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_le_eq_zle
% 0.35/0.72 thf(fact_9084_nat__eq__iff2,axiom,
% 0.35/0.72 ! [M: nat,W2: int] :
% 0.35/0.72 ( ( M
% 0.35/0.72 = ( nat2 @ W2 ) )
% 0.35/0.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( W2
% 0.35/0.72 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_eq_iff2
% 0.35/0.72 thf(fact_9085_nat__eq__iff,axiom,
% 0.35/0.72 ! [W2: int,M: nat] :
% 0.35/0.72 ( ( ( nat2 @ W2 )
% 0.35/0.72 = M )
% 0.35/0.72 = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( W2
% 0.35/0.72 = ( semiri1314217659103216013at_int @ M ) ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( M = zero_zero_nat ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_eq_iff
% 0.35/0.72 thf(fact_9086_split__nat,axiom,
% 0.35/0.72 ! [P: nat > $o,I: int] :
% 0.35/0.72 ( ( P @ ( nat2 @ I ) )
% 0.35/0.72 = ( ! [N2: nat] :
% 0.35/0.72 ( ( I
% 0.35/0.72 = ( semiri1314217659103216013at_int @ N2 ) )
% 0.35/0.72 => ( P @ N2 ) )
% 0.35/0.72 & ( ( ord_less_int @ I @ zero_zero_int )
% 0.35/0.72 => ( P @ zero_zero_nat ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % split_nat
% 0.35/0.72 thf(fact_9087_le__nat__iff,axiom,
% 0.35/0.72 ! [K: int,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.72 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 0.35/0.72 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % le_nat_iff
% 0.35/0.72 thf(fact_9088_nat__add__distrib,axiom,
% 0.35/0.72 ! [Z: int,Z7: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.35/0.72 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 0.35/0.72 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_add_distrib
% 0.35/0.72 thf(fact_9089_nat__mult__distrib,axiom,
% 0.35/0.72 ! [Z: int,Z7: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.35/0.72 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mult_distrib
% 0.35/0.72 thf(fact_9090_nat__diff__distrib_H,axiom,
% 0.35/0.72 ! [X2: int,Y3: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.72 => ( ( nat2 @ ( minus_minus_int @ X2 @ Y3 ) )
% 0.35/0.72 = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_diff_distrib'
% 0.35/0.72 thf(fact_9091_nat__diff__distrib,axiom,
% 0.35/0.72 ! [Z7: int,Z: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 0.35/0.72 => ( ( ord_less_eq_int @ Z7 @ Z )
% 0.35/0.72 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 0.35/0.72 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_diff_distrib
% 0.35/0.72 thf(fact_9092_nat__abs__triangle__ineq,axiom,
% 0.35/0.72 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_abs_triangle_ineq
% 0.35/0.72 thf(fact_9093_nat__div__distrib,axiom,
% 0.35/0.72 ! [X2: int,Y3: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.72 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 0.35/0.72 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_div_distrib
% 0.35/0.72 thf(fact_9094_nat__div__distrib_H,axiom,
% 0.35/0.72 ! [Y3: int,X2: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.72 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 0.35/0.72 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_div_distrib'
% 0.35/0.72 thf(fact_9095_nat__power__eq,axiom,
% 0.35/0.72 ! [Z: int,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 0.35/0.72 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_power_eq
% 0.35/0.72 thf(fact_9096_nat__floor__neg,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.35/0.72 = zero_zero_nat ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_floor_neg
% 0.35/0.72 thf(fact_9097_nat__mod__distrib,axiom,
% 0.35/0.72 ! [X2: int,Y3: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.72 => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y3 ) )
% 0.35/0.72 = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mod_distrib
% 0.35/0.72 thf(fact_9098_floor__eq3,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.35/0.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.35/0.72 = N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % floor_eq3
% 0.35/0.72 thf(fact_9099_dvd__mult__cancel2,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 0.35/0.72 = ( N = one_one_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_mult_cancel2
% 0.35/0.72 thf(fact_9100_dvd__mult__cancel1,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 0.35/0.72 = ( N = one_one_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_mult_cancel1
% 0.35/0.72 thf(fact_9101_le__nat__floor,axiom,
% 0.35/0.72 ! [X2: nat,A2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A2 )
% 0.35/0.72 => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % le_nat_floor
% 0.35/0.72 thf(fact_9102_dvd__minus__add,axiom,
% 0.35/0.72 ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ Q3 @ N )
% 0.35/0.72 => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 0.35/0.72 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 0.35/0.72 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_minus_add
% 0.35/0.72 thf(fact_9103_power__dvd__imp__le,axiom,
% 0.35/0.72 ! [I: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 0.35/0.72 => ( ( ord_less_nat @ one_one_nat @ I )
% 0.35/0.72 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % power_dvd_imp_le
% 0.35/0.72 thf(fact_9104_mod__nat__eqI,axiom,
% 0.35/0.72 ! [R2: nat,N: nat,M: nat] :
% 0.35/0.72 ( ( ord_less_nat @ R2 @ N )
% 0.35/0.72 => ( ( ord_less_eq_nat @ R2 @ M )
% 0.35/0.72 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 0.35/0.72 => ( ( modulo_modulo_nat @ M @ N )
% 0.35/0.72 = R2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % mod_nat_eqI
% 0.35/0.72 thf(fact_9105_mod__int__pos__iff,axiom,
% 0.35/0.72 ! [K: int,L: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.35/0.72 = ( ( dvd_dvd_int @ L @ K )
% 0.35/0.72 | ( ( L = zero_zero_int )
% 0.35/0.72 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.35/0.72 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % mod_int_pos_iff
% 0.35/0.72 thf(fact_9106_nat__2,axiom,
% 0.35/0.72 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.72 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_2
% 0.35/0.72 thf(fact_9107_Suc__nat__eq__nat__zadd1,axiom,
% 0.35/0.72 ! [Z: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 0.35/0.72 => ( ( suc @ ( nat2 @ Z ) )
% 0.35/0.72 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Suc_nat_eq_nat_zadd1
% 0.35/0.72 thf(fact_9108_nat__less__iff,axiom,
% 0.35/0.72 ! [W2: int,M: nat] :
% 0.35/0.72 ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.35/0.72 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 0.35/0.72 = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_less_iff
% 0.35/0.72 thf(fact_9109_nat__mult__distrib__neg,axiom,
% 0.35/0.72 ! [Z: int,Z7: int] :
% 0.35/0.72 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 0.35/0.72 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 0.35/0.72 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_mult_distrib_neg
% 0.35/0.72 thf(fact_9110_odd__pos,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.72
% 0.35/0.72 % odd_pos
% 0.35/0.72 thf(fact_9111_nat__abs__int__diff,axiom,
% 0.35/0.72 ! [A2: nat,B3: nat] :
% 0.35/0.72 ( ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.72 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
% 0.35/0.72 = ( minus_minus_nat @ B3 @ A2 ) ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.72 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
% 0.35/0.72 = ( minus_minus_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % nat_abs_int_diff
% 0.35/0.72 thf(fact_9112_floor__eq4,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.35/0.72 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 0.35/0.72 = N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % floor_eq4
% 0.35/0.72 thf(fact_9113_dvd__power__iff__le,axiom,
% 0.35/0.72 ! [K: nat,M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 0.35/0.72 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_power_iff_le
% 0.35/0.72 thf(fact_9114_diff__nat__eq__if,axiom,
% 0.35/0.72 ! [Z7: int,Z: int] :
% 0.35/0.72 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 0.35/0.72 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.35/0.72 = ( nat2 @ Z ) ) )
% 0.35/0.72 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 0.35/0.72 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 0.35/0.72 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % diff_nat_eq_if
% 0.35/0.72 thf(fact_9115_sum__choose__diagonal,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.72 => ( ( groups3542108847815614940at_nat
% 0.35/0.72 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.35/0.72 @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.72 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sum_choose_diagonal
% 0.35/0.72 thf(fact_9116_even__set__encode__iff,axiom,
% 0.35/0.72 ! [A3: set_nat] :
% 0.35/0.72 ( ( finite_finite_nat @ A3 )
% 0.35/0.72 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 0.35/0.72 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % even_set_encode_iff
% 0.35/0.72 thf(fact_9117_atLeast1__atMost__eq__remove0,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.72 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % atLeast1_atMost_eq_remove0
% 0.35/0.72 thf(fact_9118_polynomial__product__nat,axiom,
% 0.35/0.72 ! [M: nat,A2: nat > nat,N: nat,B3: nat > nat,X2: nat] :
% 0.35/0.72 ( ! [I2: nat] :
% 0.35/0.72 ( ( ord_less_nat @ M @ I2 )
% 0.35/0.72 => ( ( A2 @ I2 )
% 0.35/0.72 = zero_zero_nat ) )
% 0.35/0.72 => ( ! [J2: nat] :
% 0.35/0.72 ( ( ord_less_nat @ N @ J2 )
% 0.35/0.72 => ( ( B3 @ J2 )
% 0.35/0.72 = zero_zero_nat ) )
% 0.35/0.72 => ( ( times_times_nat
% 0.35/0.72 @ ( groups3542108847815614940at_nat
% 0.35/0.72 @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
% 0.35/0.72 @ ( set_ord_atMost_nat @ M ) )
% 0.35/0.72 @ ( groups3542108847815614940at_nat
% 0.35/0.72 @ ^ [J3: nat] : ( times_times_nat @ ( B3 @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
% 0.35/0.72 @ ( set_ord_atMost_nat @ N ) ) )
% 0.35/0.72 = ( groups3542108847815614940at_nat
% 0.35/0.72 @ ^ [R5: nat] :
% 0.35/0.72 ( times_times_nat
% 0.35/0.72 @ ( groups3542108847815614940at_nat
% 0.35/0.72 @ ^ [K3: nat] : ( times_times_nat @ ( A2 @ K3 ) @ ( B3 @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.35/0.72 @ ( set_ord_atMost_nat @ R5 ) )
% 0.35/0.72 @ ( power_power_nat @ X2 @ R5 ) )
% 0.35/0.72 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % polynomial_product_nat
% 0.35/0.72 thf(fact_9119_even__mod__4__div__2,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = ( suc @ zero_zero_nat ) )
% 0.35/0.72 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % even_mod_4_div_2
% 0.35/0.72 thf(fact_9120_divmod__step__nat__def,axiom,
% 0.35/0.72 ( unique5026877609467782581ep_nat
% 0.35/0.72 = ( ^ [L2: num] :
% 0.35/0.72 ( produc2626176000494625587at_nat
% 0.35/0.72 @ ^ [Q6: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % divmod_step_nat_def
% 0.35/0.72 thf(fact_9121_divmod__step__int__def,axiom,
% 0.35/0.72 ( unique5024387138958732305ep_int
% 0.35/0.72 = ( ^ [L2: num] :
% 0.35/0.72 ( produc4245557441103728435nt_int
% 0.35/0.72 @ ^ [Q6: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % divmod_step_int_def
% 0.35/0.72 thf(fact_9122_odd__mod__4__div__2,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.35/0.72 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % odd_mod_4_div_2
% 0.35/0.72 thf(fact_9123_Bernoulli__inequality__even,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Bernoulli_inequality_even
% 0.35/0.72 thf(fact_9124_sin__coeff__def,axiom,
% 0.35/0.72 ( sin_coeff
% 0.35/0.72 = ( ^ [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 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_coeff_def
% 0.35/0.72 thf(fact_9125_vebt__buildup_Oelims,axiom,
% 0.35/0.72 ! [X2: nat,Y3: vEBT_VEBT] :
% 0.35/0.72 ( ( ( vEBT_vebt_buildup @ X2 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 => ( ( ( X2 = zero_zero_nat )
% 0.35/0.72 => ( Y3
% 0.35/0.72 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.72 => ( ( ( X2
% 0.35/0.72 = ( suc @ zero_zero_nat ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.35/0.72 => ~ ! [Va2: nat] :
% 0.35/0.72 ( ( X2
% 0.35/0.72 = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.35/0.72 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % vebt_buildup.elims
% 0.35/0.72 thf(fact_9126_divmod__nat__if,axiom,
% 0.35/0.72 ( divmod_nat
% 0.35/0.72 = ( ^ [M2: nat,N2: nat] :
% 0.35/0.72 ( if_Pro6206227464963214023at_nat
% 0.35/0.72 @ ( ( N2 = zero_zero_nat )
% 0.35/0.72 | ( ord_less_nat @ M2 @ N2 ) )
% 0.35/0.72 @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 0.35/0.72 @ ( produc2626176000494625587at_nat
% 0.35/0.72 @ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
% 0.35/0.72 @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % divmod_nat_if
% 0.35/0.72 thf(fact_9127_sin__coeff__0,axiom,
% 0.35/0.72 ( ( sin_coeff @ zero_zero_nat )
% 0.35/0.72 = zero_zero_real ) ).
% 0.35/0.72
% 0.35/0.72 % sin_coeff_0
% 0.35/0.72 thf(fact_9128_dvd__antisym,axiom,
% 0.35/0.72 ! [M: nat,N: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ M @ N )
% 0.35/0.72 => ( ( dvd_dvd_nat @ N @ M )
% 0.35/0.72 => ( M = N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % dvd_antisym
% 0.35/0.72 thf(fact_9129_sin__x__le__x,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_x_le_x
% 0.35/0.72 thf(fact_9130_sin__le__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % sin_le_one
% 0.35/0.72 thf(fact_9131_cos__le__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % cos_le_one
% 0.35/0.72 thf(fact_9132_abs__sin__x__le__abs__x,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % abs_sin_x_le_abs_x
% 0.35/0.72 thf(fact_9133_sin__cos__le1,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % sin_cos_le1
% 0.35/0.72 thf(fact_9134_sin__x__ge__neg__x,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_x_ge_neg_x
% 0.35/0.72 thf(fact_9135_sin__ge__minus__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_ge_minus_one
% 0.35/0.72 thf(fact_9136_cos__ge__minus__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_ge_minus_one
% 0.35/0.72 thf(fact_9137_abs__sin__le__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % abs_sin_le_one
% 0.35/0.72 thf(fact_9138_abs__cos__le__one,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % abs_cos_le_one
% 0.35/0.72 thf(fact_9139_sin__gt__zero__02,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_gt_zero_02
% 0.35/0.72 thf(fact_9140_cos__two__less__zero,axiom,
% 0.35/0.72 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.35/0.72
% 0.35/0.72 % cos_two_less_zero
% 0.35/0.72 thf(fact_9141_cos__is__zero,axiom,
% 0.35/0.72 ? [X5: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.35/0.72 & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72 & ( ( cos_real @ X5 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 & ! [Y5: real] :
% 0.35/0.72 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.35/0.72 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72 & ( ( cos_real @ Y5 )
% 0.35/0.72 = zero_zero_real ) )
% 0.35/0.72 => ( Y5 = X5 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_is_zero
% 0.35/0.72 thf(fact_9142_cos__two__le__zero,axiom,
% 0.35/0.72 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.35/0.72
% 0.35/0.72 % cos_two_le_zero
% 0.35/0.72 thf(fact_9143_cos__double__less__one,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.72 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_double_less_one
% 0.35/0.72 thf(fact_9144_vebt__buildup_Osimps_I3_J,axiom,
% 0.35/0.72 ! [Va: nat] :
% 0.35/0.72 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.72 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.72 = ( 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 ) ) ) ) ) ) )
% 0.35/0.72 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.72 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.35/0.72 = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % vebt_buildup.simps(3)
% 0.35/0.72 thf(fact_9145_vebt__buildup_Opelims,axiom,
% 0.35/0.72 ! [X2: nat,Y3: vEBT_VEBT] :
% 0.35/0.72 ( ( ( vEBT_vebt_buildup @ X2 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
% 0.35/0.72 => ( ( ( X2 = zero_zero_nat )
% 0.35/0.72 => ( ( Y3
% 0.35/0.72 = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.72 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.35/0.72 => ( ( ( X2
% 0.35/0.72 = ( suc @ zero_zero_nat ) )
% 0.35/0.72 => ( ( Y3
% 0.35/0.72 = ( vEBT_Leaf @ $false @ $false ) )
% 0.35/0.72 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.72 => ~ ! [Va2: nat] :
% 0.35/0.72 ( ( X2
% 0.35/0.72 = ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 0.35/0.72 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.35/0.72 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % vebt_buildup.pelims
% 0.35/0.72 thf(fact_9146_Maclaurin__sin__expansion3,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_real @ T6 @ X2 )
% 0.35/0.72 & ( ( sin_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_sin_expansion3
% 0.35/0.72 thf(fact_9147_Maclaurin__sin__expansion4,axiom,
% 0.35/0.72 ! [X2: real,N: nat] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_eq_real @ T6 @ X2 )
% 0.35/0.72 & ( ( sin_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_sin_expansion4
% 0.35/0.72 thf(fact_9148_Maclaurin__sin__expansion2,axiom,
% 0.35/0.72 ! [X2: real,N: nat] :
% 0.35/0.72 ? [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.72 & ( ( sin_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_sin_expansion2
% 0.35/0.72 thf(fact_9149_pi__gt__zero,axiom,
% 0.35/0.72 ord_less_real @ zero_zero_real @ pi ).
% 0.35/0.72
% 0.35/0.72 % pi_gt_zero
% 0.35/0.72 thf(fact_9150_pi__not__less__zero,axiom,
% 0.35/0.72 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.35/0.72
% 0.35/0.72 % pi_not_less_zero
% 0.35/0.72 thf(fact_9151_sin__gt__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ pi )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_gt_zero
% 0.35/0.72 thf(fact_9152_pi__ge__zero,axiom,
% 0.35/0.72 ord_less_eq_real @ zero_zero_real @ pi ).
% 0.35/0.72
% 0.35/0.72 % pi_ge_zero
% 0.35/0.72 thf(fact_9153_cos__inj__pi,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.35/0.72 => ( ( ( cos_real @ X2 )
% 0.35/0.72 = ( cos_real @ Y3 ) )
% 0.35/0.72 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_inj_pi
% 0.35/0.72 thf(fact_9154_cos__mono__le__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_mono_le_eq
% 0.35/0.72 thf(fact_9155_cos__monotone__0__pi__le,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_monotone_0_pi_le
% 0.35/0.72 thf(fact_9156_sin__ge__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_ge_zero
% 0.35/0.72 thf(fact_9157_cos__monotone__0__pi,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_monotone_0_pi
% 0.35/0.72 thf(fact_9158_cos__mono__less__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.35/0.72 => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_mono_less_eq
% 0.35/0.72 thf(fact_9159_sin__eq__0__pi,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ pi )
% 0.35/0.72 => ( ( ( sin_real @ X2 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 => ( X2 = zero_zero_real ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_eq_0_pi
% 0.35/0.72 thf(fact_9160_cos__monotone__minus__pi__0_H,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_monotone_minus_pi_0'
% 0.35/0.72 thf(fact_9161_sincos__principal__value,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ? [Y4: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 0.35/0.72 & ( ord_less_eq_real @ Y4 @ pi )
% 0.35/0.72 & ( ( sin_real @ Y4 )
% 0.35/0.72 = ( sin_real @ X2 ) )
% 0.35/0.72 & ( ( cos_real @ Y4 )
% 0.35/0.72 = ( cos_real @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sincos_principal_value
% 0.35/0.72 thf(fact_9162_sin__zero__pi__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ pi )
% 0.35/0.72 => ( ( ( sin_real @ X2 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 = ( X2 = zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_zero_pi_iff
% 0.35/0.72 thf(fact_9163_pi__less__4,axiom,
% 0.35/0.72 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % pi_less_4
% 0.35/0.72 thf(fact_9164_pi__ge__two,axiom,
% 0.35/0.72 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 0.35/0.72
% 0.35/0.72 % pi_ge_two
% 0.35/0.72 thf(fact_9165_cos__monotone__minus__pi__0,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_monotone_minus_pi_0
% 0.35/0.72 thf(fact_9166_cos__total,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.72 => ? [X5: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.35/0.72 & ( ord_less_eq_real @ X5 @ pi )
% 0.35/0.72 & ( ( cos_real @ X5 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 & ! [Y5: real] :
% 0.35/0.72 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 0.35/0.72 & ( ord_less_eq_real @ Y5 @ pi )
% 0.35/0.72 & ( ( cos_real @ Y5 )
% 0.35/0.72 = Y3 ) )
% 0.35/0.72 => ( Y5 = X5 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_total
% 0.35/0.72 thf(fact_9167_pi__half__less__two,axiom,
% 0.35/0.72 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.72
% 0.35/0.72 % pi_half_less_two
% 0.35/0.72 thf(fact_9168_pi__half__le__two,axiom,
% 0.35/0.72 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.72
% 0.35/0.72 % pi_half_le_two
% 0.35/0.72 thf(fact_9169_sin__pi__divide__n__ge__0,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( N != zero_zero_nat )
% 0.35/0.72 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_pi_divide_n_ge_0
% 0.35/0.72 thf(fact_9170_pi__half__gt__zero,axiom,
% 0.35/0.72 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % pi_half_gt_zero
% 0.35/0.72 thf(fact_9171_cos__gt__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_gt_zero
% 0.35/0.72 thf(fact_9172_sin__gt__zero2,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_gt_zero2
% 0.35/0.72 thf(fact_9173_sin__lt__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ pi @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_lt_zero
% 0.35/0.72 thf(fact_9174_pi__half__ge__zero,axiom,
% 0.35/0.72 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % pi_half_ge_zero
% 0.35/0.72 thf(fact_9175_m2pi__less__pi,axiom,
% 0.35/0.72 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.35/0.72
% 0.35/0.72 % m2pi_less_pi
% 0.35/0.72 thf(fact_9176_sin__inj__pi,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ( sin_real @ X2 )
% 0.35/0.72 = ( sin_real @ Y3 ) )
% 0.35/0.72 => ( X2 = Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_inj_pi
% 0.35/0.72 thf(fact_9177_sin__mono__le__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_mono_le_eq
% 0.35/0.72 thf(fact_9178_sin__monotone__2pi__le,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_monotone_2pi_le
% 0.35/0.72 thf(fact_9179_arctan__ubound,axiom,
% 0.35/0.72 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan_ubound
% 0.35/0.72 thf(fact_9180_sin__le__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ pi @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_le_zero
% 0.35/0.72 thf(fact_9181_minus__pi__half__less__zero,axiom,
% 0.35/0.72 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.35/0.72
% 0.35/0.72 % minus_pi_half_less_zero
% 0.35/0.72 thf(fact_9182_cos__gt__zero__pi,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_gt_zero_pi
% 0.35/0.72 thf(fact_9183_sin__less__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_less_zero
% 0.35/0.72 thf(fact_9184_cos__ge__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_ge_zero
% 0.35/0.72 thf(fact_9185_sin__mono__less__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_mono_less_eq
% 0.35/0.72 thf(fact_9186_sin__monotone__2pi,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_monotone_2pi
% 0.35/0.72 thf(fact_9187_sin__total,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.72 => ? [X5: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.35/0.72 & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( sin_real @ X5 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 & ! [Y5: real] :
% 0.35/0.72 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.35/0.72 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( sin_real @ Y5 )
% 0.35/0.72 = Y3 ) )
% 0.35/0.72 => ( Y5 = X5 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_total
% 0.35/0.72 thf(fact_9188_arctan__lbound,axiom,
% 0.35/0.72 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan_lbound
% 0.35/0.72 thf(fact_9189_arctan__bounded,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.35/0.72 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan_bounded
% 0.35/0.72 thf(fact_9190_sin__pi__divide__n__gt__0,axiom,
% 0.35/0.72 ! [N: nat] :
% 0.35/0.72 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_pi_divide_n_gt_0
% 0.35/0.72 thf(fact_9191_sincos__total__pi,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = one_one_real )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_eq_real @ T6 @ pi )
% 0.35/0.72 & ( X2
% 0.35/0.72 = ( cos_real @ T6 ) )
% 0.35/0.72 & ( Y3
% 0.35/0.72 = ( sin_real @ T6 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sincos_total_pi
% 0.35/0.72 thf(fact_9192_sincos__total__pi__half,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = one_one_real )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( X2
% 0.35/0.72 = ( cos_real @ T6 ) )
% 0.35/0.72 & ( Y3
% 0.35/0.72 = ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sincos_total_pi_half
% 0.35/0.72 thf(fact_9193_sincos__total__2pi__le,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = one_one_real )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72 & ( X2
% 0.35/0.72 = ( cos_real @ T6 ) )
% 0.35/0.72 & ( Y3
% 0.35/0.72 = ( sin_real @ T6 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sincos_total_2pi_le
% 0.35/0.72 thf(fact_9194_cos__zero__lemma,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ( cos_real @ X2 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 => ? [N3: nat] :
% 0.35/0.72 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.35/0.72 & ( X2
% 0.35/0.72 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_zero_lemma
% 0.35/0.72 thf(fact_9195_sin__zero__lemma,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ( sin_real @ X2 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 => ? [N3: nat] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 0.35/0.72 & ( X2
% 0.35/0.72 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_zero_lemma
% 0.35/0.72 thf(fact_9196_sincos__total__2pi,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = one_one_real )
% 0.35/0.72 => ~ ! [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72 => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72 => ( ( X2
% 0.35/0.72 = ( cos_real @ T6 ) )
% 0.35/0.72 => ( Y3
% 0.35/0.72 != ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sincos_total_2pi
% 0.35/0.72 thf(fact_9197_Maclaurin__cos__expansion2,axiom,
% 0.35/0.72 ! [X2: real,N: nat] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.72 & ( ord_less_real @ T6 @ X2 )
% 0.35/0.72 & ( ( cos_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_cos_expansion2
% 0.35/0.72 thf(fact_9198_Maclaurin__minus__cos__expansion,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ? [T6: real] :
% 0.35/0.72 ( ( ord_less_real @ X2 @ T6 )
% 0.35/0.72 & ( ord_less_real @ T6 @ zero_zero_real )
% 0.35/0.72 & ( ( cos_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_minus_cos_expansion
% 0.35/0.72 thf(fact_9199_Maclaurin__cos__expansion,axiom,
% 0.35/0.72 ! [X2: real,N: nat] :
% 0.35/0.72 ? [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.72 & ( ( cos_real @ X2 )
% 0.35/0.72 = ( plus_plus_real
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.72 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_cos_expansion
% 0.35/0.72 thf(fact_9200_complex__unimodular__polar,axiom,
% 0.35/0.72 ! [Z: complex] :
% 0.35/0.72 ( ( ( real_V1022390504157884413omplex @ Z )
% 0.35/0.72 = one_one_real )
% 0.35/0.72 => ~ ! [T6: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.72 => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.35/0.72 => ( Z
% 0.35/0.72 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % complex_unimodular_polar
% 0.35/0.72 thf(fact_9201_cos__coeff__0,axiom,
% 0.35/0.72 ( ( cos_coeff @ zero_zero_nat )
% 0.35/0.72 = one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % cos_coeff_0
% 0.35/0.72 thf(fact_9202_lemma__tan__total,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ? [X5: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X5 )
% 0.35/0.72 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % lemma_tan_total
% 0.35/0.72 thf(fact_9203_tan__gt__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_gt_zero
% 0.35/0.72 thf(fact_9204_lemma__tan__total1,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ? [X5: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.35/0.72 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( tan_real @ X5 )
% 0.35/0.72 = Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % lemma_tan_total1
% 0.35/0.72 thf(fact_9205_tan__mono__lt__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_mono_lt_eq
% 0.35/0.72 thf(fact_9206_tan__monotone_H,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ X2 )
% 0.35/0.72 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_monotone'
% 0.35/0.72 thf(fact_9207_tan__monotone,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_monotone
% 0.35/0.72 thf(fact_9208_tan__total,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ? [X5: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 0.35/0.72 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( tan_real @ X5 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 & ! [Y5: real] :
% 0.35/0.72 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 0.35/0.72 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( tan_real @ Y5 )
% 0.35/0.72 = Y3 ) )
% 0.35/0.72 => ( Y5 = X5 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_total
% 0.35/0.72 thf(fact_9209_tan__pos__pi2__le,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_pos_pi2_le
% 0.35/0.72 thf(fact_9210_tan__total__pos,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ? [X5: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 0.35/0.72 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( tan_real @ X5 )
% 0.35/0.72 = Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_total_pos
% 0.35/0.72 thf(fact_9211_tan__less__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_less_zero
% 0.35/0.72 thf(fact_9212_tan__mono__le__eq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_mono_le_eq
% 0.35/0.72 thf(fact_9213_tan__mono__le,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_mono_le
% 0.35/0.72 thf(fact_9214_tan__bound__pi2,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_bound_pi2
% 0.35/0.72 thf(fact_9215_arctan,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.35/0.72 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 & ( ( tan_real @ ( arctan @ Y3 ) )
% 0.35/0.72 = Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan
% 0.35/0.72 thf(fact_9216_arctan__tan,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( arctan @ ( tan_real @ X2 ) )
% 0.35/0.72 = X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan_tan
% 0.35/0.72 thf(fact_9217_arctan__unique,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( ( tan_real @ X2 )
% 0.35/0.72 = Y3 )
% 0.35/0.72 => ( ( arctan @ Y3 )
% 0.35/0.72 = X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arctan_unique
% 0.35/0.72 thf(fact_9218_tan__total__pi4,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.72 => ? [Z4: real] :
% 0.35/0.72 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 0.35/0.72 & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72 & ( ( tan_real @ Z4 )
% 0.35/0.72 = X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % tan_total_pi4
% 0.35/0.72 thf(fact_9219_sin__tan,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( sin_real @ X2 )
% 0.35/0.72 = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sin_tan
% 0.35/0.72 thf(fact_9220_cos__tan,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ( cos_real @ X2 )
% 0.35/0.72 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cos_tan
% 0.35/0.72 thf(fact_9221_Maclaurin__sin__bound,axiom,
% 0.35/0.72 ! [X2: real,N: nat] :
% 0.35/0.72 ( ord_less_eq_real
% 0.35/0.72 @ ( abs_abs_real
% 0.35/0.72 @ ( minus_minus_real @ ( sin_real @ X2 )
% 0.35/0.72 @ ( groups6591440286371151544t_real
% 0.35/0.72 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.72 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 0.35/0.72 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % Maclaurin_sin_bound
% 0.35/0.72 thf(fact_9222_cot__less__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.72 => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % cot_less_zero
% 0.35/0.72 thf(fact_9223_real__sqrt__less__iff,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_less_iff
% 0.35/0.72 thf(fact_9224_real__sqrt__le__iff,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_le_iff
% 0.35/0.72 thf(fact_9225_real__sqrt__gt__0__iff,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_gt_0_iff
% 0.35/0.72 thf(fact_9226_real__sqrt__lt__0__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 0.35/0.72 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_lt_0_iff
% 0.35/0.72 thf(fact_9227_real__sqrt__ge__0__iff,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_0_iff
% 0.35/0.72 thf(fact_9228_real__sqrt__le__0__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_le_0_iff
% 0.35/0.72 thf(fact_9229_real__sqrt__lt__1__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
% 0.35/0.72 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_lt_1_iff
% 0.35/0.72 thf(fact_9230_real__sqrt__gt__1__iff,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_gt_1_iff
% 0.35/0.72 thf(fact_9231_real__sqrt__le__1__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
% 0.35/0.72 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_le_1_iff
% 0.35/0.72 thf(fact_9232_real__sqrt__ge__1__iff,axiom,
% 0.35/0.72 ! [Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.35/0.72 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_1_iff
% 0.35/0.72 thf(fact_9233_real__sqrt__pow2,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.72 = X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_pow2
% 0.35/0.72 thf(fact_9234_real__sqrt__pow2__iff,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.72 = X2 )
% 0.35/0.72 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_pow2_iff
% 0.35/0.72 thf(fact_9235_real__sqrt__le__mono,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_le_mono
% 0.35/0.72 thf(fact_9236_real__sqrt__less__mono,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.72 => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_less_mono
% 0.35/0.72 thf(fact_9237_sqrt__divide__self__eq,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
% 0.35/0.72 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_divide_self_eq
% 0.35/0.72 thf(fact_9238_real__sqrt__gt__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_gt_zero
% 0.35/0.72 thf(fact_9239_real__sqrt__ge__zero,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_zero
% 0.35/0.72 thf(fact_9240_real__sqrt__eq__zero__cancel,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ( sqrt @ X2 )
% 0.35/0.72 = zero_zero_real )
% 0.35/0.72 => ( X2 = zero_zero_real ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_eq_zero_cancel
% 0.35/0.72 thf(fact_9241_real__sqrt__ge__one,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_one
% 0.35/0.72 thf(fact_9242_real__inv__sqrt__pow2,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.72 = ( inverse_inverse_real @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_inv_sqrt_pow2
% 0.35/0.72 thf(fact_9243_real__div__sqrt,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
% 0.35/0.72 = ( sqrt @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_div_sqrt
% 0.35/0.72 thf(fact_9244_sqrt__add__le__add__sqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_add_le_add_sqrt
% 0.35/0.72 thf(fact_9245_le__real__sqrt__sumsq,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % le_real_sqrt_sumsq
% 0.35/0.72 thf(fact_9246_inverse__powr,axiom,
% 0.35/0.72 ! [Y3: real,A2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A2 )
% 0.35/0.72 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % inverse_powr
% 0.35/0.72 thf(fact_9247_sqrt2__less__2,axiom,
% 0.35/0.72 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt2_less_2
% 0.35/0.72 thf(fact_9248_forall__pos__mono__1,axiom,
% 0.35/0.72 ! [P: real > $o,E2: real] :
% 0.35/0.72 ( ! [D6: real,E: real] :
% 0.35/0.72 ( ( ord_less_real @ D6 @ E )
% 0.35/0.72 => ( ( P @ D6 )
% 0.35/0.72 => ( P @ E ) ) )
% 0.35/0.72 => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.72 => ( P @ E2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % forall_pos_mono_1
% 0.35/0.72 thf(fact_9249_forall__pos__mono,axiom,
% 0.35/0.72 ! [P: real > $o,E2: real] :
% 0.35/0.72 ( ! [D6: real,E: real] :
% 0.35/0.72 ( ( ord_less_real @ D6 @ E )
% 0.35/0.72 => ( ( P @ D6 )
% 0.35/0.72 => ( P @ E ) ) )
% 0.35/0.72 => ( ! [N3: nat] :
% 0.35/0.72 ( ( N3 != zero_zero_nat )
% 0.35/0.72 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.72 => ( P @ E2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % forall_pos_mono
% 0.35/0.72 thf(fact_9250_real__arch__inverse,axiom,
% 0.35/0.72 ! [E2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.35/0.72 = ( ? [N2: nat] :
% 0.35/0.72 ( ( N2 != zero_zero_nat )
% 0.35/0.72 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.35/0.72 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_arch_inverse
% 0.35/0.72 thf(fact_9251_ln__inverse,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ln_ln_real @ ( inverse_inverse_real @ X2 ) )
% 0.35/0.72 = ( uminus_uminus_real @ ( ln_ln_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % ln_inverse
% 0.35/0.72 thf(fact_9252_real__less__rsqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.35/0.72 => ( ord_less_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_less_rsqrt
% 0.35/0.72 thf(fact_9253_real__le__rsqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_le_rsqrt
% 0.35/0.72 thf(fact_9254_sqrt__le__D,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_le_D
% 0.35/0.72 thf(fact_9255_log__inverse,axiom,
% 0.35/0.72 ! [A2: real,X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ A2 )
% 0.35/0.72 => ( ( A2 != one_one_real )
% 0.35/0.72 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( log @ A2 @ ( inverse_inverse_real @ X2 ) )
% 0.35/0.72 = ( uminus_uminus_real @ ( log @ A2 @ X2 ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % log_inverse
% 0.35/0.72 thf(fact_9256_real__le__lsqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_le_lsqrt
% 0.35/0.72 thf(fact_9257_real__sqrt__unique,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] :
% 0.35/0.72 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.72 = X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( sqrt @ X2 )
% 0.35/0.72 = Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_unique
% 0.35/0.72 thf(fact_9258_lemma__real__divide__sqrt__less,axiom,
% 0.35/0.72 ! [U: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ U )
% 0.35/0.72 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.35/0.72
% 0.35/0.72 % lemma_real_divide_sqrt_less
% 0.35/0.72 thf(fact_9259_real__sqrt__sum__squares__ge1,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_sum_squares_ge1
% 0.35/0.72 thf(fact_9260_real__sqrt__sum__squares__ge2,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_sum_squares_ge2
% 0.35/0.72 thf(fact_9261_real__sqrt__sum__squares__triangle__ineq,axiom,
% 0.35/0.72 ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A2 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B3 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( 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 ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_sum_squares_triangle_ineq
% 0.35/0.72 thf(fact_9262_sqrt__ge__absD,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y3 ) )
% 0.35/0.72 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_ge_absD
% 0.35/0.72 thf(fact_9263_exp__plus__inverse__exp,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % exp_plus_inverse_exp
% 0.35/0.72 thf(fact_9264_real__less__lsqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ( ord_less_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.72 => ( ord_less_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_less_lsqrt
% 0.35/0.72 thf(fact_9265_sqrt__sum__squares__le__sum,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_sum_squares_le_sum
% 0.35/0.72 thf(fact_9266_real__sqrt__ge__abs1,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_abs1
% 0.35/0.72 thf(fact_9267_real__sqrt__ge__abs2,axiom,
% 0.35/0.72 ! [Y3: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_ge_abs2
% 0.35/0.72 thf(fact_9268_sqrt__sum__squares__le__sum__abs,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % sqrt_sum_squares_le_sum_abs
% 0.35/0.72 thf(fact_9269_ln__sqrt,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ln_ln_real @ ( sqrt @ X2 ) )
% 0.35/0.72 = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % ln_sqrt
% 0.35/0.72 thf(fact_9270_plus__inverse__ge__2,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % plus_inverse_ge_2
% 0.35/0.72 thf(fact_9271_arsinh__real__aux,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arsinh_real_aux
% 0.35/0.72 thf(fact_9272_real__sqrt__power__even,axiom,
% 0.35/0.72 ! [N: nat,X2: real] :
% 0.35/0.72 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( power_power_real @ ( sqrt @ X2 ) @ N )
% 0.35/0.72 = ( power_power_real @ X2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_power_even
% 0.35/0.72 thf(fact_9273_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 0.35/0.72 ! [X2: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_sum_squares_mult_ge_zero
% 0.35/0.72 thf(fact_9274_arith__geo__mean__sqrt,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.72 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arith_geo_mean_sqrt
% 0.35/0.72 thf(fact_9275_powr__half__sqrt,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.72 = ( sqrt @ X2 ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % powr_half_sqrt
% 0.35/0.72 thf(fact_9276_real__le__x__sinh,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_le_x_sinh
% 0.35/0.72 thf(fact_9277_real__le__abs__sinh,axiom,
% 0.35/0.72 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_le_abs_sinh
% 0.35/0.72 thf(fact_9278_cos__x__y__le__one,axiom,
% 0.35/0.72 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 0.35/0.72
% 0.35/0.72 % cos_x_y_le_one
% 0.35/0.72 thf(fact_9279_real__sqrt__sum__squares__less,axiom,
% 0.35/0.72 ! [X2: real,U: real,Y3: real] :
% 0.35/0.72 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.72 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % real_sqrt_sum_squares_less
% 0.35/0.72 thf(fact_9280_arcosh__real__def,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 0.35/0.72 => ( ( arcosh_real @ X2 )
% 0.35/0.72 = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % arcosh_real_def
% 0.35/0.72 thf(fact_9281_powr__real__of__int,axiom,
% 0.35/0.72 ! [X2: real,N: int] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 0.35/0.72 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.35/0.72 = ( power_power_real @ X2 @ ( nat2 @ N ) ) ) )
% 0.35/0.72 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 0.35/0.72 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.35/0.72 = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 0.35/0.72
% 0.35/0.72 % powr_real_of_int
% 0.35/0.72 thf(fact_9282_sinh__ln__real,axiom,
% 0.35/0.72 ! [X2: real] :
% 0.35/0.72 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.72 => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.72 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sinh_ln_real
% 0.35/0.73 thf(fact_9283_cot__gt__zero,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cot_gt_zero
% 0.35/0.73 thf(fact_9284_sqrt__sum__squares__half__less,axiom,
% 0.35/0.73 ! [X2: real,U: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.73 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sqrt_sum_squares_half_less
% 0.35/0.73 thf(fact_9285_sin__cos__sqrt,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
% 0.35/0.73 => ( ( sin_real @ X2 )
% 0.35/0.73 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sin_cos_sqrt
% 0.35/0.73 thf(fact_9286_cos__arcsin,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( cos_real @ ( arcsin @ X2 ) )
% 0.35/0.73 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cos_arcsin
% 0.35/0.73 thf(fact_9287_sin__arccos__abs,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( sin_real @ ( arccos @ Y3 ) )
% 0.35/0.73 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sin_arccos_abs
% 0.35/0.73 thf(fact_9288_sin__arccos,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( sin_real @ ( arccos @ X2 ) )
% 0.35/0.73 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sin_arccos
% 0.35/0.73 thf(fact_9289_le__arcsin__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X2 ) )
% 0.35/0.73 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % le_arcsin_iff
% 0.35/0.73 thf(fact_9290_cos__arccos,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cos_arccos
% 0.35/0.73 thf(fact_9291_sin__arcsin,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sin_arcsin
% 0.35/0.73 thf(fact_9292_cosh__real__pos,axiom,
% 0.35/0.73 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_pos
% 0.35/0.73 thf(fact_9293_arcosh__cosh__real,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( arcosh_real @ ( cosh_real @ X2 ) )
% 0.35/0.73 = X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcosh_cosh_real
% 0.35/0.73 thf(fact_9294_cosh__real__nonpos__le__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.35/0.73 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_nonpos_le_iff
% 0.35/0.73 thf(fact_9295_cosh__real__nonneg__le__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.35/0.73 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_nonneg_le_iff
% 0.35/0.73 thf(fact_9296_cosh__real__nonneg,axiom,
% 0.35/0.73 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_nonneg
% 0.35/0.73 thf(fact_9297_cosh__real__ge__1,axiom,
% 0.35/0.73 ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_ge_1
% 0.35/0.73 thf(fact_9298_sinh__less__cosh__real,axiom,
% 0.35/0.73 ! [X2: real] : ( ord_less_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % sinh_less_cosh_real
% 0.35/0.73 thf(fact_9299_sinh__le__cosh__real,axiom,
% 0.35/0.73 ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % sinh_le_cosh_real
% 0.35/0.73 thf(fact_9300_cosh__real__strict__mono,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.73 => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_strict_mono
% 0.35/0.73 thf(fact_9301_cosh__real__nonneg__less__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.73 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.35/0.73 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_nonneg_less_iff
% 0.35/0.73 thf(fact_9302_cosh__real__nonpos__less__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.35/0.73 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 0.35/0.73 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_real_nonpos_less_iff
% 0.35/0.73 thf(fact_9303_arccos__le__arccos,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_le_arccos
% 0.35/0.73 thf(fact_9304_arccos__eq__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 0.35/0.73 => ( ( ( arccos @ X2 )
% 0.35/0.73 = ( arccos @ Y3 ) )
% 0.35/0.73 = ( X2 = Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_eq_iff
% 0.35/0.73 thf(fact_9305_arccos__le__mono,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 0.35/0.73 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_le_mono
% 0.35/0.73 thf(fact_9306_arcsin__minus,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.73 = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_minus
% 0.35/0.73 thf(fact_9307_arcsin__le__arcsin,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_le_arcsin
% 0.35/0.73 thf(fact_9308_arcsin__eq__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( ( arcsin @ X2 )
% 0.35/0.73 = ( arcsin @ Y3 ) )
% 0.35/0.73 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_eq_iff
% 0.35/0.73 thf(fact_9309_arcsin__le__mono,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_le_mono
% 0.35/0.73 thf(fact_9310_arccos__lbound,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_lbound
% 0.35/0.73 thf(fact_9311_arccos__less__arccos,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_less_arccos
% 0.35/0.73 thf(fact_9312_arccos__less__mono,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 0.35/0.73 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_less_mono
% 0.35/0.73 thf(fact_9313_arccos__ubound,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_ubound
% 0.35/0.73 thf(fact_9314_arccos__cos,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.73 => ( ( arccos @ ( cos_real @ X2 ) )
% 0.35/0.73 = X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_cos
% 0.35/0.73 thf(fact_9315_arcsin__less__arcsin,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_less_arcsin
% 0.35/0.73 thf(fact_9316_arcsin__less__mono,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_less_mono
% 0.35/0.73 thf(fact_9317_cos__arccos__abs,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.35/0.73 => ( ( cos_real @ ( arccos @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ).
% 0.35/0.73
% 0.35/0.73 % cos_arccos_abs
% 0.35/0.73 thf(fact_9318_arccos__cos__eq__abs,axiom,
% 0.35/0.73 ! [Theta: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 0.35/0.73 => ( ( arccos @ ( cos_real @ Theta ) )
% 0.35/0.73 = ( abs_abs_real @ Theta ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_cos_eq_abs
% 0.35/0.73 thf(fact_9319_arccos__lt__bounded,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.35/0.73 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_lt_bounded
% 0.35/0.73 thf(fact_9320_arccos__bounded,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.35/0.73 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_bounded
% 0.35/0.73 thf(fact_9321_sin__arccos__nonzero,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( sin_real @ ( arccos @ X2 ) )
% 0.35/0.73 != zero_zero_real ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sin_arccos_nonzero
% 0.35/0.73 thf(fact_9322_arccos__cos2,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 0.35/0.73 => ( ( arccos @ ( cos_real @ X2 ) )
% 0.35/0.73 = ( uminus_uminus_real @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_cos2
% 0.35/0.73 thf(fact_9323_arccos__minus,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.73 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_minus
% 0.35/0.73 thf(fact_9324_cos__arcsin__nonzero,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( cos_real @ ( arcsin @ X2 ) )
% 0.35/0.73 != zero_zero_real ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cos_arcsin_nonzero
% 0.35/0.73 thf(fact_9325_arccos,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.35/0.73 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 0.35/0.73 & ( ( cos_real @ ( arccos @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos
% 0.35/0.73 thf(fact_9326_arccos__minus__abs,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 0.35/0.73 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_minus_abs
% 0.35/0.73 thf(fact_9327_arccos__le__pi2,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_le_pi2
% 0.35/0.73 thf(fact_9328_cosh__ln__real,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
% 0.35/0.73 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % cosh_ln_real
% 0.35/0.73 thf(fact_9329_arcsin__lt__bounded,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_lt_bounded
% 0.35/0.73 thf(fact_9330_arcsin__lbound,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_lbound
% 0.35/0.73 thf(fact_9331_arcsin__ubound,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_ubound
% 0.35/0.73 thf(fact_9332_arcsin__bounded,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_bounded
% 0.35/0.73 thf(fact_9333_arcsin__sin,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ( arcsin @ ( sin_real @ X2 ) )
% 0.35/0.73 = X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_sin
% 0.35/0.73 thf(fact_9334_arcsin,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin
% 0.35/0.73 thf(fact_9335_arcsin__pi,axiom,
% 0.35/0.73 ! [Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.35/0.73 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 0.35/0.73 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 0.35/0.73 = Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_pi
% 0.35/0.73 thf(fact_9336_arcsin__le__iff,axiom,
% 0.35/0.73 ! [X2: real,Y3: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y3 )
% 0.35/0.73 = ( ord_less_eq_real @ X2 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_le_iff
% 0.35/0.73 thf(fact_9337_int__ge__less__than2__def,axiom,
% 0.35/0.73 ( int_ge_less_than2
% 0.35/0.73 = ( ^ [D5: int] :
% 0.35/0.73 ( collec213857154873943460nt_int
% 0.35/0.73 @ ( produc4947309494688390418_int_o
% 0.35/0.73 @ ^ [Z8: int,Z2: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ D5 @ Z2 )
% 0.35/0.73 & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % int_ge_less_than2_def
% 0.35/0.73 thf(fact_9338_int__ge__less__than__def,axiom,
% 0.35/0.73 ( int_ge_less_than
% 0.35/0.73 = ( ^ [D5: int] :
% 0.35/0.73 ( collec213857154873943460nt_int
% 0.35/0.73 @ ( produc4947309494688390418_int_o
% 0.35/0.73 @ ^ [Z8: int,Z2: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ D5 @ Z8 )
% 0.35/0.73 & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % int_ge_less_than_def
% 0.35/0.73 thf(fact_9339_set__decode__0,axiom,
% 0.35/0.73 ! [X2: nat] :
% 0.35/0.73 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
% 0.35/0.73 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % set_decode_0
% 0.35/0.73 thf(fact_9340_arctan__def,axiom,
% 0.35/0.73 ( arctan
% 0.35/0.73 = ( ^ [Y: real] :
% 0.35/0.73 ( the_real
% 0.35/0.73 @ ^ [X: real] :
% 0.35/0.73 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.73 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 & ( ( tan_real @ X )
% 0.35/0.73 = Y ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arctan_def
% 0.35/0.73 thf(fact_9341_set__decode__zero,axiom,
% 0.35/0.73 ( ( nat_set_decode @ zero_zero_nat )
% 0.35/0.73 = bot_bot_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % set_decode_zero
% 0.35/0.73 thf(fact_9342_set__encode__inverse,axiom,
% 0.35/0.73 ! [A3: set_nat] :
% 0.35/0.73 ( ( finite_finite_nat @ A3 )
% 0.35/0.73 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
% 0.35/0.73 = A3 ) ) ).
% 0.35/0.73
% 0.35/0.73 % set_encode_inverse
% 0.35/0.73 thf(fact_9343_finite__set__decode,axiom,
% 0.35/0.73 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_set_decode
% 0.35/0.73 thf(fact_9344_ln__neg__is__const,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( ( ln_ln_real @ X2 )
% 0.35/0.73 = ( the_real
% 0.35/0.73 @ ^ [X: real] : $false ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % ln_neg_is_const
% 0.35/0.73 thf(fact_9345_subset__decode__imp__le,axiom,
% 0.35/0.73 ! [M: nat,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 0.35/0.73 => ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % subset_decode_imp_le
% 0.35/0.73 thf(fact_9346_arccos__def,axiom,
% 0.35/0.73 ( arccos
% 0.35/0.73 = ( ^ [Y: real] :
% 0.35/0.73 ( the_real
% 0.35/0.73 @ ^ [X: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.73 & ( ord_less_eq_real @ X @ pi )
% 0.35/0.73 & ( ( cos_real @ X )
% 0.35/0.73 = Y ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arccos_def
% 0.35/0.73 thf(fact_9347_pi__half,axiom,
% 0.35/0.73 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.73 = ( the_real
% 0.35/0.73 @ ^ [X: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.73 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.73 & ( ( cos_real @ X )
% 0.35/0.73 = zero_zero_real ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % pi_half
% 0.35/0.73 thf(fact_9348_pi__def,axiom,
% 0.35/0.73 ( pi
% 0.35/0.73 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.35/0.73 @ ( the_real
% 0.35/0.73 @ ^ [X: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 0.35/0.73 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.35/0.73 & ( ( cos_real @ X )
% 0.35/0.73 = zero_zero_real ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % pi_def
% 0.35/0.73 thf(fact_9349_arcsin__def,axiom,
% 0.35/0.73 ( arcsin
% 0.35/0.73 = ( ^ [Y: real] :
% 0.35/0.73 ( the_real
% 0.35/0.73 @ ^ [X: real] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.35/0.73 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.35/0.73 & ( ( sin_real @ X )
% 0.35/0.73 = Y ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % arcsin_def
% 0.35/0.73 thf(fact_9350_num_Osize__gen_I3_J,axiom,
% 0.35/0.73 ! [X32: num] :
% 0.35/0.73 ( ( size_num @ ( bit1 @ X32 ) )
% 0.35/0.73 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % num.size_gen(3)
% 0.35/0.73 thf(fact_9351_modulo__int__unfold,axiom,
% 0.35/0.73 ! [L: int,K: int,N: nat,M: nat] :
% 0.35/0.73 ( ( ( ( ( sgn_sgn_int @ L )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( ( sgn_sgn_int @ K )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( N = zero_zero_nat ) )
% 0.35/0.73 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.35/0.73 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( ( sgn_sgn_int @ K )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( N = zero_zero_nat ) )
% 0.35/0.73 => ( ( ( ( sgn_sgn_int @ K )
% 0.35/0.73 = ( sgn_sgn_int @ L ) )
% 0.35/0.73 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 0.35/0.73 & ( ( ( sgn_sgn_int @ K )
% 0.35/0.73 != ( sgn_sgn_int @ L ) )
% 0.35/0.73 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.35/0.73 @ ( minus_minus_int
% 0.35/0.73 @ ( semiri1314217659103216013at_int
% 0.35/0.73 @ ( times_times_nat @ N
% 0.35/0.73 @ ( zero_n2687167440665602831ol_nat
% 0.35/0.73 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 0.35/0.73 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % modulo_int_unfold
% 0.35/0.73 thf(fact_9352_num_Osize__gen_I2_J,axiom,
% 0.35/0.73 ! [X22: num] :
% 0.35/0.73 ( ( size_num @ ( bit0 @ X22 ) )
% 0.35/0.73 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % num.size_gen(2)
% 0.35/0.73 thf(fact_9353_take__bit__of__Suc__0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_of_Suc_0
% 0.35/0.73 thf(fact_9354_Suc__0__mod__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( zero_n2687167440665602831ol_nat
% 0.35/0.73 @ ( N
% 0.35/0.73 != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Suc_0_mod_eq
% 0.35/0.73 thf(fact_9355_take__bit__tightened__less__eq__nat,axiom,
% 0.35/0.73 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_tightened_less_eq_nat
% 0.35/0.73 thf(fact_9356_take__bit__nat__less__eq__self,axiom,
% 0.35/0.73 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_less_eq_self
% 0.35/0.73 thf(fact_9357_take__bit__nat__eq,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 0.35/0.73 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_eq
% 0.35/0.73 thf(fact_9358_nat__take__bit__eq,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.35/0.73 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % nat_take_bit_eq
% 0.35/0.73 thf(fact_9359_take__bit__tightened__less__eq__int,axiom,
% 0.35/0.73 ! [M: nat,N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_tightened_less_eq_int
% 0.35/0.73 thf(fact_9360_take__bit__nonnegative,axiom,
% 0.35/0.73 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nonnegative
% 0.35/0.73 thf(fact_9361_take__bit__int__less__eq__self__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.35/0.73 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_less_eq_self_iff
% 0.35/0.73 thf(fact_9362_not__take__bit__negative,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 0.35/0.73
% 0.35/0.73 % not_take_bit_negative
% 0.35/0.73 thf(fact_9363_take__bit__int__greater__self__iff,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.35/0.73 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_greater_self_iff
% 0.35/0.73 thf(fact_9364_take__bit__nat__eq__self__iff,axiom,
% 0.35/0.73 ! [N: nat,M: nat] :
% 0.35/0.73 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.35/0.73 = M )
% 0.35/0.73 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_eq_self_iff
% 0.35/0.73 thf(fact_9365_take__bit__nat__less__exp,axiom,
% 0.35/0.73 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_less_exp
% 0.35/0.73 thf(fact_9366_take__bit__nat__eq__self,axiom,
% 0.35/0.73 ! [M: nat,N: nat] :
% 0.35/0.73 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 0.35/0.73 = M ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_eq_self
% 0.35/0.73 thf(fact_9367_take__bit__int__less__exp,axiom,
% 0.35/0.73 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_less_exp
% 0.35/0.73 thf(fact_9368_num_Osize__gen_I1_J,axiom,
% 0.35/0.73 ( ( size_num @ one )
% 0.35/0.73 = zero_zero_nat ) ).
% 0.35/0.73
% 0.35/0.73 % num.size_gen(1)
% 0.35/0.73 thf(fact_9369_take__bit__nat__less__self__iff,axiom,
% 0.35/0.73 ! [N: nat,M: nat] :
% 0.35/0.73 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 0.35/0.73 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_nat_less_self_iff
% 0.35/0.73 thf(fact_9370_take__bit__int__greater__eq__self__iff,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 0.35/0.73 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_greater_eq_self_iff
% 0.35/0.73 thf(fact_9371_take__bit__int__less__self__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 0.35/0.73 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_less_self_iff
% 0.35/0.73 thf(fact_9372_take__bit__int__eq__self__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.35/0.73 = K )
% 0.35/0.73 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_eq_self_iff
% 0.35/0.73 thf(fact_9373_take__bit__int__eq__self,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 0.35/0.73 = K ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_eq_self
% 0.35/0.73 thf(fact_9374_floor__real__def,axiom,
% 0.35/0.73 ( archim6058952711729229775r_real
% 0.35/0.73 = ( ^ [X: real] :
% 0.35/0.73 ( the_int
% 0.35/0.73 @ ^ [Z2: int] :
% 0.35/0.73 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 0.35/0.73 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % floor_real_def
% 0.35/0.73 thf(fact_9375_take__bit__int__less__eq,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 0.35/0.73 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_less_eq
% 0.35/0.73 thf(fact_9376_take__bit__int__greater__eq,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.73 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_int_greater_eq
% 0.35/0.73 thf(fact_9377_take__bit__minus__small__eq,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( ord_less_int @ zero_zero_int @ K )
% 0.35/0.73 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 0.35/0.73 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % take_bit_minus_small_eq
% 0.35/0.73 thf(fact_9378_divide__int__unfold,axiom,
% 0.35/0.73 ! [L: int,K: int,N: nat,M: nat] :
% 0.35/0.73 ( ( ( ( ( sgn_sgn_int @ L )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( ( sgn_sgn_int @ K )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( N = zero_zero_nat ) )
% 0.35/0.73 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = zero_zero_int ) )
% 0.35/0.73 & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( ( sgn_sgn_int @ K )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 | ( N = zero_zero_nat ) )
% 0.35/0.73 => ( ( ( ( sgn_sgn_int @ K )
% 0.35/0.73 = ( sgn_sgn_int @ L ) )
% 0.35/0.73 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 0.35/0.73 & ( ( ( sgn_sgn_int @ K )
% 0.35/0.73 != ( sgn_sgn_int @ L ) )
% 0.35/0.73 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 0.35/0.73 = ( uminus_uminus_int
% 0.35/0.73 @ ( semiri1314217659103216013at_int
% 0.35/0.73 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 0.35/0.73 @ ( zero_n2687167440665602831ol_nat
% 0.35/0.73 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % divide_int_unfold
% 0.35/0.73 thf(fact_9379_flip__bit__nonnegative__int__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 0.35/0.73 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.73
% 0.35/0.73 % flip_bit_nonnegative_int_iff
% 0.35/0.73 thf(fact_9380_flip__bit__negative__int__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.73 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.73
% 0.35/0.73 % flip_bit_negative_int_iff
% 0.35/0.73 thf(fact_9381_and__int_Opsimps,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.35/0.73 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.35/0.73 = ( uminus_uminus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.35/0.73 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.35/0.73 = ( plus_plus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.35/0.73 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_int.psimps
% 0.35/0.73 thf(fact_9382_and__int_Opelims,axiom,
% 0.35/0.73 ! [X2: int,Xa2: int,Y3: int] :
% 0.35/0.73 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 0.35/0.73 = Y3 )
% 0.35/0.73 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 0.35/0.73 => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( uminus_uminus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.35/0.73 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( plus_plus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.35/0.73 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.35/0.73 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_int.pelims
% 0.35/0.73 thf(fact_9383_floor__rat__def,axiom,
% 0.35/0.73 ( archim3151403230148437115or_rat
% 0.35/0.73 = ( ^ [X: rat] :
% 0.35/0.73 ( the_int
% 0.35/0.73 @ ^ [Z2: int] :
% 0.35/0.73 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 0.35/0.73 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % floor_rat_def
% 0.35/0.73 thf(fact_9384_and__int_Osimps,axiom,
% 0.35/0.73 ( bit_se725231765392027082nd_int
% 0.35/0.73 = ( ^ [K3: int,L2: int] :
% 0.35/0.73 ( if_int
% 0.35/0.73 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 @ ( uminus_uminus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.35/0.73 @ ( plus_plus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.35/0.73 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_int.simps
% 0.35/0.73 thf(fact_9385_and__nonnegative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 0.35/0.73 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_nonnegative_int_iff
% 0.35/0.73 thf(fact_9386_and__negative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.35/0.73 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.73 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_negative_int_iff
% 0.35/0.73 thf(fact_9387_abs__rat__def,axiom,
% 0.35/0.73 ( abs_abs_rat
% 0.35/0.73 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % abs_rat_def
% 0.35/0.73 thf(fact_9388_sgn__rat__def,axiom,
% 0.35/0.73 ( sgn_sgn_rat
% 0.35/0.73 = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sgn_rat_def
% 0.35/0.73 thf(fact_9389_less__eq__rat__def,axiom,
% 0.35/0.73 ( ord_less_eq_rat
% 0.35/0.73 = ( ^ [X: rat,Y: rat] :
% 0.35/0.73 ( ( ord_less_rat @ X @ Y )
% 0.35/0.73 | ( X = Y ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % less_eq_rat_def
% 0.35/0.73 thf(fact_9390_obtain__pos__sum,axiom,
% 0.35/0.73 ! [R2: rat] :
% 0.35/0.73 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.73 => ~ ! [S3: rat] :
% 0.35/0.73 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 0.35/0.73 => ! [T6: rat] :
% 0.35/0.73 ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 0.35/0.73 => ( R2
% 0.35/0.73 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % obtain_pos_sum
% 0.35/0.73 thf(fact_9391_AND__lower,axiom,
% 0.35/0.73 ! [X2: int,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_lower
% 0.35/0.73 thf(fact_9392_AND__upper1,axiom,
% 0.35/0.73 ! [X2: int,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper1
% 0.35/0.73 thf(fact_9393_AND__upper2,axiom,
% 0.35/0.73 ! [Y3: int,X2: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Y3 ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper2
% 0.35/0.73 thf(fact_9394_AND__upper1_H,axiom,
% 0.35/0.73 ! [Y3: int,Z: int,Ya: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper1'
% 0.35/0.73 thf(fact_9395_AND__upper2_H,axiom,
% 0.35/0.73 ! [Y3: int,Z: int,X2: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ( ord_less_eq_int @ Y3 @ Z )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper2'
% 0.35/0.73 thf(fact_9396_AND__upper2_H_H,axiom,
% 0.35/0.73 ! [Y3: int,Z: int,X2: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ( ord_less_int @ Y3 @ Z )
% 0.35/0.73 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper2''
% 0.35/0.73 thf(fact_9397_AND__upper1_H_H,axiom,
% 0.35/0.73 ! [Y3: int,Z: int,Ya: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ( ord_less_int @ Y3 @ Z )
% 0.35/0.73 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % AND_upper1''
% 0.35/0.73 thf(fact_9398_and__less__eq,axiom,
% 0.35/0.73 ! [L: int,K: int] :
% 0.35/0.73 ( ( ord_less_int @ L @ zero_zero_int )
% 0.35/0.73 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_less_eq
% 0.35/0.73 thf(fact_9399_and__int_Oelims,axiom,
% 0.35/0.73 ! [X2: int,Xa2: int,Y3: int] :
% 0.35/0.73 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 0.35/0.73 = Y3 )
% 0.35/0.73 => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( uminus_uminus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.35/0.73 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.35/0.73 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( plus_plus_int
% 0.35/0.73 @ ( zero_n2684676970156552555ol_int
% 0.35/0.73 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 0.35/0.73 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.35/0.73 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_int.elims
% 0.35/0.73 thf(fact_9400_normalize__negative,axiom,
% 0.35/0.73 ! [Q3: int,P6: int] :
% 0.35/0.73 ( ( ord_less_int @ Q3 @ zero_zero_int )
% 0.35/0.73 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 0.35/0.73 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % normalize_negative
% 0.35/0.73 thf(fact_9401_xor__Suc__0__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.73 @ ( zero_n2687167440665602831ol_nat
% 0.35/0.73 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_Suc_0_eq
% 0.35/0.73 thf(fact_9402_Suc__0__xor__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.73 @ ( zero_n2687167440665602831ol_nat
% 0.35/0.73 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Suc_0_xor_eq
% 0.35/0.73 thf(fact_9403_and__nat__numerals_I3_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = zero_zero_nat ) ).
% 0.35/0.73
% 0.35/0.73 % and_nat_numerals(3)
% 0.35/0.73 thf(fact_9404_and__nat__numerals_I1_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.35/0.73 = zero_zero_nat ) ).
% 0.35/0.73
% 0.35/0.73 % and_nat_numerals(1)
% 0.35/0.73 thf(fact_9405_and__nat__numerals_I4_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = one_one_nat ) ).
% 0.35/0.73
% 0.35/0.73 % and_nat_numerals(4)
% 0.35/0.73 thf(fact_9406_and__nat__numerals_I2_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.35/0.73 = one_one_nat ) ).
% 0.35/0.73
% 0.35/0.73 % and_nat_numerals(2)
% 0.35/0.73 thf(fact_9407_and__Suc__0__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_Suc_0_eq
% 0.35/0.73 thf(fact_9408_Suc__0__and__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Suc_0_and_eq
% 0.35/0.73 thf(fact_9409_xor__nat__numerals_I1_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nat_numerals(1)
% 0.35/0.73 thf(fact_9410_xor__nat__numerals_I2_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nat_numerals(2)
% 0.35/0.73 thf(fact_9411_xor__nat__numerals_I3_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nat_numerals(3)
% 0.35/0.73 thf(fact_9412_xor__nat__numerals_I4_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nat_numerals(4)
% 0.35/0.73 thf(fact_9413_normalize__denom__pos,axiom,
% 0.35/0.73 ! [R2: product_prod_int_int,P6: int,Q3: int] :
% 0.35/0.73 ( ( ( normalize @ R2 )
% 0.35/0.73 = ( product_Pair_int_int @ P6 @ Q3 ) )
% 0.35/0.73 => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 0.35/0.73
% 0.35/0.73 % normalize_denom_pos
% 0.35/0.73 thf(fact_9414_and__nat__unfold,axiom,
% 0.35/0.73 ( bit_se727722235901077358nd_nat
% 0.35/0.73 = ( ^ [M2: nat,N2: nat] :
% 0.35/0.73 ( if_nat
% 0.35/0.73 @ ( ( M2 = zero_zero_nat )
% 0.35/0.73 | ( N2 = zero_zero_nat ) )
% 0.35/0.73 @ zero_zero_nat
% 0.35/0.73 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % and_nat_unfold
% 0.35/0.73 thf(fact_9415_xor__nat__unfold,axiom,
% 0.35/0.73 ( bit_se6528837805403552850or_nat
% 0.35/0.73 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nat_unfold
% 0.35/0.73 thf(fact_9416_horner__sum__of__bool__2__less,axiom,
% 0.35/0.73 ! [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 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % horner_sum_of_bool_2_less
% 0.35/0.73 thf(fact_9417_Least__eq__0,axiom,
% 0.35/0.73 ! [P: nat > $o] :
% 0.35/0.73 ( ( P @ zero_zero_nat )
% 0.35/0.73 => ( ( ord_Least_nat @ P )
% 0.35/0.73 = zero_zero_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % Least_eq_0
% 0.35/0.73 thf(fact_9418_xor__nonnegative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 0.35/0.73 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_nonnegative_int_iff
% 0.35/0.73 thf(fact_9419_xor__negative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.35/0.73 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.73 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % xor_negative_int_iff
% 0.35/0.73 thf(fact_9420_XOR__lower,axiom,
% 0.35/0.73 ! [X2: int,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % XOR_lower
% 0.35/0.73 thf(fact_9421_Least__Suc2,axiom,
% 0.35/0.73 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 0.35/0.73 ( ( P @ N )
% 0.35/0.73 => ( ( Q @ M )
% 0.35/0.73 => ( ~ ( P @ zero_zero_nat )
% 0.35/0.73 => ( ! [K2: nat] :
% 0.35/0.73 ( ( P @ ( suc @ K2 ) )
% 0.35/0.73 = ( Q @ K2 ) )
% 0.35/0.73 => ( ( ord_Least_nat @ P )
% 0.35/0.73 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Least_Suc2
% 0.35/0.73 thf(fact_9422_Least__Suc,axiom,
% 0.35/0.73 ! [P: nat > $o,N: nat] :
% 0.35/0.73 ( ( P @ N )
% 0.35/0.73 => ( ~ ( P @ zero_zero_nat )
% 0.35/0.73 => ( ( ord_Least_nat @ P )
% 0.35/0.73 = ( suc
% 0.35/0.73 @ ( ord_Least_nat
% 0.35/0.73 @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Least_Suc
% 0.35/0.73 thf(fact_9423_fact__eq__fact__times,axiom,
% 0.35/0.73 ! [N: nat,M: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.73 => ( ( semiri1408675320244567234ct_nat @ M )
% 0.35/0.73 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 0.35/0.73 @ ( groups708209901874060359at_nat
% 0.35/0.73 @ ^ [X: nat] : X
% 0.35/0.73 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % fact_eq_fact_times
% 0.35/0.73 thf(fact_9424_fact__div__fact,axiom,
% 0.35/0.73 ! [N: nat,M: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.73 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 0.35/0.73 = ( groups708209901874060359at_nat
% 0.35/0.73 @ ^ [X: nat] : X
% 0.35/0.73 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % fact_div_fact
% 0.35/0.73 thf(fact_9425_XOR__upper,axiom,
% 0.35/0.73 ! [X2: int,N: nat,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % XOR_upper
% 0.35/0.73 thf(fact_9426_or__nat__unfold,axiom,
% 0.35/0.73 ( bit_se1412395901928357646or_nat
% 0.35/0.73 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nat_unfold
% 0.35/0.73 thf(fact_9427_Sum__Ico__nat,axiom,
% 0.35/0.73 ! [M: nat,N: nat] :
% 0.35/0.73 ( ( groups3542108847815614940at_nat
% 0.35/0.73 @ ^ [X: nat] : X
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 0.35/0.73 = ( 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 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Sum_Ico_nat
% 0.35/0.73 thf(fact_9428_finite__atLeastLessThan,axiom,
% 0.35/0.73 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_atLeastLessThan
% 0.35/0.73 thf(fact_9429_card__atLeastLessThan,axiom,
% 0.35/0.73 ! [L: nat,U: nat] :
% 0.35/0.73 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 0.35/0.73 = ( minus_minus_nat @ U @ L ) ) ).
% 0.35/0.73
% 0.35/0.73 % card_atLeastLessThan
% 0.35/0.73 thf(fact_9430_atLeastLessThan__singleton,axiom,
% 0.35/0.73 ! [M: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.35/0.73 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThan_singleton
% 0.35/0.73 thf(fact_9431_or__nat__numerals_I4_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nat_numerals(4)
% 0.35/0.73 thf(fact_9432_or__nat__numerals_I2_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nat_numerals(2)
% 0.35/0.73 thf(fact_9433_or__nat__numerals_I1_J,axiom,
% 0.35/0.73 ! [Y3: num] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nat_numerals(1)
% 0.35/0.73 thf(fact_9434_or__nat__numerals_I3_J,axiom,
% 0.35/0.73 ! [X2: num] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nat_numerals(3)
% 0.35/0.73 thf(fact_9435_all__nat__less__eq,axiom,
% 0.35/0.73 ! [N: nat,P: nat > $o] :
% 0.35/0.73 ( ( ! [M2: nat] :
% 0.35/0.73 ( ( ord_less_nat @ M2 @ N )
% 0.35/0.73 => ( P @ M2 ) ) )
% 0.35/0.73 = ( ! [X: nat] :
% 0.35/0.73 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73 => ( P @ X ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % all_nat_less_eq
% 0.35/0.73 thf(fact_9436_ex__nat__less__eq,axiom,
% 0.35/0.73 ! [N: nat,P: nat > $o] :
% 0.35/0.73 ( ( ? [M2: nat] :
% 0.35/0.73 ( ( ord_less_nat @ M2 @ N )
% 0.35/0.73 & ( P @ M2 ) ) )
% 0.35/0.73 = ( ? [X: nat] :
% 0.35/0.73 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73 & ( P @ X ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % ex_nat_less_eq
% 0.35/0.73 thf(fact_9437_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.35/0.73 ! [L: nat,U: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.35/0.73 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThanSuc_atLeastAtMost
% 0.35/0.73 thf(fact_9438_lessThan__atLeast0,axiom,
% 0.35/0.73 ( set_ord_lessThan_nat
% 0.35/0.73 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % lessThan_atLeast0
% 0.35/0.73 thf(fact_9439_atLeastLessThan0,axiom,
% 0.35/0.73 ! [M: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.35/0.73 = bot_bot_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThan0
% 0.35/0.73 thf(fact_9440_atLeast0__lessThan__Suc,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeast0_lessThan_Suc
% 0.35/0.73 thf(fact_9441_prod__Suc__fact,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % prod_Suc_fact
% 0.35/0.73 thf(fact_9442_prod__Suc__Suc__fact,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 0.35/0.73 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % prod_Suc_Suc_fact
% 0.35/0.73 thf(fact_9443_prod__int__eq,axiom,
% 0.35/0.73 ! [I: nat,J: nat] :
% 0.35/0.73 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.35/0.73 = ( groups1705073143266064639nt_int
% 0.35/0.73 @ ^ [X: int] : X
% 0.35/0.73 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % prod_int_eq
% 0.35/0.73 thf(fact_9444_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.35/0.73 ! [N6: set_nat,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73 => ( finite_finite_nat @ N6 ) ) ).
% 0.35/0.73
% 0.35/0.73 % subset_eq_atLeast0_lessThan_finite
% 0.35/0.73 thf(fact_9445_subset__card__intvl__is__intvl,axiom,
% 0.35/0.73 ! [A3: set_nat,K: nat] :
% 0.35/0.73 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 0.35/0.73 => ( A3
% 0.35/0.73 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % subset_card_intvl_is_intvl
% 0.35/0.73 thf(fact_9446_atLeastLessThan__add__Un,axiom,
% 0.35/0.73 ! [I: nat,J: nat,K: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.73 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.73 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThan_add_Un
% 0.35/0.73 thf(fact_9447_prod__int__plus__eq,axiom,
% 0.35/0.73 ! [I: nat,J: nat] :
% 0.35/0.73 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.35/0.73 = ( groups1705073143266064639nt_int
% 0.35/0.73 @ ^ [X: int] : X
% 0.35/0.73 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % prod_int_plus_eq
% 0.35/0.73 thf(fact_9448_atLeastLessThanSuc,axiom,
% 0.35/0.73 ! [M: nat,N: nat] :
% 0.35/0.73 ( ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.73 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_nat @ M @ N )
% 0.35/0.73 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 0.35/0.73 = bot_bot_set_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThanSuc
% 0.35/0.73 thf(fact_9449_subset__eq__atLeast0__lessThan__card,axiom,
% 0.35/0.73 ! [N6: set_nat,N: nat] :
% 0.35/0.73 ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 0.35/0.73 => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % subset_eq_atLeast0_lessThan_card
% 0.35/0.73 thf(fact_9450_card__sum__le__nat__sum,axiom,
% 0.35/0.73 ! [S: set_nat] :
% 0.35/0.73 ( ord_less_eq_nat
% 0.35/0.73 @ ( groups3542108847815614940at_nat
% 0.35/0.73 @ ^ [X: nat] : X
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 0.35/0.73 @ ( groups3542108847815614940at_nat
% 0.35/0.73 @ ^ [X: nat] : X
% 0.35/0.73 @ S ) ) ).
% 0.35/0.73
% 0.35/0.73 % card_sum_le_nat_sum
% 0.35/0.73 thf(fact_9451_atLeast1__lessThan__eq__remove0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeast1_lessThan_eq_remove0
% 0.35/0.73 thf(fact_9452_Suc__0__or__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Suc_0_or_eq
% 0.35/0.73 thf(fact_9453_or__Suc__0__eq,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.73 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_Suc_0_eq
% 0.35/0.73 thf(fact_9454_sum__power2,axiom,
% 0.35/0.73 ! [K: nat] :
% 0.35/0.73 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.35/0.73 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % sum_power2
% 0.35/0.73 thf(fact_9455_Chebyshev__sum__upper__nat,axiom,
% 0.35/0.73 ! [N: nat,A2: nat > nat,B3: nat > nat] :
% 0.35/0.73 ( ! [I2: nat,J2: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.35/0.73 => ( ( ord_less_nat @ J2 @ N )
% 0.35/0.73 => ( ord_less_eq_nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
% 0.35/0.73 => ( ! [I2: nat,J2: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ I2 @ J2 )
% 0.35/0.73 => ( ( ord_less_nat @ J2 @ N )
% 0.35/0.73 => ( ord_less_eq_nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
% 0.35/0.73 => ( ord_less_eq_nat
% 0.35/0.73 @ ( times_times_nat @ N
% 0.35/0.73 @ ( groups3542108847815614940at_nat
% 0.35/0.73 @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( B3 @ I4 ) )
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.35/0.73 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Chebyshev_sum_upper_nat
% 0.35/0.73 thf(fact_9456_Cauchy__iff2,axiom,
% 0.35/0.73 ( topolo4055970368930404560y_real
% 0.35/0.73 = ( ^ [X8: nat > real] :
% 0.35/0.73 ! [J3: nat] :
% 0.35/0.73 ? [M8: nat] :
% 0.35/0.73 ! [M2: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.35/0.73 => ! [N2: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ M8 @ N2 )
% 0.35/0.73 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % Cauchy_iff2
% 0.35/0.73 thf(fact_9457_finite__atLeastLessThan__int,axiom,
% 0.35/0.73 ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_atLeastLessThan_int
% 0.35/0.73 thf(fact_9458_or__nonnegative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 0.35/0.73 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_nonnegative_int_iff
% 0.35/0.73 thf(fact_9459_or__negative__int__iff,axiom,
% 0.35/0.73 ! [K: int,L: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.35/0.73 = ( ( ord_less_int @ K @ zero_zero_int )
% 0.35/0.73 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_negative_int_iff
% 0.35/0.73 thf(fact_9460_card__atLeastLessThan__int,axiom,
% 0.35/0.73 ! [L: int,U: int] :
% 0.35/0.73 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 0.35/0.73 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % card_atLeastLessThan_int
% 0.35/0.73 thf(fact_9461_OR__lower,axiom,
% 0.35/0.73 ! [X2: int,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.73 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % OR_lower
% 0.35/0.73 thf(fact_9462_or__greater__eq,axiom,
% 0.35/0.73 ! [L: int,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 0.35/0.73 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % or_greater_eq
% 0.35/0.73 thf(fact_9463_finite__atLeastZeroLessThan__int,axiom,
% 0.35/0.73 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_atLeastZeroLessThan_int
% 0.35/0.73 thf(fact_9464_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 0.35/0.73 ! [L: int,U: int] :
% 0.35/0.73 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 0.35/0.73 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeastLessThanPlusOne_atLeastAtMost_int
% 0.35/0.73 thf(fact_9465_card__atLeastZeroLessThan__int,axiom,
% 0.35/0.73 ! [U: int] :
% 0.35/0.73 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 0.35/0.73 = ( nat2 @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % card_atLeastZeroLessThan_int
% 0.35/0.73 thf(fact_9466_OR__upper,axiom,
% 0.35/0.73 ! [X2: int,N: nat,Y3: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.73 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 0.35/0.73 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % OR_upper
% 0.35/0.73 thf(fact_9467_VEBT_Osize__gen_I1_J,axiom,
% 0.35/0.73 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.35/0.73 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.35/0.73 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % VEBT.size_gen(1)
% 0.35/0.73 thf(fact_9468_VEBT_Osize_I3_J,axiom,
% 0.35/0.73 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.35/0.73 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.35/0.73 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % VEBT.size(3)
% 0.35/0.73 thf(fact_9469_upto_Opsimps,axiom,
% 0.35/0.73 ! [I: int,J: int] :
% 0.35/0.73 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.35/0.73 => ( ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( upto @ I @ J )
% 0.35/0.73 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( upto @ I @ J )
% 0.35/0.73 = nil_int ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto.psimps
% 0.35/0.73 thf(fact_9470_upto_Opelims,axiom,
% 0.35/0.73 ! [X2: int,Xa2: int,Y3: list_int] :
% 0.35/0.73 ( ( ( upto @ X2 @ Xa2 )
% 0.35/0.73 = Y3 )
% 0.35/0.73 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 0.35/0.73 => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 0.35/0.73 => ( Y3 = nil_int ) ) )
% 0.35/0.73 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto.pelims
% 0.35/0.73 thf(fact_9471_upto__rec__numeral_I4_J,axiom,
% 0.35/0.73 ! [M: num,N: num] :
% 0.35/0.73 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 = ( 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 ) ) ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 = nil_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec_numeral(4)
% 0.35/0.73 thf(fact_9472_upto__empty,axiom,
% 0.35/0.73 ! [J: int,I: int] :
% 0.35/0.73 ( ( ord_less_int @ J @ I )
% 0.35/0.73 => ( ( upto @ I @ J )
% 0.35/0.73 = nil_int ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_empty
% 0.35/0.73 thf(fact_9473_upto__Nil2,axiom,
% 0.35/0.73 ! [I: int,J: int] :
% 0.35/0.73 ( ( nil_int
% 0.35/0.73 = ( upto @ I @ J ) )
% 0.35/0.73 = ( ord_less_int @ J @ I ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_Nil2
% 0.35/0.73 thf(fact_9474_upto__Nil,axiom,
% 0.35/0.73 ! [I: int,J: int] :
% 0.35/0.73 ( ( ( upto @ I @ J )
% 0.35/0.73 = nil_int )
% 0.35/0.73 = ( ord_less_int @ J @ I ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_Nil
% 0.35/0.73 thf(fact_9475_nth__upto,axiom,
% 0.35/0.73 ! [I: int,K: nat,J: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.35/0.73 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.35/0.73 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % nth_upto
% 0.35/0.73 thf(fact_9476_upto__rec__numeral_I1_J,axiom,
% 0.35/0.73 ! [M: num,N: num] :
% 0.35/0.73 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 = nil_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec_numeral(1)
% 0.35/0.73 thf(fact_9477_upto__rec__numeral_I2_J,axiom,
% 0.35/0.73 ! [M: num,N: num] :
% 0.35/0.73 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 0.35/0.73 = nil_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec_numeral(2)
% 0.35/0.73 thf(fact_9478_upto__rec__numeral_I3_J,axiom,
% 0.35/0.73 ! [M: num,N: num] :
% 0.35/0.73 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 = ( 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 ) ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 0.35/0.73 = nil_int ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec_numeral(3)
% 0.35/0.73 thf(fact_9479_upto__split2,axiom,
% 0.35/0.73 ! [I: int,J: int,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.73 => ( ( upto @ I @ K )
% 0.35/0.73 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_split2
% 0.35/0.73 thf(fact_9480_upto__split1,axiom,
% 0.35/0.73 ! [I: int,J: int,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.73 => ( ( upto @ I @ K )
% 0.35/0.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_split1
% 0.35/0.73 thf(fact_9481_upto__rec2,axiom,
% 0.35/0.73 ! [I: int,J: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( upto @ I @ J )
% 0.35/0.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec2
% 0.35/0.73 thf(fact_9482_upto__split3,axiom,
% 0.35/0.73 ! [I: int,J: int,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( ord_less_eq_int @ J @ K )
% 0.35/0.73 => ( ( upto @ I @ K )
% 0.35/0.73 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_split3
% 0.35/0.73 thf(fact_9483_upto__rec1,axiom,
% 0.35/0.73 ! [I: int,J: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ I @ J )
% 0.35/0.73 => ( ( upto @ I @ J )
% 0.35/0.73 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto_rec1
% 0.35/0.73 thf(fact_9484_upto_Oelims,axiom,
% 0.35/0.73 ! [X2: int,Xa2: int,Y3: list_int] :
% 0.35/0.73 ( ( ( upto @ X2 @ Xa2 )
% 0.35/0.73 = Y3 )
% 0.35/0.73 => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 0.35/0.73 => ( Y3
% 0.35/0.73 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 0.35/0.73 => ( Y3 = nil_int ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto.elims
% 0.35/0.73 thf(fact_9485_upto_Osimps,axiom,
% 0.35/0.73 ( upto
% 0.35/0.73 = ( ^ [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 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % upto.simps
% 0.35/0.73 thf(fact_9486_divmod__step__integer__def,axiom,
% 0.35/0.73 ( unique4921790084139445826nteger
% 0.35/0.73 = ( ^ [L2: num] :
% 0.35/0.73 ( produc6916734918728496179nteger
% 0.35/0.73 @ ^ [Q6: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % divmod_step_integer_def
% 0.35/0.73 thf(fact_9487_less__eq__integer__code_I1_J,axiom,
% 0.35/0.73 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 0.35/0.73
% 0.35/0.73 % less_eq_integer_code(1)
% 0.35/0.73 thf(fact_9488_sgn__integer__code,axiom,
% 0.35/0.73 ( sgn_sgn_Code_integer
% 0.35/0.73 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % sgn_integer_code
% 0.35/0.73 thf(fact_9489_zero__natural_Orsp,axiom,
% 0.35/0.73 zero_zero_nat = zero_zero_nat ).
% 0.35/0.73
% 0.35/0.73 % zero_natural.rsp
% 0.35/0.73 thf(fact_9490_integer__of__int__code,axiom,
% 0.35/0.73 ( code_integer_of_int
% 0.35/0.73 = ( ^ [K3: int] :
% 0.35/0.73 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.35/0.73 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.35/0.73 @ ( if_Code_integer
% 0.35/0.73 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.35/0.73 = zero_zero_int )
% 0.35/0.73 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.35/0.73 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % integer_of_int_code
% 0.35/0.73 thf(fact_9491_signed__take__bit__nonnegative__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 0.35/0.73 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % signed_take_bit_nonnegative_iff
% 0.35/0.73 thf(fact_9492_signed__take__bit__negative__iff,axiom,
% 0.35/0.73 ! [N: nat,K: int] :
% 0.35/0.73 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.73 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % signed_take_bit_negative_iff
% 0.35/0.73 thf(fact_9493_bit__nat__iff,axiom,
% 0.35/0.73 ! [K: int,N: nat] :
% 0.35/0.73 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 0.35/0.73 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.35/0.73 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % bit_nat_iff
% 0.35/0.73 thf(fact_9494_less__integer_Oabs__eq,axiom,
% 0.35/0.73 ! [Xa2: int,X2: int] :
% 0.35/0.73 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 0.35/0.73 = ( ord_less_int @ Xa2 @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % less_integer.abs_eq
% 0.35/0.73 thf(fact_9495_less__integer__code_I1_J,axiom,
% 0.35/0.73 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.35/0.73
% 0.35/0.73 % less_integer_code(1)
% 0.35/0.73 thf(fact_9496_abs__integer__code,axiom,
% 0.35/0.73 ( abs_abs_Code_integer
% 0.35/0.73 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % abs_integer_code
% 0.35/0.73 thf(fact_9497_not__bit__Suc__0__Suc,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % not_bit_Suc_0_Suc
% 0.35/0.73 thf(fact_9498_bit__Suc__0__iff,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.73 = ( N = zero_zero_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % bit_Suc_0_iff
% 0.35/0.73 thf(fact_9499_less__eq__integer_Oabs__eq,axiom,
% 0.35/0.73 ! [Xa2: int,X2: int] :
% 0.35/0.73 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 0.35/0.73 = ( ord_less_eq_int @ Xa2 @ X2 ) ) ).
% 0.35/0.73
% 0.35/0.73 % less_eq_integer.abs_eq
% 0.35/0.73 thf(fact_9500_not__bit__Suc__0__numeral,axiom,
% 0.35/0.73 ! [N: num] :
% 0.35/0.73 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % not_bit_Suc_0_numeral
% 0.35/0.73 thf(fact_9501_bit__imp__take__bit__positive,axiom,
% 0.35/0.73 ! [N: nat,M: nat,K: int] :
% 0.35/0.73 ( ( ord_less_nat @ N @ M )
% 0.35/0.73 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 0.35/0.73 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % bit_imp_take_bit_positive
% 0.35/0.73 thf(fact_9502_int__bit__bound,axiom,
% 0.35/0.73 ! [K: int] :
% 0.35/0.73 ~ ! [N3: nat] :
% 0.35/0.73 ( ! [M3: nat] :
% 0.35/0.73 ( ( ord_less_eq_nat @ N3 @ M3 )
% 0.35/0.73 => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 0.35/0.73 = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 0.35/0.73 => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.35/0.73 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 0.35/0.73 = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % int_bit_bound
% 0.35/0.73 thf(fact_9503_binomial__def,axiom,
% 0.35/0.73 ( binomial
% 0.35/0.73 = ( ^ [N2: nat,K3: nat] :
% 0.35/0.73 ( finite_card_set_nat
% 0.35/0.73 @ ( collect_set_nat
% 0.35/0.73 @ ^ [K6: set_nat] :
% 0.35/0.73 ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 0.35/0.73 & ( ( finite_card_nat @ K6 )
% 0.35/0.73 = K3 ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % binomial_def
% 0.35/0.73 thf(fact_9504_int__of__integer__code,axiom,
% 0.35/0.73 ( code_int_of_integer
% 0.35/0.73 = ( ^ [K3: code_integer] :
% 0.35/0.73 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.35/0.73 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.35/0.73 @ ( produc1553301316500091796er_int
% 0.35/0.73 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 0.35/0.73 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % int_of_integer_code
% 0.35/0.73 thf(fact_9505_finite__enumerate,axiom,
% 0.35/0.73 ! [S: set_nat] :
% 0.35/0.73 ( ( finite_finite_nat @ S )
% 0.35/0.73 => ? [R3: nat > nat] :
% 0.35/0.73 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
% 0.35/0.73 & ! [N5: nat] :
% 0.35/0.73 ( ( ord_less_nat @ N5 @ ( finite_card_nat @ S ) )
% 0.35/0.73 => ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_enumerate
% 0.35/0.73 thf(fact_9506_image__Suc__atLeastAtMost,axiom,
% 0.35/0.73 ! [I: nat,J: nat] :
% 0.35/0.73 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.35/0.73 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_Suc_atLeastAtMost
% 0.35/0.73 thf(fact_9507_image__Suc__atLeastLessThan,axiom,
% 0.35/0.73 ! [I: nat,J: nat] :
% 0.35/0.73 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.35/0.73 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_Suc_atLeastLessThan
% 0.35/0.73 thf(fact_9508_zero__notin__Suc__image,axiom,
% 0.35/0.73 ! [A3: set_nat] :
% 0.35/0.73 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 0.35/0.73
% 0.35/0.73 % zero_notin_Suc_image
% 0.35/0.73 thf(fact_9509_less__integer_Orep__eq,axiom,
% 0.35/0.73 ( ord_le6747313008572928689nteger
% 0.35/0.73 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % less_integer.rep_eq
% 0.35/0.73 thf(fact_9510_integer__less__iff,axiom,
% 0.35/0.73 ( ord_le6747313008572928689nteger
% 0.35/0.73 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % integer_less_iff
% 0.35/0.73 thf(fact_9511_less__eq__integer_Orep__eq,axiom,
% 0.35/0.73 ( ord_le3102999989581377725nteger
% 0.35/0.73 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % less_eq_integer.rep_eq
% 0.35/0.73 thf(fact_9512_integer__less__eq__iff,axiom,
% 0.35/0.73 ( ord_le3102999989581377725nteger
% 0.35/0.73 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % integer_less_eq_iff
% 0.35/0.73 thf(fact_9513_finite__int__iff__bounded,axiom,
% 0.35/0.73 ( finite_finite_int
% 0.35/0.73 = ( ^ [S6: set_int] :
% 0.35/0.73 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_int_iff_bounded
% 0.35/0.73 thf(fact_9514_finite__int__iff__bounded__le,axiom,
% 0.35/0.73 ( finite_finite_int
% 0.35/0.73 = ( ^ [S6: set_int] :
% 0.35/0.73 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % finite_int_iff_bounded_le
% 0.35/0.73 thf(fact_9515_image__int__atLeastAtMost,axiom,
% 0.35/0.73 ! [A2: nat,B3: nat] :
% 0.35/0.73 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 0.35/0.73 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_int_atLeastAtMost
% 0.35/0.73 thf(fact_9516_image__int__atLeastLessThan,axiom,
% 0.35/0.73 ! [A2: nat,B3: nat] :
% 0.35/0.73 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B3 ) )
% 0.35/0.73 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_int_atLeastLessThan
% 0.35/0.73 thf(fact_9517_image__Suc__lessThan,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.73 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_Suc_lessThan
% 0.35/0.73 thf(fact_9518_image__Suc__atMost,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 0.35/0.73 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_Suc_atMost
% 0.35/0.73 thf(fact_9519_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeast0_atMost_Suc_eq_insert_0
% 0.35/0.73 thf(fact_9520_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atLeast0_lessThan_Suc_eq_insert_0
% 0.35/0.73 thf(fact_9521_lessThan__Suc__eq__insert__0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % lessThan_Suc_eq_insert_0
% 0.35/0.73 thf(fact_9522_atMost__Suc__eq__insert__0,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % atMost_Suc_eq_insert_0
% 0.35/0.73 thf(fact_9523_image__add__int__atLeastLessThan,axiom,
% 0.35/0.73 ! [L: int,U: int] :
% 0.35/0.73 ( ( image_int_int
% 0.35/0.73 @ ^ [X: int] : ( plus_plus_int @ X @ L )
% 0.35/0.73 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 0.35/0.73 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_add_int_atLeastLessThan
% 0.35/0.73 thf(fact_9524_image__atLeastZeroLessThan__int,axiom,
% 0.35/0.73 ! [U: int] :
% 0.35/0.73 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.35/0.73 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 0.35/0.73 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_atLeastZeroLessThan_int
% 0.35/0.73 thf(fact_9525_image__minus__const__atLeastLessThan__nat,axiom,
% 0.35/0.73 ! [C: nat,Y3: nat,X2: nat] :
% 0.35/0.73 ( ( ( ord_less_nat @ C @ Y3 )
% 0.35/0.73 => ( ( image_nat_nat
% 0.35/0.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.35/0.73 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.35/0.73 & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.35/0.73 => ( ( ( ord_less_nat @ X2 @ Y3 )
% 0.35/0.73 => ( ( image_nat_nat
% 0.35/0.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.35/0.73 & ( ~ ( ord_less_nat @ X2 @ Y3 )
% 0.35/0.73 => ( ( image_nat_nat
% 0.35/0.73 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 0.35/0.73 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 0.35/0.73 = bot_bot_set_nat ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % image_minus_const_atLeastLessThan_nat
% 0.35/0.73 thf(fact_9526_Sup__nat__empty,axiom,
% 0.35/0.73 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.35/0.73 = zero_zero_nat ) ).
% 0.35/0.73
% 0.35/0.73 % Sup_nat_empty
% 0.35/0.73 thf(fact_9527_num__of__integer__code,axiom,
% 0.35/0.73 ( code_num_of_integer
% 0.35/0.73 = ( ^ [K3: code_integer] :
% 0.35/0.73 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 0.35/0.73 @ ( produc7336495610019696514er_num
% 0.35/0.73 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 0.35/0.73 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % num_of_integer_code
% 0.35/0.73 thf(fact_9528_nat__of__integer__code,axiom,
% 0.35/0.73 ( code_nat_of_integer
% 0.35/0.73 = ( ^ [K3: code_integer] :
% 0.35/0.73 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.35/0.73 @ ( produc1555791787009142072er_nat
% 0.35/0.73 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 0.35/0.73 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % nat_of_integer_code
% 0.35/0.73 thf(fact_9529_nat__of__integer__non__positive,axiom,
% 0.35/0.73 ! [K: code_integer] :
% 0.35/0.73 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.35/0.73 => ( ( code_nat_of_integer @ K )
% 0.35/0.73 = zero_zero_nat ) ) ).
% 0.35/0.73
% 0.35/0.73 % nat_of_integer_non_positive
% 0.35/0.73 thf(fact_9530_nat__of__integer__code__post_I1_J,axiom,
% 0.35/0.73 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.35/0.73 = zero_zero_nat ) ).
% 0.35/0.73
% 0.35/0.73 % nat_of_integer_code_post(1)
% 0.35/0.73 thf(fact_9531_Inf__nat__def1,axiom,
% 0.35/0.73 ! [K4: set_nat] :
% 0.35/0.73 ( ( K4 != bot_bot_set_nat )
% 0.35/0.73 => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 0.35/0.73
% 0.35/0.73 % Inf_nat_def1
% 0.35/0.73 thf(fact_9532_nat__not__finite,axiom,
% 0.35/0.73 ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % nat_not_finite
% 0.35/0.73 thf(fact_9533_infinite__UNIV__nat,axiom,
% 0.35/0.73 ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % infinite_UNIV_nat
% 0.35/0.73 thf(fact_9534_UN__lessThan__UNIV,axiom,
% 0.35/0.73 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 0.35/0.73 = top_top_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % UN_lessThan_UNIV
% 0.35/0.73 thf(fact_9535_UN__atMost__UNIV,axiom,
% 0.35/0.73 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 0.35/0.73 = top_top_set_nat ) ).
% 0.35/0.73
% 0.35/0.73 % UN_atMost_UNIV
% 0.35/0.73 thf(fact_9536_range__enumerate,axiom,
% 0.35/0.73 ! [S: set_nat] :
% 0.35/0.73 ( ~ ( finite_finite_nat @ S )
% 0.35/0.73 => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
% 0.35/0.73 = S ) ) ).
% 0.35/0.73
% 0.35/0.73 % range_enumerate
% 0.35/0.73 thf(fact_9537_UNIV__nat__eq,axiom,
% 0.35/0.73 ( top_top_set_nat
% 0.35/0.73 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % UNIV_nat_eq
% 0.35/0.73 thf(fact_9538_range__mod,axiom,
% 0.35/0.73 ! [N: nat] :
% 0.35/0.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.73 => ( ( image_nat_nat
% 0.35/0.73 @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
% 0.35/0.73 @ top_top_set_nat )
% 0.35/0.73 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % range_mod
% 0.35/0.73 thf(fact_9539_suminf__eq__SUP__real,axiom,
% 0.35/0.73 ! [X6: nat > real] :
% 0.35/0.73 ( ( summable_real @ X6 )
% 0.35/0.73 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I2 ) )
% 0.35/0.73 => ( ( suminf_real @ X6 )
% 0.35/0.73 = ( comple1385675409528146559p_real
% 0.35/0.73 @ ( image_nat_real
% 0.35/0.73 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X6 @ ( set_ord_lessThan_nat @ I4 ) )
% 0.35/0.73 @ top_top_set_nat ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % suminf_eq_SUP_real
% 0.35/0.73 thf(fact_9540_card__UNIV__unit,axiom,
% 0.35/0.73 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 0.35/0.73 = one_one_nat ) ).
% 0.35/0.73
% 0.35/0.73 % card_UNIV_unit
% 0.35/0.73 thf(fact_9541_range__mult,axiom,
% 0.35/0.73 ! [A2: real] :
% 0.35/0.73 ( ( ( A2 = zero_zero_real )
% 0.35/0.73 => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
% 0.35/0.73 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.35/0.73 & ( ( A2 != zero_zero_real )
% 0.35/0.73 => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
% 0.35/0.73 = top_top_set_real ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % range_mult
% 0.35/0.73 thf(fact_9542_root__def,axiom,
% 0.35/0.73 ( root
% 0.35/0.73 = ( ^ [N2: nat,X: real] :
% 0.35/0.73 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 0.35/0.73 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.35/0.73 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.35/0.73 @ X ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % root_def
% 0.35/0.73 thf(fact_9543_DERIV__even__real__root,axiom,
% 0.35/0.73 ! [N: nat,X2: real] :
% 0.35/0.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.73 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % DERIV_even_real_root
% 0.35/0.73 thf(fact_9544_DERIV__real__root__generic,axiom,
% 0.35/0.73 ! [N: nat,X2: real,D4: real] :
% 0.35/0.73 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.73 => ( ( X2 != zero_zero_real )
% 0.35/0.73 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.73 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.73 => ( D4
% 0.35/0.73 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.35/0.73 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.73 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.73 => ( D4
% 0.35/0.73 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.35/0.73 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.73 => ( D4
% 0.35/0.73 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.35/0.73 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % DERIV_real_root_generic
% 0.35/0.73 thf(fact_9545_DERIV__arctan__series,axiom,
% 0.35/0.73 ! [X2: real] :
% 0.35/0.73 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.73 => ( has_fi5821293074295781190e_real
% 0.35/0.73 @ ^ [X9: real] :
% 0.35/0.73 ( suminf_real
% 0.35/0.73 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.35/0.73 @ ( suminf_real
% 0.35/0.73 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.73 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % DERIV_arctan_series
% 0.35/0.73 thf(fact_9546_has__real__derivative__pos__inc__left,axiom,
% 0.35/0.73 ! [F: real > real,L: real,X2: real,S: set_real] :
% 0.35/0.73 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 0.35/0.73 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.73 => ? [D6: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.73 & ! [H4: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.73 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
% 0.35/0.73 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.73 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % has_real_derivative_pos_inc_left
% 0.35/0.73 thf(fact_9547_has__real__derivative__neg__dec__left,axiom,
% 0.35/0.73 ! [F: real > real,L: real,X2: real,S: set_real] :
% 0.35/0.73 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 0.35/0.73 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.73 => ? [D6: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.73 & ! [H4: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.73 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
% 0.35/0.73 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.73 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.73
% 0.35/0.73 % has_real_derivative_neg_dec_left
% 0.35/0.73 thf(fact_9548_has__real__derivative__neg__dec__right,axiom,
% 0.35/0.73 ! [F: real > real,L: real,X2: real,S: set_real] :
% 0.35/0.73 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 0.35/0.73 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.73 => ? [D6: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.73 & ! [H4: real] :
% 0.35/0.73 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.73 => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
% 0.35/0.73 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.73 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % has_real_derivative_neg_dec_right
% 0.35/0.74 thf(fact_9549_has__real__derivative__pos__inc__right,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real,S: set_real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74 => ? [D6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.74 & ! [H4: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74 => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
% 0.35/0.74 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.74 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % has_real_derivative_pos_inc_right
% 0.35/0.74 thf(fact_9550_deriv__nonneg__imp__mono,axiom,
% 0.35/0.74 ! [A2: real,B3: real,G3: real > real,G4: real > real] :
% 0.35/0.74 ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ord_less_eq_real @ zero_zero_real @ ( G4 @ X5 ) ) )
% 0.35/0.74 => ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.74 => ( ord_less_eq_real @ ( G3 @ A2 ) @ ( G3 @ B3 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % deriv_nonneg_imp_mono
% 0.35/0.74 thf(fact_9551_DERIV__nonneg__imp__nondecreasing,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 0.35/0.74 => ( ord_less_eq_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_nonneg_imp_nondecreasing
% 0.35/0.74 thf(fact_9552_DERIV__nonpos__imp__nonincreasing,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 0.35/0.74 => ( ord_less_eq_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_nonpos_imp_nonincreasing
% 0.35/0.74 thf(fact_9553_DERIV__neg__imp__decreasing,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.35/0.74 => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_neg_imp_decreasing
% 0.35/0.74 thf(fact_9554_DERIV__pos__imp__increasing,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.35/0.74 => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pos_imp_increasing
% 0.35/0.74 thf(fact_9555_DERIV__neg__dec__right,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74 => ? [D6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.74 & ! [H4: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.74 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_neg_dec_right
% 0.35/0.74 thf(fact_9556_DERIV__pos__inc__right,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74 => ? [D6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.74 & ! [H4: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.74 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pos_inc_right
% 0.35/0.74 thf(fact_9557_DERIV__pos__inc__left,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74 => ? [D6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.74 & ! [H4: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.74 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pos_inc_left
% 0.35/0.74 thf(fact_9558_DERIV__neg__dec__left,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74 => ? [D6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D6 )
% 0.35/0.74 & ! [H4: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.35/0.74 => ( ( ord_less_real @ H4 @ D6 )
% 0.35/0.74 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_neg_dec_left
% 0.35/0.74 thf(fact_9559_MVT2,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,F6: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ? [Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 & ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 0.35/0.74 = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ ( F6 @ Z4 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % MVT2
% 0.35/0.74 thf(fact_9560_DERIV__local__const,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real,D: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74 => ( ! [Y4: real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.35/0.74 => ( ( F @ X2 )
% 0.35/0.74 = ( F @ Y4 ) ) )
% 0.35/0.74 => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_local_const
% 0.35/0.74 thf(fact_9561_DERIV__ln,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_ln
% 0.35/0.74 thf(fact_9562_DERIV__local__min,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real,D: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74 => ( ! [Y4: real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.35/0.74 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 0.35/0.74 => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_local_min
% 0.35/0.74 thf(fact_9563_DERIV__local__max,axiom,
% 0.35/0.74 ! [F: real > real,L: real,X2: real,D: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74 => ( ! [Y4: real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 0.35/0.74 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 0.35/0.74 => ( L = zero_zero_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_local_max
% 0.35/0.74 thf(fact_9564_DERIV__ln__divide,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_ln_divide
% 0.35/0.74 thf(fact_9565_DERIV__pow,axiom,
% 0.35/0.74 ! [N: nat,X2: real,S2: set_real] :
% 0.35/0.74 ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] : ( power_power_real @ X @ N )
% 0.35/0.74 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pow
% 0.35/0.74 thf(fact_9566_has__real__derivative__powr,axiom,
% 0.35/0.74 ! [Z: real,R2: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ Z )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [Z2: real] : ( powr_real @ Z2 @ R2 )
% 0.35/0.74 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % has_real_derivative_powr
% 0.35/0.74 thf(fact_9567_DERIV__log,axiom,
% 0.35/0.74 ! [X2: real,B3: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( log @ B3 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B3 ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_log
% 0.35/0.74 thf(fact_9568_DERIV__fun__powr,axiom,
% 0.35/0.74 ! [G3: real > real,M: real,X2: real,R2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ R2 )
% 0.35/0.74 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G3 @ X2 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_fun_powr
% 0.35/0.74 thf(fact_9569_DERIV__powr,axiom,
% 0.35/0.74 ! [G3: real > real,M: real,X2: real,F: real > real,R2: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
% 0.35/0.74 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ ( F @ X ) )
% 0.35/0.74 @ ( times_times_real @ ( powr_real @ ( G3 @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G3 @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G3 @ X2 ) ) ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_powr
% 0.35/0.74 thf(fact_9570_DERIV__real__sqrt,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_real_sqrt
% 0.35/0.74 thf(fact_9571_DERIV__real__sqrt__generic,axiom,
% 0.35/0.74 ! [X2: real,D4: real] :
% 0.35/0.74 ( ( X2 != zero_zero_real )
% 0.35/0.74 => ( ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( D4
% 0.35/0.74 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 => ( ( ( ord_less_real @ X2 @ zero_zero_real )
% 0.35/0.74 => ( D4
% 0.35/0.74 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_real_sqrt_generic
% 0.35/0.74 thf(fact_9572_arcosh__real__has__field__derivative,axiom,
% 0.35/0.74 ! [X2: real,A3: set_real] :
% 0.35/0.74 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % arcosh_real_has_field_derivative
% 0.35/0.74 thf(fact_9573_artanh__real__has__field__derivative,axiom,
% 0.35/0.74 ! [X2: real,A3: set_real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % artanh_real_has_field_derivative
% 0.35/0.74 thf(fact_9574_DERIV__real__root,axiom,
% 0.35/0.74 ! [N: nat,X2: real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_real_root
% 0.35/0.74 thf(fact_9575_DERIV__arccos,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_arccos
% 0.35/0.74 thf(fact_9576_DERIV__arcsin,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_arcsin
% 0.35/0.74 thf(fact_9577_Maclaurin__all__le,axiom,
% 0.35/0.74 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 0.35/0.74 ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.74 & ( ( F @ X2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_all_le
% 0.35/0.74 thf(fact_9578_Maclaurin__all__le__objl,axiom,
% 0.35/0.74 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 0.35/0.74 ( ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.74 & ( ( F @ X2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_all_le_objl
% 0.35/0.74 thf(fact_9579_DERIV__odd__real__root,axiom,
% 0.35/0.74 ! [N: nat,X2: real] :
% 0.35/0.74 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74 => ( ( X2 != zero_zero_real )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_odd_real_root
% 0.35/0.74 thf(fact_9580_Maclaurin,axiom,
% 0.35/0.74 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.74 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ H2 )
% 0.35/0.74 & ( ( F @ H2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin
% 0.35/0.74 thf(fact_9581_Maclaurin2,axiom,
% 0.35/0.74 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ H2 )
% 0.35/0.74 & ( ( F @ H2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin2
% 0.35/0.74 thf(fact_9582_Maclaurin__minus,axiom,
% 0.35/0.74 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.35/0.74 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ H2 @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ H2 @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ zero_zero_real )
% 0.35/0.74 & ( ( F @ H2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_minus
% 0.35/0.74 thf(fact_9583_Maclaurin__all__lt,axiom,
% 0.35/0.74 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 0.35/0.74 ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( X2 != zero_zero_real )
% 0.35/0.74 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.35/0.74 & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.74 & ( ( F @ X2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_all_lt
% 0.35/0.74 thf(fact_9584_Maclaurin__bi__le,axiom,
% 0.35/0.74 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 0.35/0.74 ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 0.35/0.74 & ( ( F @ X2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_bi_le
% 0.35/0.74 thf(fact_9585_Taylor,axiom,
% 0.35/0.74 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real,X2: real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ A2 @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ B3 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ( ord_less_eq_real @ A2 @ C )
% 0.35/0.74 => ( ( ord_less_eq_real @ C @ B3 )
% 0.35/0.74 => ( ( ord_less_eq_real @ A2 @ X2 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X2 @ B3 )
% 0.35/0.74 => ( ( X2 != C )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ( ord_less_real @ X2 @ C )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ C ) ) )
% 0.35/0.74 & ( ~ ( ord_less_real @ X2 @ C )
% 0.35/0.74 => ( ( ord_less_real @ C @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ X2 ) ) )
% 0.35/0.74 & ( ( F @ X2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Taylor
% 0.35/0.74 thf(fact_9586_Taylor__up,axiom,
% 0.35/0.74 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ A2 @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ B3 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ( ord_less_eq_real @ A2 @ C )
% 0.35/0.74 => ( ( ord_less_real @ C @ B3 )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ C @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ B3 )
% 0.35/0.74 & ( ( F @ B3 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Taylor_up
% 0.35/0.74 thf(fact_9587_Taylor__down,axiom,
% 0.35/0.74 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( ( Diff @ zero_zero_nat )
% 0.35/0.74 = F )
% 0.35/0.74 => ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ A2 @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ B3 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ( ord_less_real @ A2 @ C )
% 0.35/0.74 => ( ( ord_less_eq_real @ C @ B3 )
% 0.35/0.74 => ? [T6: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ T6 )
% 0.35/0.74 & ( ord_less_real @ T6 @ C )
% 0.35/0.74 & ( ( F @ A2 )
% 0.35/0.74 = ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ M2 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ N ) )
% 0.35/0.74 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Taylor_down
% 0.35/0.74 thf(fact_9588_Maclaurin__lemma2,axiom,
% 0.35/0.74 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 0.35/0.74 ( ! [M4: nat,T6: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M4 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.35/0.74 & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ( N
% 0.35/0.74 = ( suc @ K ) )
% 0.35/0.74 => ! [M3: nat,T7: real] :
% 0.35/0.74 ( ( ( ord_less_nat @ M3 @ N )
% 0.35/0.74 & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 0.35/0.74 & ( ord_less_eq_real @ T7 @ H2 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [U2: real] :
% 0.35/0.74 ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 0.35/0.74 @ ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 0.35/0.74 @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 0.35/0.74 @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T7 )
% 0.35/0.74 @ ( plus_plus_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T7 @ P5 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 0.35/0.74 @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Maclaurin_lemma2
% 0.35/0.74 thf(fact_9589_DERIV__power__series_H,axiom,
% 0.35/0.74 ! [R: real,F: nat > real,X0: real] :
% 0.35/0.74 ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.35/0.74 => ( summable_real
% 0.35/0.74 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 0.35/0.74 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ R )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] :
% 0.35/0.74 ( suminf_real
% 0.35/0.74 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_power_series'
% 0.35/0.74 thf(fact_9590_DERIV__isconst3,axiom,
% 0.35/0.74 ! [A2: real,B3: real,X2: real,Y3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ( F @ X2 )
% 0.35/0.74 = ( F @ Y3 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_isconst3
% 0.35/0.74 thf(fact_9591_DERIV__series_H,axiom,
% 0.35/0.74 ! [F: real > nat > real,F6: real > nat > real,X0: real,A2: real,B3: real,L5: nat > real] :
% 0.35/0.74 ( ! [N3: nat] :
% 0.35/0.74 ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] : ( F @ X @ N3 )
% 0.35/0.74 @ ( F6 @ X0 @ N3 )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( summable_real @ ( F @ X5 ) ) )
% 0.35/0.74 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ( summable_real @ ( F6 @ X0 ) )
% 0.35/0.74 => ( ( summable_real @ L5 )
% 0.35/0.74 => ( ! [N3: nat,X5: real,Y4: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 0.35/0.74 => ( has_fi5821293074295781190e_real
% 0.35/0.74 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 0.35/0.74 @ ( suminf_real @ ( F6 @ X0 ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_series'
% 0.35/0.74 thf(fact_9592_Gcd__eq__Max,axiom,
% 0.35/0.74 ! [M5: set_nat] :
% 0.35/0.74 ( ( finite_finite_nat @ M5 )
% 0.35/0.74 => ( ( M5 != bot_bot_set_nat )
% 0.35/0.74 => ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 0.35/0.74 => ( ( gcd_Gcd_nat @ M5 )
% 0.35/0.74 = ( lattic8265883725875713057ax_nat
% 0.35/0.74 @ ( comple7806235888213564991et_nat
% 0.35/0.74 @ ( image_nat_set_nat
% 0.35/0.74 @ ^ [M2: nat] :
% 0.35/0.74 ( collect_nat
% 0.35/0.74 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
% 0.35/0.74 @ M5 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Gcd_eq_Max
% 0.35/0.74 thf(fact_9593_finite__greaterThanLessThan,axiom,
% 0.35/0.74 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % finite_greaterThanLessThan
% 0.35/0.74 thf(fact_9594_finite__greaterThanLessThan__int,axiom,
% 0.35/0.74 ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % finite_greaterThanLessThan_int
% 0.35/0.74 thf(fact_9595_card__greaterThanLessThan,axiom,
% 0.35/0.74 ! [L: nat,U: nat] :
% 0.35/0.74 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.35/0.74 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % card_greaterThanLessThan
% 0.35/0.74 thf(fact_9596_Max__divisors__self__nat,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( N != zero_zero_nat )
% 0.35/0.74 => ( ( lattic8265883725875713057ax_nat
% 0.35/0.74 @ ( collect_nat
% 0.35/0.74 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
% 0.35/0.74 = N ) ) ).
% 0.35/0.74
% 0.35/0.74 % Max_divisors_self_nat
% 0.35/0.74 thf(fact_9597_card__greaterThanLessThan__int,axiom,
% 0.35/0.74 ! [L: int,U: int] :
% 0.35/0.74 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 0.35/0.74 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % card_greaterThanLessThan_int
% 0.35/0.74 thf(fact_9598_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.35/0.74 ! [L: nat,U: nat] :
% 0.35/0.74 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.35/0.74 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeastSucLessThan_greaterThanLessThan
% 0.35/0.74 thf(fact_9599_isCont__Lb__Ub,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_eq_real @ X5 @ B3 ) )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.35/0.74 => ? [L6: real,M9: real] :
% 0.35/0.74 ( ! [X4: real] :
% 0.35/0.74 ( ( ( ord_less_eq_real @ A2 @ X4 )
% 0.35/0.74 & ( ord_less_eq_real @ X4 @ B3 ) )
% 0.35/0.74 => ( ( ord_less_eq_real @ L6 @ ( F @ X4 ) )
% 0.35/0.74 & ( ord_less_eq_real @ ( F @ X4 ) @ M9 ) ) )
% 0.35/0.74 & ! [Y5: real] :
% 0.35/0.74 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 0.35/0.74 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 0.35/0.74 => ? [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 & ( ( F @ X5 )
% 0.35/0.74 = Y5 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_Lb_Ub
% 0.35/0.74 thf(fact_9600_LIM__fun__gt__zero,axiom,
% 0.35/0.74 ! [F: real > real,L: real,C: real] :
% 0.35/0.74 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ L )
% 0.35/0.74 => ? [R3: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74 & ! [X4: real] :
% 0.35/0.74 ( ( ( X4 != C )
% 0.35/0.74 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 0.35/0.74 => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIM_fun_gt_zero
% 0.35/0.74 thf(fact_9601_LIM__fun__not__zero,axiom,
% 0.35/0.74 ! [F: real > real,L: real,C: real] :
% 0.35/0.74 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74 => ( ( L != zero_zero_real )
% 0.35/0.74 => ? [R3: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74 & ! [X4: real] :
% 0.35/0.74 ( ( ( X4 != C )
% 0.35/0.74 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 0.35/0.74 => ( ( F @ X4 )
% 0.35/0.74 != zero_zero_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIM_fun_not_zero
% 0.35/0.74 thf(fact_9602_LIM__fun__less__zero,axiom,
% 0.35/0.74 ! [F: real > real,L: real,C: real] :
% 0.35/0.74 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.35/0.74 => ( ( ord_less_real @ L @ zero_zero_real )
% 0.35/0.74 => ? [R3: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ R3 )
% 0.35/0.74 & ! [X4: real] :
% 0.35/0.74 ( ( ( X4 != C )
% 0.35/0.74 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 0.35/0.74 => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIM_fun_less_zero
% 0.35/0.74 thf(fact_9603_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 0.35/0.74 ! [L: int,U: int] :
% 0.35/0.74 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.35/0.74 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeastPlusOneLessThan_greaterThanLessThan_int
% 0.35/0.74 thf(fact_9604_isCont__inverse__function2,axiom,
% 0.35/0.74 ! [A2: real,X2: real,B3: real,G3: real > real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ B3 )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 0.35/0.74 => ( ( G3 @ ( F @ Z4 ) )
% 0.35/0.74 = Z4 ) ) )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_inverse_function2
% 0.35/0.74 thf(fact_9605_Sup__nat__def,axiom,
% 0.35/0.74 ( complete_Sup_Sup_nat
% 0.35/0.74 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Sup_nat_def
% 0.35/0.74 thf(fact_9606_card__le__Suc__Max,axiom,
% 0.35/0.74 ! [S: set_nat] :
% 0.35/0.74 ( ( finite_finite_nat @ S )
% 0.35/0.74 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % card_le_Suc_Max
% 0.35/0.74 thf(fact_9607_divide__nat__def,axiom,
% 0.35/0.74 ( divide_divide_nat
% 0.35/0.74 = ( ^ [M2: nat,N2: nat] :
% 0.35/0.74 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 0.35/0.74 @ ( lattic8265883725875713057ax_nat
% 0.35/0.74 @ ( collect_nat
% 0.35/0.74 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % divide_nat_def
% 0.35/0.74 thf(fact_9608_isCont__arcosh,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_arcosh
% 0.35/0.74 thf(fact_9609_DERIV__inverse__function,axiom,
% 0.35/0.74 ! [F: real > real,D4: real,G3: real > real,X2: real,A2: real,B3: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G3 @ X2 ) @ top_top_set_real ) )
% 0.35/0.74 => ( ( D4 != zero_zero_real )
% 0.35/0.74 => ( ( ord_less_real @ A2 @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ B3 )
% 0.35/0.74 => ( ! [Y4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Y4 )
% 0.35/0.74 => ( ( ord_less_real @ Y4 @ B3 )
% 0.35/0.74 => ( ( F @ ( G3 @ Y4 ) )
% 0.35/0.74 = Y4 ) ) )
% 0.35/0.74 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ G3 @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_inverse_function
% 0.35/0.74 thf(fact_9610_isCont__arccos,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_arccos
% 0.35/0.74 thf(fact_9611_isCont__arcsin,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_arcsin
% 0.35/0.74 thf(fact_9612_LIM__less__bound,axiom,
% 0.35/0.74 ! [B3: real,X2: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ B3 @ X2 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B3 @ X2 ) )
% 0.35/0.74 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 0.35/0.74 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
% 0.35/0.74 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIM_less_bound
% 0.35/0.74 thf(fact_9613_isCont__artanh,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_artanh
% 0.35/0.74 thf(fact_9614_isCont__inverse__function,axiom,
% 0.35/0.74 ! [D: real,X2: real,G3: real > real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ D )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
% 0.35/0.74 => ( ( G3 @ ( F @ Z4 ) )
% 0.35/0.74 = Z4 ) )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % isCont_inverse_function
% 0.35/0.74 thf(fact_9615_GMVT_H,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,G3: real > real,G4: real > real,F6: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G3 ) ) )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ( ! [Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 => ( ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ? [C3: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ C3 )
% 0.35/0.74 & ( ord_less_real @ C3 @ B3 )
% 0.35/0.74 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ ( G4 @ C3 ) )
% 0.35/0.74 = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % GMVT'
% 0.35/0.74 thf(fact_9616_summable__Leibniz_I3_J,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ( topolo6980174941875973593q_real @ A2 )
% 0.35/0.74 => ( ( ord_less_real @ ( A2 @ zero_zero_nat ) @ zero_zero_real )
% 0.35/0.74 => ! [N5: nat] :
% 0.35/0.74 ( member_real
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 0.35/0.74 @ ( set_or1222579329274155063t_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz(3)
% 0.35/0.74 thf(fact_9617_summable__Leibniz_I2_J,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ( topolo6980174941875973593q_real @ A2 )
% 0.35/0.74 => ( ( ord_less_real @ zero_zero_real @ ( A2 @ zero_zero_nat ) )
% 0.35/0.74 => ! [N5: nat] :
% 0.35/0.74 ( member_real
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 0.35/0.74 @ ( set_or1222579329274155063t_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz(2)
% 0.35/0.74 thf(fact_9618_summable__Leibniz_H_I4_J,axiom,
% 0.35/0.74 ! [A2: nat > real,N: nat] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ord_less_eq_real
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz'(4)
% 0.35/0.74 thf(fact_9619_trivial__limit__sequentially,axiom,
% 0.35/0.74 at_top_nat != bot_bot_filter_nat ).
% 0.35/0.74
% 0.35/0.74 % trivial_limit_sequentially
% 0.35/0.74 thf(fact_9620_mult__nat__right__at__top,axiom,
% 0.35/0.74 ! [C: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.74 => ( filterlim_nat_nat
% 0.35/0.74 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 0.35/0.74 @ at_top_nat
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % mult_nat_right_at_top
% 0.35/0.74 thf(fact_9621_mult__nat__left__at__top,axiom,
% 0.35/0.74 ! [C: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.35/0.74 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % mult_nat_left_at_top
% 0.35/0.74 thf(fact_9622_monoseq__convergent,axiom,
% 0.35/0.74 ! [X6: nat > real,B2: real] :
% 0.35/0.74 ( ( topolo6980174941875973593q_real @ X6 )
% 0.35/0.74 => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X6 @ I2 ) ) @ B2 )
% 0.35/0.74 => ~ ! [L6: real] :
% 0.35/0.74 ~ ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % monoseq_convergent
% 0.35/0.74 thf(fact_9623_nested__sequence__unique,axiom,
% 0.35/0.74 ! [F: nat > real,G3: nat > real] :
% 0.35/0.74 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( G3 @ ( suc @ N3 ) ) @ ( G3 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 0.35/0.74 => ( ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74 @ at_top_nat )
% 0.35/0.74 => ? [L4: real] :
% 0.35/0.74 ( ! [N5: nat] : ( ord_less_eq_real @ ( F @ N5 ) @ L4 )
% 0.35/0.74 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.35/0.74 & ! [N5: nat] : ( ord_less_eq_real @ L4 @ ( G3 @ N5 ) )
% 0.35/0.74 & ( filterlim_nat_real @ G3 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % nested_sequence_unique
% 0.35/0.74 thf(fact_9624_LIMSEQ__inverse__zero,axiom,
% 0.35/0.74 ! [X6: nat > real] :
% 0.35/0.74 ( ! [R3: real] :
% 0.35/0.74 ? [N8: nat] :
% 0.35/0.74 ! [N3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.35/0.74 => ( ord_less_real @ R3 @ ( X6 @ N3 ) ) )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X6 @ N2 ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_inverse_zero
% 0.35/0.74 thf(fact_9625_LIMSEQ__root__const,axiom,
% 0.35/0.74 ! [C: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ C )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] : ( root @ N2 @ C )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_root_const
% 0.35/0.74 thf(fact_9626_increasing__LIMSEQ,axiom,
% 0.35/0.74 ! [F: nat > real,L: real] :
% 0.35/0.74 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 0.35/0.74 => ( ! [E: real] :
% 0.35/0.74 ( ( ord_less_real @ zero_zero_real @ E )
% 0.35/0.74 => ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
% 0.35/0.74 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % increasing_LIMSEQ
% 0.35/0.74 thf(fact_9627_LIMSEQ__realpow__zero,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( ( ord_less_real @ X2 @ one_one_real )
% 0.35/0.74 => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_realpow_zero
% 0.35/0.74 thf(fact_9628_LIMSEQ__divide__realpow__zero,axiom,
% 0.35/0.74 ! [X2: real,A2: real] :
% 0.35/0.74 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] : ( divide_divide_real @ A2 @ ( power_power_real @ X2 @ N2 ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_divide_realpow_zero
% 0.35/0.74 thf(fact_9629_LIMSEQ__abs__realpow__zero,axiom,
% 0.35/0.74 ! [C: real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.35/0.74 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_abs_realpow_zero
% 0.35/0.74 thf(fact_9630_LIMSEQ__abs__realpow__zero2,axiom,
% 0.35/0.74 ! [C: real] :
% 0.35/0.74 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.35/0.74 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_abs_realpow_zero2
% 0.35/0.74 thf(fact_9631_LIMSEQ__inverse__realpow__zero,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_real @ one_one_real @ X2 )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N2 ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % LIMSEQ_inverse_realpow_zero
% 0.35/0.74 thf(fact_9632_summable,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( summable_real
% 0.35/0.74 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable
% 0.35/0.74 thf(fact_9633_zeroseq__arctan__series,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.35/0.74 @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % zeroseq_arctan_series
% 0.35/0.74 thf(fact_9634_summable__Leibniz_H_I3_J,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] :
% 0.35/0.74 ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
% 0.35/0.74 @ at_top_nat ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz'(3)
% 0.35/0.74 thf(fact_9635_summable__Leibniz_H_I2_J,axiom,
% 0.35/0.74 ! [A2: nat > real,N: nat] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ord_less_eq_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz'(2)
% 0.35/0.74 thf(fact_9636_sums__alternating__upper__lower,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ? [L4: real] :
% 0.35/0.74 ( ! [N5: nat] :
% 0.35/0.74 ( ord_less_eq_real
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 0.35/0.74 @ L4 )
% 0.35/0.74 & ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] :
% 0.35/0.74 ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ L4 )
% 0.35/0.74 @ at_top_nat )
% 0.35/0.74 & ! [N5: nat] :
% 0.35/0.74 ( ord_less_eq_real @ L4
% 0.35/0.74 @ ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) )
% 0.35/0.74 & ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] :
% 0.35/0.74 ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real @ L4 )
% 0.35/0.74 @ at_top_nat ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % sums_alternating_upper_lower
% 0.35/0.74 thf(fact_9637_summable__Leibniz_H_I5_J,axiom,
% 0.35/0.74 ! [A2: nat > real] :
% 0.35/0.74 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 0.35/0.74 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 0.35/0.74 => ( filterlim_nat_real
% 0.35/0.74 @ ^ [N2: nat] :
% 0.35/0.74 ( groups6591440286371151544t_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 0.35/0.74 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 0.35/0.74 @ ( topolo2815343760600316023s_real
% 0.35/0.74 @ ( suminf_real
% 0.35/0.74 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
% 0.35/0.74 @ at_top_nat ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_Leibniz'(5)
% 0.35/0.74 thf(fact_9638_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.35/0.74 ! [B3: real,F: real > real,Flim: real] :
% 0.35/0.74 ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ B3 @ X5 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 0.35/0.74 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.35/0.74 => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_neg_imp_decreasing_at_top
% 0.35/0.74 thf(fact_9639_filterlim__pow__at__bot__even,axiom,
% 0.35/0.74 ! [N: nat,F: real > real,F2: filter_real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
% 0.35/0.74 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74 => ( filterlim_real_real
% 0.35/0.74 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.35/0.74 @ at_top_real
% 0.35/0.74 @ F2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % filterlim_pow_at_bot_even
% 0.35/0.74 thf(fact_9640_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.35/0.74 ! [B3: real,F: real > real,Flim: real] :
% 0.35/0.74 ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 0.35/0.74 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.35/0.74 => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pos_imp_increasing_at_bot
% 0.35/0.74 thf(fact_9641_filterlim__pow__at__bot__odd,axiom,
% 0.35/0.74 ! [N: nat,F: real > real,F2: filter_real] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
% 0.35/0.74 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 0.35/0.74 => ( filterlim_real_real
% 0.35/0.74 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 0.35/0.74 @ at_bot_real
% 0.35/0.74 @ F2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % filterlim_pow_at_bot_odd
% 0.35/0.74 thf(fact_9642_GMVT,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,G3: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_eq_real @ X5 @ B3 ) )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_real @ X5 @ B3 ) )
% 0.35/0.74 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_eq_real @ X5 @ B3 ) )
% 0.35/0.74 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G3 ) )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 & ( ord_less_real @ X5 @ B3 ) )
% 0.35/0.74 => ( differ6690327859849518006l_real @ G3 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 0.35/0.74 => ? [G_c: real,F_c: real,C3: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ G3 @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.35/0.74 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ A2 @ C3 )
% 0.35/0.74 & ( ord_less_real @ C3 @ B3 )
% 0.35/0.74 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ G_c )
% 0.35/0.74 = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % GMVT
% 0.35/0.74 thf(fact_9643_at__bot__le__at__infinity,axiom,
% 0.35/0.74 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 0.35/0.74
% 0.35/0.74 % at_bot_le_at_infinity
% 0.35/0.74 thf(fact_9644_at__top__le__at__infinity,axiom,
% 0.35/0.74 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 0.35/0.74
% 0.35/0.74 % at_top_le_at_infinity
% 0.35/0.74 thf(fact_9645_MVT,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ? [L4: real,Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 & ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 0.35/0.74 & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 0.35/0.74 = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ L4 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % MVT
% 0.35/0.74 thf(fact_9646_eventually__sequentiallyI,axiom,
% 0.35/0.74 ! [C: nat,P: nat > $o] :
% 0.35/0.74 ( ! [X5: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ C @ X5 )
% 0.35/0.74 => ( P @ X5 ) )
% 0.35/0.74 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % eventually_sequentiallyI
% 0.35/0.74 thf(fact_9647_eventually__sequentially,axiom,
% 0.35/0.74 ! [P: nat > $o] :
% 0.35/0.74 ( ( eventually_nat @ P @ at_top_nat )
% 0.35/0.74 = ( ? [N4: nat] :
% 0.35/0.74 ! [N2: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.35/0.74 => ( P @ N2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % eventually_sequentially
% 0.35/0.74 thf(fact_9648_le__sequentially,axiom,
% 0.35/0.74 ! [F2: filter_nat] :
% 0.35/0.74 ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
% 0.35/0.74 = ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F2 ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % le_sequentially
% 0.35/0.74 thf(fact_9649_continuous__on__arcosh_H,axiom,
% 0.35/0.74 ! [A3: set_real,F: real > real] :
% 0.35/0.74 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( member_real @ X5 @ A3 )
% 0.35/0.74 => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 0.35/0.74 => ( topolo5044208981011980120l_real @ A3
% 0.35/0.74 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % continuous_on_arcosh'
% 0.35/0.74 thf(fact_9650_eventually__at__left__real,axiom,
% 0.35/0.74 ! [B3: real,A2: real] :
% 0.35/0.74 ( ( ord_less_real @ B3 @ A2 )
% 0.35/0.74 => ( eventually_real
% 0.35/0.74 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B3 @ A2 ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % eventually_at_left_real
% 0.35/0.74 thf(fact_9651_continuous__image__closed__interval,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_eq_real @ A2 @ B3 )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ? [C3: real,D6: real] :
% 0.35/0.74 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 0.35/0.74 = ( set_or1222579329274155063t_real @ C3 @ D6 ) )
% 0.35/0.74 & ( ord_less_eq_real @ C3 @ D6 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % continuous_image_closed_interval
% 0.35/0.74 thf(fact_9652_continuous__on__artanh,axiom,
% 0.35/0.74 ! [A3: set_real] :
% 0.35/0.74 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 0.35/0.74 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 0.35/0.74
% 0.35/0.74 % continuous_on_artanh
% 0.35/0.74 thf(fact_9653_Rolle__deriv,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( ( F @ A2 )
% 0.35/0.74 = ( F @ B3 ) )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ? [Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 & ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 & ( ( F6 @ Z4 )
% 0.35/0.74 = ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Rolle_deriv
% 0.35/0.74 thf(fact_9654_Bseq__eq__bounded,axiom,
% 0.35/0.74 ! [F: nat > real,A2: real,B3: real] :
% 0.35/0.74 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 0.35/0.74 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % Bseq_eq_bounded
% 0.35/0.74 thf(fact_9655_mvt,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ~ ! [Xi: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Xi )
% 0.35/0.74 => ( ( ord_less_real @ Xi @ B3 )
% 0.35/0.74 => ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 0.35/0.74 != ( F6 @ Xi @ ( minus_minus_real @ B3 @ A2 ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % mvt
% 0.35/0.74 thf(fact_9656_Bseq__realpow,axiom,
% 0.35/0.74 ! [X2: real] :
% 0.35/0.74 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.74 => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Bseq_realpow
% 0.35/0.74 thf(fact_9657_DERIV__pos__imp__increasing__open,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_pos_imp_increasing_open
% 0.35/0.74 thf(fact_9658_DERIV__neg__imp__decreasing__open,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ? [Y5: real] :
% 0.35/0.74 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_neg_imp_decreasing_open
% 0.35/0.74 thf(fact_9659_DERIV__isconst__end,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ( ( F @ B3 )
% 0.35/0.74 = ( F @ A2 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_isconst_end
% 0.35/0.74 thf(fact_9660_DERIV__isconst2,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real,X2: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ( ( ord_less_eq_real @ A2 @ X2 )
% 0.35/0.74 => ( ( ord_less_eq_real @ X2 @ B3 )
% 0.35/0.74 => ( ( F @ X2 )
% 0.35/0.74 = ( F @ A2 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % DERIV_isconst2
% 0.35/0.74 thf(fact_9661_Rolle,axiom,
% 0.35/0.74 ! [A2: real,B3: real,F: real > real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( ( ( F @ A2 )
% 0.35/0.74 = ( F @ B3 ) )
% 0.35/0.74 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 0.35/0.74 => ( ! [X5: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ X5 )
% 0.35/0.74 => ( ( ord_less_real @ X5 @ B3 )
% 0.35/0.74 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 0.35/0.74 => ? [Z4: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ Z4 )
% 0.35/0.74 & ( ord_less_real @ Z4 @ B3 )
% 0.35/0.74 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Rolle
% 0.35/0.74 thf(fact_9662_eventually__at__right__real,axiom,
% 0.35/0.74 ! [A2: real,B3: real] :
% 0.35/0.74 ( ( ord_less_real @ A2 @ B3 )
% 0.35/0.74 => ( eventually_real
% 0.35/0.74 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 0.35/0.74 @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % eventually_at_right_real
% 0.35/0.74 thf(fact_9663_greaterThan__0,axiom,
% 0.35/0.74 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.35/0.74 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.35/0.74
% 0.35/0.74 % greaterThan_0
% 0.35/0.74 thf(fact_9664_greaterThan__Suc,axiom,
% 0.35/0.74 ! [K: nat] :
% 0.35/0.74 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.35/0.74 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % greaterThan_Suc
% 0.35/0.74 thf(fact_9665_INT__greaterThan__UNIV,axiom,
% 0.35/0.74 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.35/0.74 = bot_bot_set_nat ) ).
% 0.35/0.74
% 0.35/0.74 % INT_greaterThan_UNIV
% 0.35/0.74 thf(fact_9666_finite__greaterThanAtMost,axiom,
% 0.35/0.74 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % finite_greaterThanAtMost
% 0.35/0.74 thf(fact_9667_card__greaterThanAtMost,axiom,
% 0.35/0.74 ! [L: nat,U: nat] :
% 0.35/0.74 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 0.35/0.74 = ( minus_minus_nat @ U @ L ) ) ).
% 0.35/0.74
% 0.35/0.74 % card_greaterThanAtMost
% 0.35/0.74 thf(fact_9668_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.35/0.74 ! [L: nat,U: nat] :
% 0.35/0.74 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.35/0.74 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeastSucAtMost_greaterThanAtMost
% 0.35/0.74 thf(fact_9669_GreatestI__ex__nat,axiom,
% 0.35/0.74 ! [P: nat > $o,B3: nat] :
% 0.35/0.74 ( ? [X_12: nat] : ( P @ X_12 )
% 0.35/0.74 => ( ! [Y4: nat] :
% 0.35/0.74 ( ( P @ Y4 )
% 0.35/0.74 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 0.35/0.74 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % GreatestI_ex_nat
% 0.35/0.74 thf(fact_9670_Greatest__le__nat,axiom,
% 0.35/0.74 ! [P: nat > $o,K: nat,B3: nat] :
% 0.35/0.74 ( ( P @ K )
% 0.35/0.74 => ( ! [Y4: nat] :
% 0.35/0.74 ( ( P @ Y4 )
% 0.35/0.74 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 0.35/0.74 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Greatest_le_nat
% 0.35/0.74 thf(fact_9671_GreatestI__nat,axiom,
% 0.35/0.74 ! [P: nat > $o,K: nat,B3: nat] :
% 0.35/0.74 ( ( P @ K )
% 0.35/0.74 => ( ! [Y4: nat] :
% 0.35/0.74 ( ( P @ Y4 )
% 0.35/0.74 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 0.35/0.74 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % GreatestI_nat
% 0.35/0.74 thf(fact_9672_finite__greaterThanAtMost__int,axiom,
% 0.35/0.74 ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % finite_greaterThanAtMost_int
% 0.35/0.74 thf(fact_9673_atLeast__0,axiom,
% 0.35/0.74 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.35/0.74 = top_top_set_nat ) ).
% 0.35/0.74
% 0.35/0.74 % atLeast_0
% 0.35/0.74 thf(fact_9674_card__greaterThanAtMost__int,axiom,
% 0.35/0.74 ! [L: int,U: int] :
% 0.35/0.74 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 0.35/0.74 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % card_greaterThanAtMost_int
% 0.35/0.74 thf(fact_9675_atLeast__Suc__greaterThan,axiom,
% 0.35/0.74 ! [K: nat] :
% 0.35/0.74 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.35/0.74 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeast_Suc_greaterThan
% 0.35/0.74 thf(fact_9676_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 0.35/0.74 ! [L: int,U: int] :
% 0.35/0.74 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 0.35/0.74 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeastPlusOneAtMost_greaterThanAtMost_int
% 0.35/0.74 thf(fact_9677_decseq__bounded,axiom,
% 0.35/0.74 ! [X6: nat > real,B2: real] :
% 0.35/0.74 ( ( order_9091379641038594480t_real @ X6 )
% 0.35/0.74 => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
% 0.35/0.74 => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % decseq_bounded
% 0.35/0.74 thf(fact_9678_decseq__convergent,axiom,
% 0.35/0.74 ! [X6: nat > real,B2: real] :
% 0.35/0.74 ( ( order_9091379641038594480t_real @ X6 )
% 0.35/0.74 => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
% 0.35/0.74 => ~ ! [L6: real] :
% 0.35/0.74 ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.35/0.74 => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X6 @ I3 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % decseq_convergent
% 0.35/0.74 thf(fact_9679_UN__atLeast__UNIV,axiom,
% 0.35/0.74 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 0.35/0.74 = top_top_set_nat ) ).
% 0.35/0.74
% 0.35/0.74 % UN_atLeast_UNIV
% 0.35/0.74 thf(fact_9680_atLeast__Suc,axiom,
% 0.35/0.74 ! [K: nat] :
% 0.35/0.74 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.35/0.74 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % atLeast_Suc
% 0.35/0.74 thf(fact_9681_continuous__on__arcosh,axiom,
% 0.35/0.74 ! [A3: set_real] :
% 0.35/0.74 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 0.35/0.74 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 0.35/0.74
% 0.35/0.74 % continuous_on_arcosh
% 0.35/0.74 thf(fact_9682_bdd__above__nat,axiom,
% 0.35/0.74 condit2214826472909112428ve_nat = finite_finite_nat ).
% 0.35/0.74
% 0.35/0.74 % bdd_above_nat
% 0.35/0.74 thf(fact_9683_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.74 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 = Y3 )
% 0.35/0.74 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( Y3
% 0.35/0.74 = ( Xa2 != one_one_nat ) ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ( Y3
% 0.35/0.74 = ( ~ ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.elims(1)
% 0.35/0.74 thf(fact_9684_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.74 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( Xa2 != one_one_nat ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ~ ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X4: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.elims(2)
% 0.35/0.74 thf(fact_9685_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.74 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( Xa2 = one_one_nat ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X5: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.elims(3)
% 0.35/0.74 thf(fact_9686_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.35/0.74 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 0.35/0.74 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 0.35/0.74 = ( ( Deg = Deg3 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima2 ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.simps(2)
% 0.35/0.74 thf(fact_9687_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.74 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.74 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.35/0.74 => ( Xa2 = one_one_nat ) ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
% 0.35/0.74 => ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X5: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.pelims(3)
% 0.35/0.74 thf(fact_9688_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat] :
% 0.35/0.74 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.74 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.35/0.74 => ( Xa2 != one_one_nat ) ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
% 0.35/0.74 => ~ ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X4: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.pelims(2)
% 0.35/0.74 thf(fact_9689_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.35/0.74 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.35/0.74 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 0.35/0.74 = Y3 )
% 0.35/0.74 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 0.35/0.74 => ( ! [Uu2: $o,Uv2: $o] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.35/0.74 => ( ( Y3
% 0.35/0.74 = ( Xa2 = one_one_nat ) )
% 0.35/0.74 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.35/0.74 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.35/0.74 ( ( X2
% 0.35/0.74 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 0.35/0.74 => ( ( Y3
% 0.35/0.74 = ( ( Deg2 = Xa2 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.35/0.74 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.35/0.74 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.35/0.74 & ( case_o184042715313410164at_nat
% 0.35/0.74 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 0.35/0.74 & ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 @ ( produc6081775807080527818_nat_o
% 0.35/0.74 @ ^ [Mi3: nat,Ma3: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.35/0.74 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 & ! [I4: nat] :
% 0.35/0.74 ( ( 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 ) ) ) ) ) )
% 0.35/0.74 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 0.35/0.74 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.35/0.74 & ( ( Mi3 = Ma3 )
% 0.35/0.74 => ! [X: vEBT_VEBT] :
% 0.35/0.74 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.35/0.74 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 0.35/0.74 & ( ( Mi3 != Ma3 )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 0.35/0.74 & ! [X: nat] :
% 0.35/0.74 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.35/0.74 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 0.35/0.74 => ( ( ord_less_nat @ Mi3 @ X )
% 0.35/0.74 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.35/0.74 @ Mima ) ) )
% 0.35/0.74 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % VEBT_internal.valid'.pelims(1)
% 0.35/0.74 thf(fact_9690_Sup__real__def,axiom,
% 0.35/0.74 ( comple1385675409528146559p_real
% 0.35/0.74 = ( ^ [X8: set_real] :
% 0.35/0.74 ( ord_Least_real
% 0.35/0.74 @ ^ [Z2: real] :
% 0.35/0.74 ! [X: real] :
% 0.35/0.74 ( ( member_real @ X @ X8 )
% 0.35/0.74 => ( ord_less_eq_real @ X @ Z2 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Sup_real_def
% 0.35/0.74 thf(fact_9691_Sup__int__def,axiom,
% 0.35/0.74 ( complete_Sup_Sup_int
% 0.35/0.74 = ( ^ [X8: set_int] :
% 0.35/0.74 ( the_int
% 0.35/0.74 @ ^ [X: int] :
% 0.35/0.74 ( ( member_int @ X @ X8 )
% 0.35/0.74 & ! [Y: int] :
% 0.35/0.74 ( ( member_int @ Y @ X8 )
% 0.35/0.74 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % Sup_int_def
% 0.35/0.74 thf(fact_9692_uniformity__real__def,axiom,
% 0.35/0.74 ( topolo1511823702728130853y_real
% 0.35/0.74 = ( comple2936214249959783750l_real
% 0.35/0.74 @ ( image_2178119161166701260l_real
% 0.35/0.74 @ ^ [E3: real] :
% 0.35/0.74 ( princi6114159922880469582l_real
% 0.35/0.74 @ ( collec3799799289383736868l_real
% 0.35/0.74 @ ( produc5414030515140494994real_o
% 0.35/0.74 @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 0.35/0.74 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % uniformity_real_def
% 0.35/0.74 thf(fact_9693_uniformity__complex__def,axiom,
% 0.35/0.74 ( topolo896644834953643431omplex
% 0.35/0.74 = ( comple8358262395181532106omplex
% 0.35/0.74 @ ( image_5971271580939081552omplex
% 0.35/0.74 @ ^ [E3: real] :
% 0.35/0.74 ( princi3496590319149328850omplex
% 0.35/0.74 @ ( collec8663557070575231912omplex
% 0.35/0.74 @ ( produc6771430404735790350plex_o
% 0.35/0.74 @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 0.35/0.74 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % uniformity_complex_def
% 0.35/0.74 thf(fact_9694_bit__cut__integer__code,axiom,
% 0.35/0.74 ( code_bit_cut_integer
% 0.35/0.74 = ( ^ [K3: code_integer] :
% 0.35/0.74 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.35/0.74 @ ( produc9125791028180074456eger_o
% 0.35/0.74 @ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
% 0.35/0.74 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % bit_cut_integer_code
% 0.35/0.74 thf(fact_9695_divmod__integer__code,axiom,
% 0.35/0.74 ( code_divmod_integer
% 0.35/0.74 = ( ^ [K3: code_integer,L2: code_integer] :
% 0.35/0.74 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.35/0.74 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.35/0.74 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.35/0.74 @ ( produc6916734918728496179nteger
% 0.35/0.74 @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S7 ) ) )
% 0.35/0.74 @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 0.35/0.74 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.35/0.74 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.35/0.74 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.35/0.74 @ ( produc6916734918728496179nteger
% 0.35/0.74 @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S7 ) ) )
% 0.35/0.74 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % divmod_integer_code
% 0.35/0.74 thf(fact_9696_eventually__prod__sequentially,axiom,
% 0.35/0.74 ! [P: product_prod_nat_nat > $o] :
% 0.35/0.74 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.35/0.74 = ( ? [N4: nat] :
% 0.35/0.74 ! [M2: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ N4 @ M2 )
% 0.35/0.74 => ! [N2: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ N4 @ N2 )
% 0.35/0.74 => ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % eventually_prod_sequentially
% 0.35/0.74 thf(fact_9697_card_Ocomp__fun__commute__on,axiom,
% 0.35/0.74 ( ( comp_nat_nat_nat @ suc @ suc )
% 0.35/0.74 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.35/0.74
% 0.35/0.74 % card.comp_fun_commute_on
% 0.35/0.74 thf(fact_9698_drop__bit__nonnegative__int__iff,axiom,
% 0.35/0.74 ! [N: nat,K: int] :
% 0.35/0.74 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 0.35/0.74 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.74
% 0.35/0.74 % drop_bit_nonnegative_int_iff
% 0.35/0.74 thf(fact_9699_drop__bit__negative__int__iff,axiom,
% 0.35/0.74 ! [N: nat,K: int] :
% 0.35/0.74 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.74 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.74
% 0.35/0.74 % drop_bit_negative_int_iff
% 0.35/0.74 thf(fact_9700_drop__bit__of__Suc__0,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.74 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % drop_bit_of_Suc_0
% 0.35/0.74 thf(fact_9701_mono__Suc,axiom,
% 0.35/0.74 order_mono_nat_nat @ suc ).
% 0.35/0.74
% 0.35/0.74 % mono_Suc
% 0.35/0.74 thf(fact_9702_mono__times__nat,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % mono_times_nat
% 0.35/0.74 thf(fact_9703_incseq__bounded,axiom,
% 0.35/0.74 ! [X6: nat > real,B2: real] :
% 0.35/0.74 ( ( order_mono_nat_real @ X6 )
% 0.35/0.74 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
% 0.35/0.74 => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % incseq_bounded
% 0.35/0.74 thf(fact_9704_incseq__convergent,axiom,
% 0.35/0.74 ! [X6: nat > real,B2: real] :
% 0.35/0.74 ( ( order_mono_nat_real @ X6 )
% 0.35/0.74 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
% 0.35/0.74 => ~ ! [L6: real] :
% 0.35/0.74 ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 0.35/0.74 => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X6 @ I3 ) @ L6 ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % incseq_convergent
% 0.35/0.74 thf(fact_9705_infinite__int__iff__infinite__nat__abs,axiom,
% 0.35/0.74 ! [S: set_int] :
% 0.35/0.74 ( ( ~ ( finite_finite_int @ S ) )
% 0.35/0.74 = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % infinite_int_iff_infinite_nat_abs
% 0.35/0.74 thf(fact_9706_mono__ge2__power__minus__self,axiom,
% 0.35/0.74 ! [K: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.35/0.74 => ( order_mono_nat_nat
% 0.35/0.74 @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % mono_ge2_power_minus_self
% 0.35/0.74 thf(fact_9707_take__bit__num__simps_I1_J,axiom,
% 0.35/0.74 ! [M: num] :
% 0.35/0.74 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.35/0.74 = none_num ) ).
% 0.35/0.74
% 0.35/0.74 % take_bit_num_simps(1)
% 0.35/0.74 thf(fact_9708_nonneg__incseq__Bseq__subseq__iff,axiom,
% 0.35/0.74 ! [F: nat > real,G3: nat > nat] :
% 0.35/0.74 ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.35/0.74 => ( ( order_mono_nat_real @ F )
% 0.35/0.74 => ( ( order_5726023648592871131at_nat @ G3 )
% 0.35/0.74 => ( ( bfun_nat_real
% 0.35/0.74 @ ^ [X: nat] : ( F @ ( G3 @ X ) )
% 0.35/0.74 @ at_top_nat )
% 0.35/0.74 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % nonneg_incseq_Bseq_subseq_iff
% 0.35/0.74 thf(fact_9709_strict__mono__imp__increasing,axiom,
% 0.35/0.74 ! [F: nat > nat,N: nat] :
% 0.35/0.74 ( ( order_5726023648592871131at_nat @ F )
% 0.35/0.74 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % strict_mono_imp_increasing
% 0.35/0.74 thf(fact_9710_infinite__enumerate,axiom,
% 0.35/0.74 ! [S: set_nat] :
% 0.35/0.74 ( ~ ( finite_finite_nat @ S )
% 0.35/0.74 => ? [R3: nat > nat] :
% 0.35/0.74 ( ( order_5726023648592871131at_nat @ R3 )
% 0.35/0.74 & ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % infinite_enumerate
% 0.35/0.74 thf(fact_9711_strict__mono__enumerate,axiom,
% 0.35/0.74 ! [S: set_nat] :
% 0.35/0.74 ( ~ ( finite_finite_nat @ S )
% 0.35/0.74 => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % strict_mono_enumerate
% 0.35/0.74 thf(fact_9712_take__bit__num__def,axiom,
% 0.35/0.74 ( bit_take_bit_num
% 0.35/0.74 = ( ^ [N2: nat,M2: num] :
% 0.35/0.74 ( if_option_num
% 0.35/0.74 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
% 0.35/0.74 = zero_zero_nat )
% 0.35/0.74 @ none_num
% 0.35/0.74 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % take_bit_num_def
% 0.35/0.74 thf(fact_9713_num__of__nat_Osimps_I1_J,axiom,
% 0.35/0.74 ( ( num_of_nat @ zero_zero_nat )
% 0.35/0.74 = one ) ).
% 0.35/0.74
% 0.35/0.74 % num_of_nat.simps(1)
% 0.35/0.74 thf(fact_9714_numeral__num__of__nat,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 0.35/0.74 = N ) ) ).
% 0.35/0.74
% 0.35/0.74 % numeral_num_of_nat
% 0.35/0.74 thf(fact_9715_num__of__nat__One,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 0.35/0.74 => ( ( num_of_nat @ N )
% 0.35/0.74 = one ) ) ).
% 0.35/0.74
% 0.35/0.74 % num_of_nat_One
% 0.35/0.74 thf(fact_9716_num__of__nat__double,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 0.35/0.74 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % num_of_nat_double
% 0.35/0.74 thf(fact_9717_num__of__nat__plus__distrib,axiom,
% 0.35/0.74 ! [M: nat,N: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.74 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 0.35/0.74 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % num_of_nat_plus_distrib
% 0.35/0.74 thf(fact_9718_pos__deriv__imp__strict__mono,axiom,
% 0.35/0.74 ! [F: real > real,F6: real > real] :
% 0.35/0.74 ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 0.35/0.74 => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X5 ) )
% 0.35/0.74 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % pos_deriv_imp_strict_mono
% 0.35/0.74 thf(fact_9719_Suc__funpow,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( compow_nat_nat @ N @ suc )
% 0.35/0.74 = ( plus_plus_nat @ N ) ) ).
% 0.35/0.74
% 0.35/0.74 % Suc_funpow
% 0.35/0.74 thf(fact_9720_inj__sgn__power,axiom,
% 0.35/0.74 ! [N: nat] :
% 0.35/0.74 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.74 => ( inj_on_real_real
% 0.35/0.74 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.35/0.74 @ top_top_set_real ) ) ).
% 0.35/0.74
% 0.35/0.74 % inj_sgn_power
% 0.35/0.74 thf(fact_9721_log__inj,axiom,
% 0.35/0.74 ! [B3: real] :
% 0.35/0.74 ( ( ord_less_real @ one_one_real @ B3 )
% 0.35/0.74 => ( inj_on_real_real @ ( log @ B3 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % log_inj
% 0.35/0.74 thf(fact_9722_inj__Suc,axiom,
% 0.35/0.74 ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% 0.35/0.74
% 0.35/0.74 % inj_Suc
% 0.35/0.74 thf(fact_9723_inj__on__diff__nat,axiom,
% 0.35/0.74 ! [N6: set_nat,K: nat] :
% 0.35/0.74 ( ! [N3: nat] :
% 0.35/0.74 ( ( member_nat @ N3 @ N6 )
% 0.35/0.74 => ( ord_less_eq_nat @ K @ N3 ) )
% 0.35/0.74 => ( inj_on_nat_nat
% 0.35/0.74 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 0.35/0.74 @ N6 ) ) ).
% 0.35/0.74
% 0.35/0.74 % inj_on_diff_nat
% 0.35/0.74 thf(fact_9724_inj__on__set__encode,axiom,
% 0.35/0.74 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 0.35/0.74
% 0.35/0.74 % inj_on_set_encode
% 0.35/0.74 thf(fact_9725_summable__reindex,axiom,
% 0.35/0.74 ! [F: nat > real,G3: nat > nat] :
% 0.35/0.74 ( ( summable_real @ F )
% 0.35/0.74 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 0.35/0.74 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.35/0.74 => ( summable_real @ ( comp_nat_real_nat @ F @ G3 ) ) ) ) ) ).
% 0.35/0.74
% 0.35/0.74 % summable_reindex
% 0.35/0.74 thf(fact_9726_suminf__reindex__mono,axiom,
% 0.35/0.74 ! [F: nat > real,G3: nat > nat] :
% 0.35/0.74 ( ( summable_real @ F )
% 0.35/0.74 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 0.35/0.74 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.35/0.75 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % suminf_reindex_mono
% 0.35/0.75 thf(fact_9727_suminf__reindex,axiom,
% 0.35/0.75 ! [F: nat > real,G3: nat > nat] :
% 0.35/0.75 ( ( summable_real @ F )
% 0.35/0.75 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 0.35/0.75 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 0.35/0.75 => ( ! [X5: nat] :
% 0.35/0.75 ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G3 @ top_top_set_nat ) )
% 0.35/0.75 => ( ( F @ X5 )
% 0.35/0.75 = zero_zero_real ) )
% 0.35/0.75 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) )
% 0.35/0.75 = ( suminf_real @ F ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % suminf_reindex
% 0.35/0.75 thf(fact_9728_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.35/0.75 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.35/0.75 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.35/0.75 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 0.35/0.75
% 0.35/0.75 % max_nat.semilattice_neutr_order_axioms
% 0.35/0.75 thf(fact_9729_push__bit__of__Suc__0,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.35/0.75 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % push_bit_of_Suc_0
% 0.35/0.75 thf(fact_9730_push__bit__nonnegative__int__iff,axiom,
% 0.35/0.75 ! [N: nat,K: int] :
% 0.35/0.75 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 0.35/0.75 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.75
% 0.35/0.75 % push_bit_nonnegative_int_iff
% 0.35/0.75 thf(fact_9731_push__bit__negative__int__iff,axiom,
% 0.35/0.75 ! [N: nat,K: int] :
% 0.35/0.75 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 0.35/0.75 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.75
% 0.35/0.75 % push_bit_negative_int_iff
% 0.35/0.75 thf(fact_9732_bit__push__bit__iff__int,axiom,
% 0.35/0.75 ! [M: nat,K: int,N: nat] :
% 0.35/0.75 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 0.35/0.75 = ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bit_push_bit_iff_int
% 0.35/0.75 thf(fact_9733_bit__push__bit__iff__nat,axiom,
% 0.35/0.75 ! [M: nat,Q3: nat,N: nat] :
% 0.35/0.75 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 0.35/0.75 = ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75 & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bit_push_bit_iff_nat
% 0.35/0.75 thf(fact_9734_powr__real__of__int_H,axiom,
% 0.35/0.75 ! [X2: real,N: int] :
% 0.35/0.75 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.75 => ( ( ( X2 != zero_zero_real )
% 0.35/0.75 | ( ord_less_int @ zero_zero_int @ N ) )
% 0.35/0.75 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 0.35/0.75 = ( power_int_real @ X2 @ N ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % powr_real_of_int'
% 0.35/0.75 thf(fact_9735_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.35/0.75 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.35/0.75 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_list_of_set_greaterThanAtMost
% 0.35/0.75 thf(fact_9736_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.35/0.75 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.35/0.75 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_list_of_set_greaterThanLessThan
% 0.35/0.75 thf(fact_9737_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.35/0.75 ! [N: nat,J: nat,I: nat] :
% 0.35/0.75 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 0.35/0.75 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 0.35/0.75 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nth_sorted_list_of_set_greaterThanAtMost
% 0.35/0.75 thf(fact_9738_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.35/0.75 ! [N: nat,J: nat,I: nat] :
% 0.35/0.75 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.35/0.75 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 0.35/0.75 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nth_sorted_list_of_set_greaterThanLessThan
% 0.35/0.75 thf(fact_9739_num__of__nat_Osimps_I2_J,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( ( num_of_nat @ ( suc @ N ) )
% 0.35/0.75 = ( inc @ ( num_of_nat @ N ) ) ) )
% 0.35/0.75 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( ( num_of_nat @ ( suc @ N ) )
% 0.35/0.75 = one ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % num_of_nat.simps(2)
% 0.35/0.75 thf(fact_9740_Rats__eq__int__div__nat,axiom,
% 0.35/0.75 ( field_5140801741446780682s_real
% 0.35/0.75 = ( collect_real
% 0.35/0.75 @ ^ [Uu3: real] :
% 0.35/0.75 ? [I4: int,N2: nat] :
% 0.35/0.75 ( ( Uu3
% 0.35/0.75 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.35/0.75 & ( N2 != zero_zero_nat ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rats_eq_int_div_nat
% 0.35/0.75 thf(fact_9741_rat__floor__lemma,axiom,
% 0.35/0.75 ! [A2: int,B3: int] :
% 0.35/0.75 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A2 @ B3 ) ) @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 & ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % rat_floor_lemma
% 0.35/0.75 thf(fact_9742_less__rat,axiom,
% 0.35/0.75 ! [B3: int,D: int,A2: int,C: int] :
% 0.35/0.75 ( ( B3 != zero_zero_int )
% 0.35/0.75 => ( ( D != zero_zero_int )
% 0.35/0.75 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
% 0.35/0.75 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_rat
% 0.35/0.75 thf(fact_9743_le__rat,axiom,
% 0.35/0.75 ! [B3: int,D: int,A2: int,C: int] :
% 0.35/0.75 ( ( B3 != zero_zero_int )
% 0.35/0.75 => ( ( D != zero_zero_int )
% 0.35/0.75 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
% 0.35/0.75 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % le_rat
% 0.35/0.75 thf(fact_9744_Rats__dense__in__real,axiom,
% 0.35/0.75 ! [X2: real,Y3: real] :
% 0.35/0.75 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.75 => ? [X5: real] :
% 0.35/0.75 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.35/0.75 & ( ord_less_real @ X2 @ X5 )
% 0.35/0.75 & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rats_dense_in_real
% 0.35/0.75 thf(fact_9745_Rats__no__bot__less,axiom,
% 0.35/0.75 ! [X2: real] :
% 0.35/0.75 ? [X5: real] :
% 0.35/0.75 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.35/0.75 & ( ord_less_real @ X5 @ X2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rats_no_bot_less
% 0.35/0.75 thf(fact_9746_Rat__induct__pos,axiom,
% 0.35/0.75 ! [P: rat > $o,Q3: rat] :
% 0.35/0.75 ( ! [A: int,B: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.75 => ( P @ ( fract @ A @ B ) ) )
% 0.35/0.75 => ( P @ Q3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat_induct_pos
% 0.35/0.75 thf(fact_9747_Rats__no__top__le,axiom,
% 0.35/0.75 ! [X2: real] :
% 0.35/0.75 ? [X5: real] :
% 0.35/0.75 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 0.35/0.75 & ( ord_less_eq_real @ X2 @ X5 ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rats_no_top_le
% 0.35/0.75 thf(fact_9748_zero__less__Fract__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % zero_less_Fract_iff
% 0.35/0.75 thf(fact_9749_Fract__less__zero__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
% 0.35/0.75 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Fract_less_zero_iff
% 0.35/0.75 thf(fact_9750_one__less__Fract__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 = ( ord_less_int @ B3 @ A2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % one_less_Fract_iff
% 0.35/0.75 thf(fact_9751_Fract__less__one__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
% 0.35/0.75 = ( ord_less_int @ A2 @ B3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Fract_less_one_iff
% 0.35/0.75 thf(fact_9752_zero__le__Fract__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % zero_le_Fract_iff
% 0.35/0.75 thf(fact_9753_Fract__le__zero__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
% 0.35/0.75 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Fract_le_zero_iff
% 0.35/0.75 thf(fact_9754_one__le__Fract__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % one_le_Fract_iff
% 0.35/0.75 thf(fact_9755_Fract__le__one__iff,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
% 0.35/0.75 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Fract_le_one_iff
% 0.35/0.75 thf(fact_9756_positive__rat,axiom,
% 0.35/0.75 ! [A2: int,B3: int] :
% 0.35/0.75 ( ( positive @ ( fract @ A2 @ B3 ) )
% 0.35/0.75 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % positive_rat
% 0.35/0.75 thf(fact_9757_less__rat__def,axiom,
% 0.35/0.75 ( ord_less_rat
% 0.35/0.75 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_rat_def
% 0.35/0.75 thf(fact_9758_min__Suc__Suc,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 0.35/0.75 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % min_Suc_Suc
% 0.35/0.75 thf(fact_9759_min__0L,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_min_nat @ zero_zero_nat @ N )
% 0.35/0.75 = zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % min_0L
% 0.35/0.75 thf(fact_9760_min__0R,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_min_nat @ N @ zero_zero_nat )
% 0.35/0.75 = zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % min_0R
% 0.35/0.75 thf(fact_9761_nat__mult__min__right,axiom,
% 0.35/0.75 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.75 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
% 0.35/0.75 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nat_mult_min_right
% 0.35/0.75 thf(fact_9762_nat__mult__min__left,axiom,
% 0.35/0.75 ! [M: nat,N: nat,Q3: nat] :
% 0.35/0.75 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
% 0.35/0.75 = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nat_mult_min_left
% 0.35/0.75 thf(fact_9763_min__diff,axiom,
% 0.35/0.75 ! [M: nat,I: nat,N: nat] :
% 0.35/0.75 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 0.35/0.75 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 0.35/0.75
% 0.35/0.75 % min_diff
% 0.35/0.75 thf(fact_9764_inf__nat__def,axiom,
% 0.35/0.75 inf_inf_nat = ord_min_nat ).
% 0.35/0.75
% 0.35/0.75 % inf_nat_def
% 0.35/0.75 thf(fact_9765_Arg__bounded,axiom,
% 0.35/0.75 ! [Z: complex] :
% 0.35/0.75 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.35/0.75 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 0.35/0.75
% 0.35/0.75 % Arg_bounded
% 0.35/0.75 thf(fact_9766_Arg__correct,axiom,
% 0.35/0.75 ! [Z: complex] :
% 0.35/0.75 ( ( Z != zero_zero_complex )
% 0.35/0.75 => ( ( ( sgn_sgn_complex @ Z )
% 0.35/0.75 = ( cis @ ( arg @ Z ) ) )
% 0.35/0.75 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 0.35/0.75 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Arg_correct
% 0.35/0.75 thf(fact_9767_bij__betw__roots__unity,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( bij_betw_nat_complex
% 0.35/0.75 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.35/0.75 @ ( set_ord_lessThan_nat @ N )
% 0.35/0.75 @ ( collect_complex
% 0.35/0.75 @ ^ [Z2: complex] :
% 0.35/0.75 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.75 = one_one_complex ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bij_betw_roots_unity
% 0.35/0.75 thf(fact_9768_cis__Arg__unique,axiom,
% 0.35/0.75 ! [Z: complex,X2: real] :
% 0.35/0.75 ( ( ( sgn_sgn_complex @ Z )
% 0.35/0.75 = ( cis @ X2 ) )
% 0.35/0.75 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 0.35/0.75 => ( ( ord_less_eq_real @ X2 @ pi )
% 0.35/0.75 => ( ( arg @ Z )
% 0.35/0.75 = X2 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cis_Arg_unique
% 0.35/0.75 thf(fact_9769_bij__betw__nth__root__unity,axiom,
% 0.35/0.75 ! [C: complex,N: nat] :
% 0.35/0.75 ( ( C != zero_zero_complex )
% 0.35/0.75 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 0.35/0.75 @ ( collect_complex
% 0.35/0.75 @ ^ [Z2: complex] :
% 0.35/0.75 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.75 = one_one_complex ) )
% 0.35/0.75 @ ( collect_complex
% 0.35/0.75 @ ^ [Z2: complex] :
% 0.35/0.75 ( ( power_power_complex @ Z2 @ N )
% 0.35/0.75 = C ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bij_betw_nth_root_unity
% 0.35/0.75 thf(fact_9770_bij__betw__Suc,axiom,
% 0.35/0.75 ! [M5: set_nat,N6: set_nat] :
% 0.35/0.75 ( ( bij_betw_nat_nat @ suc @ M5 @ N6 )
% 0.35/0.75 = ( ( image_nat_nat @ suc @ M5 )
% 0.35/0.75 = N6 ) ) ).
% 0.35/0.75
% 0.35/0.75 % bij_betw_Suc
% 0.35/0.75 thf(fact_9771_bij__enumerate,axiom,
% 0.35/0.75 ! [S: set_nat] :
% 0.35/0.75 ( ~ ( finite_finite_nat @ S )
% 0.35/0.75 => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).
% 0.35/0.75
% 0.35/0.75 % bij_enumerate
% 0.35/0.75 thf(fact_9772_Arg__def,axiom,
% 0.35/0.75 ( arg
% 0.35/0.75 = ( ^ [Z2: complex] :
% 0.35/0.75 ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 0.35/0.75 @ ( fChoice_real
% 0.35/0.75 @ ^ [A4: real] :
% 0.35/0.75 ( ( ( sgn_sgn_complex @ Z2 )
% 0.35/0.75 = ( cis @ A4 ) )
% 0.35/0.75 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 0.35/0.75 & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Arg_def
% 0.35/0.75 thf(fact_9773_less__eq__int_Orep__eq,axiom,
% 0.35/0.75 ( ord_less_eq_int
% 0.35/0.75 = ( ^ [X: int,Xa4: int] :
% 0.35/0.75 ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [Y: nat,Z2: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 0.35/0.75 @ ( rep_Integ @ X )
% 0.35/0.75 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_int.rep_eq
% 0.35/0.75 thf(fact_9774_less__int_Orep__eq,axiom,
% 0.35/0.75 ( ord_less_int
% 0.35/0.75 = ( ^ [X: int,Xa4: int] :
% 0.35/0.75 ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [Y: nat,Z2: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 0.35/0.75 @ ( rep_Integ @ X )
% 0.35/0.75 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_int.rep_eq
% 0.35/0.75 thf(fact_9775_less__eq__int_Oabs__eq,axiom,
% 0.35/0.75 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.35/0.75 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.35/0.75 = ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.35/0.75 @ Xa2
% 0.35/0.75 @ X2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_int.abs_eq
% 0.35/0.75 thf(fact_9776_zero__int__def,axiom,
% 0.35/0.75 ( zero_zero_int
% 0.35/0.75 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % zero_int_def
% 0.35/0.75 thf(fact_9777_int__def,axiom,
% 0.35/0.75 ( semiri1314217659103216013at_int
% 0.35/0.75 = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % int_def
% 0.35/0.75 thf(fact_9778_one__int__def,axiom,
% 0.35/0.75 ( one_one_int
% 0.35/0.75 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % one_int_def
% 0.35/0.75 thf(fact_9779_less__int_Oabs__eq,axiom,
% 0.35/0.75 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 0.35/0.75 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 0.35/0.75 = ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.35/0.75 @ Xa2
% 0.35/0.75 @ X2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_int.abs_eq
% 0.35/0.75 thf(fact_9780_card__length__sum__list__rec,axiom,
% 0.35/0.75 ! [M: nat,N6: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.35/0.75 => ( ( finite_card_list_nat
% 0.35/0.75 @ ( collect_list_nat
% 0.35/0.75 @ ^ [L2: list_nat] :
% 0.35/0.75 ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75 = M )
% 0.35/0.75 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.35/0.75 = N6 ) ) ) )
% 0.35/0.75 = ( plus_plus_nat
% 0.35/0.75 @ ( finite_card_list_nat
% 0.35/0.75 @ ( collect_list_nat
% 0.35/0.75 @ ^ [L2: list_nat] :
% 0.35/0.75 ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75 = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.35/0.75 & ( ( groups4561878855575611511st_nat @ L2 )
% 0.35/0.75 = N6 ) ) ) )
% 0.35/0.75 @ ( finite_card_list_nat
% 0.35/0.75 @ ( collect_list_nat
% 0.35/0.75 @ ^ [L2: list_nat] :
% 0.35/0.75 ( ( ( size_size_list_nat @ L2 )
% 0.35/0.75 = M )
% 0.35/0.75 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.35/0.75 = N6 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % card_length_sum_list_rec
% 0.35/0.75 thf(fact_9781_hd__upt,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_nat @ I @ J )
% 0.35/0.75 => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.35/0.75 = I ) ) ).
% 0.35/0.75
% 0.35/0.75 % hd_upt
% 0.35/0.75 thf(fact_9782_upt__conv__Nil,axiom,
% 0.35/0.75 ! [J: nat,I: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ J @ I )
% 0.35/0.75 => ( ( upt @ I @ J )
% 0.35/0.75 = nil_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_conv_Nil
% 0.35/0.75 thf(fact_9783_upt__eq__Nil__conv,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ( upt @ I @ J )
% 0.35/0.75 = nil_nat )
% 0.35/0.75 = ( ( J = zero_zero_nat )
% 0.35/0.75 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_eq_Nil_conv
% 0.35/0.75 thf(fact_9784_take__upt,axiom,
% 0.35/0.75 ! [I: nat,M: nat,N: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 0.35/0.75 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 0.35/0.75 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % take_upt
% 0.35/0.75 thf(fact_9785_nth__upt,axiom,
% 0.35/0.75 ! [I: nat,K: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.35/0.75 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.35/0.75 = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nth_upt
% 0.35/0.75 thf(fact_9786_upt__rec__numeral,axiom,
% 0.35/0.75 ! [M: num,N: num] :
% 0.35/0.75 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 0.35/0.75 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 0.35/0.75 = nil_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_rec_numeral
% 0.35/0.75 thf(fact_9787_sum__list__upt,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 0.35/0.75 = ( groups3542108847815614940at_nat
% 0.35/0.75 @ ^ [X: nat] : X
% 0.35/0.75 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % sum_list_upt
% 0.35/0.75 thf(fact_9788_map__add__upt,axiom,
% 0.35/0.75 ! [N: nat,M: nat] :
% 0.35/0.75 ( ( map_nat_nat
% 0.35/0.75 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 0.35/0.75 @ ( upt @ zero_zero_nat @ M ) )
% 0.35/0.75 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % map_add_upt
% 0.35/0.75 thf(fact_9789_atLeast__upt,axiom,
% 0.35/0.75 ( set_ord_lessThan_nat
% 0.35/0.75 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % atLeast_upt
% 0.35/0.75 thf(fact_9790_upt__0,axiom,
% 0.35/0.75 ! [I: nat] :
% 0.35/0.75 ( ( upt @ I @ zero_zero_nat )
% 0.35/0.75 = nil_nat ) ).
% 0.35/0.75
% 0.35/0.75 % upt_0
% 0.35/0.75 thf(fact_9791_sorted__wrt__upt,axiom,
% 0.35/0.75 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_wrt_upt
% 0.35/0.75 thf(fact_9792_upt__add__eq__append,axiom,
% 0.35/0.75 ! [I: nat,J: nat,K: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.35/0.75 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_add_eq_append
% 0.35/0.75 thf(fact_9793_sorted__upt,axiom,
% 0.35/0.75 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_upt
% 0.35/0.75 thf(fact_9794_atMost__upto,axiom,
% 0.35/0.75 ( set_ord_atMost_nat
% 0.35/0.75 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % atMost_upto
% 0.35/0.75 thf(fact_9795_upt__rec,axiom,
% 0.35/0.75 ( upt
% 0.35/0.75 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_rec
% 0.35/0.75 thf(fact_9796_upt__conv__Cons,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_nat @ I @ J )
% 0.35/0.75 => ( ( upt @ I @ J )
% 0.35/0.75 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_conv_Cons
% 0.35/0.75 thf(fact_9797_upt__Suc__append,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75 => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_Suc_append
% 0.35/0.75 thf(fact_9798_upt__Suc,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ( ord_less_eq_nat @ I @ J )
% 0.35/0.75 => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.35/0.75 & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.35/0.75 => ( ( upt @ I @ ( suc @ J ) )
% 0.35/0.75 = nil_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_Suc
% 0.35/0.75 thf(fact_9799_map__decr__upt,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( map_nat_nat
% 0.35/0.75 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.35/0.75 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 0.35/0.75 = ( upt @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % map_decr_upt
% 0.35/0.75 thf(fact_9800_upt__eq__Cons__conv,axiom,
% 0.35/0.75 ! [I: nat,J: nat,X2: nat,Xs: list_nat] :
% 0.35/0.75 ( ( ( upt @ I @ J )
% 0.35/0.75 = ( cons_nat @ X2 @ Xs ) )
% 0.35/0.75 = ( ( ord_less_nat @ I @ J )
% 0.35/0.75 & ( I = X2 )
% 0.35/0.75 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.35/0.75 = Xs ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % upt_eq_Cons_conv
% 0.35/0.75 thf(fact_9801_sorted__wrt__less__idx,axiom,
% 0.35/0.75 ! [Ns: list_nat,I: nat] :
% 0.35/0.75 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.35/0.75 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.35/0.75 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_wrt_less_idx
% 0.35/0.75 thf(fact_9802_sorted__wrt__upto,axiom,
% 0.35/0.75 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_wrt_upto
% 0.35/0.75 thf(fact_9803_sorted__upto,axiom,
% 0.35/0.75 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % sorted_upto
% 0.35/0.75 thf(fact_9804_Suc__0__div__numeral,axiom,
% 0.35/0.75 ! [K: num] :
% 0.35/0.75 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Suc_0_div_numeral
% 0.35/0.75 thf(fact_9805_bezw_Oelims,axiom,
% 0.35/0.75 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.35/0.75 ( ( ( bezw @ X2 @ Xa2 )
% 0.35/0.75 = Y3 )
% 0.35/0.75 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.35/0.75 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezw.elims
% 0.35/0.75 thf(fact_9806_bezw_Osimps,axiom,
% 0.35/0.75 ( bezw
% 0.35/0.75 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezw.simps
% 0.35/0.75 thf(fact_9807_bezw__non__0,axiom,
% 0.35/0.75 ! [Y3: nat,X2: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.35/0.75 => ( ( bezw @ X2 @ Y3 )
% 0.35/0.75 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezw_non_0
% 0.35/0.75 thf(fact_9808_bezw_Opelims,axiom,
% 0.35/0.75 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.35/0.75 ( ( ( bezw @ X2 @ Xa2 )
% 0.35/0.75 = Y3 )
% 0.35/0.75 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.35/0.75 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.35/0.75 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
% 0.35/0.75 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezw.pelims
% 0.35/0.75 thf(fact_9809_Suc__0__mod__numeral,axiom,
% 0.35/0.75 ! [K: num] :
% 0.35/0.75 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Suc_0_mod_numeral
% 0.35/0.75 thf(fact_9810_Rat_Opositive_Orep__eq,axiom,
% 0.35/0.75 ( positive
% 0.35/0.75 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat.positive.rep_eq
% 0.35/0.75 thf(fact_9811_normalize__def,axiom,
% 0.35/0.75 ( normalize
% 0.35/0.75 = ( ^ [P5: product_prod_int_int] :
% 0.35/0.75 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
% 0.35/0.75 @ ( if_Pro3027730157355071871nt_int
% 0.35/0.75 @ ( ( product_snd_int_int @ P5 )
% 0.35/0.75 = zero_zero_int )
% 0.35/0.75 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.35/0.75 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % normalize_def
% 0.35/0.75 thf(fact_9812_gcd__pos__int,axiom,
% 0.35/0.75 ! [M: int,N: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 0.35/0.75 = ( ( M != zero_zero_int )
% 0.35/0.75 | ( N != zero_zero_int ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_pos_int
% 0.35/0.75 thf(fact_9813_gcd__ge__0__int,axiom,
% 0.35/0.75 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_ge_0_int
% 0.35/0.75 thf(fact_9814_gcd__le2__int,axiom,
% 0.35/0.75 ! [B3: int,A2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B3 )
% 0.35/0.75 => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ B3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_le2_int
% 0.35/0.75 thf(fact_9815_gcd__le1__int,axiom,
% 0.35/0.75 ! [A2: int,B3: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ A2 )
% 0.35/0.75 => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ A2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_le1_int
% 0.35/0.75 thf(fact_9816_gcd__cases__int,axiom,
% 0.35/0.75 ! [X2: int,Y3: int,P: int > $o] :
% 0.35/0.75 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.75 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.75 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) )
% 0.35/0.75 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.75 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.35/0.75 => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.35/0.75 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.35/0.75 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.75 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 0.35/0.75 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.35/0.75 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.35/0.75 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.35/0.75 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_cases_int
% 0.35/0.75 thf(fact_9817_gcd__unique__int,axiom,
% 0.35/0.75 ! [D: int,A2: int,B3: int] :
% 0.35/0.75 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.35/0.75 & ( dvd_dvd_int @ D @ A2 )
% 0.35/0.75 & ( dvd_dvd_int @ D @ B3 )
% 0.35/0.75 & ! [E3: int] :
% 0.35/0.75 ( ( ( dvd_dvd_int @ E3 @ A2 )
% 0.35/0.75 & ( dvd_dvd_int @ E3 @ B3 ) )
% 0.35/0.75 => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.35/0.75 = ( D
% 0.35/0.75 = ( gcd_gcd_int @ A2 @ B3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_unique_int
% 0.35/0.75 thf(fact_9818_gcd__non__0__int,axiom,
% 0.35/0.75 ! [Y3: int,X2: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 0.35/0.75 => ( ( gcd_gcd_int @ X2 @ Y3 )
% 0.35/0.75 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_non_0_int
% 0.35/0.75 thf(fact_9819_gcd__nat_Oeq__neutr__iff,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat] :
% 0.35/0.75 ( ( ( gcd_gcd_nat @ A2 @ B3 )
% 0.35/0.75 = zero_zero_nat )
% 0.35/0.75 = ( ( A2 = zero_zero_nat )
% 0.35/0.75 & ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.eq_neutr_iff
% 0.35/0.75 thf(fact_9820_gcd__nat_Oleft__neutral,axiom,
% 0.35/0.75 ! [A2: nat] :
% 0.35/0.75 ( ( gcd_gcd_nat @ zero_zero_nat @ A2 )
% 0.35/0.75 = A2 ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.left_neutral
% 0.35/0.75 thf(fact_9821_gcd__nat_Oneutr__eq__iff,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat] :
% 0.35/0.75 ( ( zero_zero_nat
% 0.35/0.75 = ( gcd_gcd_nat @ A2 @ B3 ) )
% 0.35/0.75 = ( ( A2 = zero_zero_nat )
% 0.35/0.75 & ( B3 = zero_zero_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.neutr_eq_iff
% 0.35/0.75 thf(fact_9822_gcd__nat_Oright__neutral,axiom,
% 0.35/0.75 ! [A2: nat] :
% 0.35/0.75 ( ( gcd_gcd_nat @ A2 @ zero_zero_nat )
% 0.35/0.75 = A2 ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.right_neutral
% 0.35/0.75 thf(fact_9823_gcd__0__nat,axiom,
% 0.35/0.75 ! [X2: nat] :
% 0.35/0.75 ( ( gcd_gcd_nat @ X2 @ zero_zero_nat )
% 0.35/0.75 = X2 ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_0_nat
% 0.35/0.75 thf(fact_9824_gcd__0__left__nat,axiom,
% 0.35/0.75 ! [X2: nat] :
% 0.35/0.75 ( ( gcd_gcd_nat @ zero_zero_nat @ X2 )
% 0.35/0.75 = X2 ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_0_left_nat
% 0.35/0.75 thf(fact_9825_gcd__Suc__0,axiom,
% 0.35/0.75 ! [M: nat] :
% 0.35/0.75 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.35/0.75 = ( suc @ zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_Suc_0
% 0.35/0.75 thf(fact_9826_gcd__pos__nat,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 0.35/0.75 = ( ( M != zero_zero_nat )
% 0.35/0.75 | ( N != zero_zero_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_pos_nat
% 0.35/0.75 thf(fact_9827_Gcd__in,axiom,
% 0.35/0.75 ! [A3: set_nat] :
% 0.35/0.75 ( ! [A: nat,B: nat] :
% 0.35/0.75 ( ( member_nat @ A @ A3 )
% 0.35/0.75 => ( ( member_nat @ B @ A3 )
% 0.35/0.75 => ( member_nat @ ( gcd_gcd_nat @ A @ B ) @ A3 ) ) )
% 0.35/0.75 => ( ( A3 != bot_bot_set_nat )
% 0.35/0.75 => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Gcd_in
% 0.35/0.75 thf(fact_9828_gcd__non__0__nat,axiom,
% 0.35/0.75 ! [Y3: nat,X2: nat] :
% 0.35/0.75 ( ( Y3 != zero_zero_nat )
% 0.35/0.75 => ( ( gcd_gcd_nat @ X2 @ Y3 )
% 0.35/0.75 = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_non_0_nat
% 0.35/0.75 thf(fact_9829_gcd__nat_Osimps,axiom,
% 0.35/0.75 ( gcd_gcd_nat
% 0.35/0.75 = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.simps
% 0.35/0.75 thf(fact_9830_gcd__nat_Oelims,axiom,
% 0.35/0.75 ! [X2: nat,Xa2: nat,Y3: nat] :
% 0.35/0.75 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 0.35/0.75 = Y3 )
% 0.35/0.75 => ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.75 => ( Y3 = X2 ) )
% 0.35/0.75 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.elims
% 0.35/0.75 thf(fact_9831_gcd__le2__nat,axiom,
% 0.35/0.75 ! [B3: nat,A2: nat] :
% 0.35/0.75 ( ( B3 != zero_zero_nat )
% 0.35/0.75 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ B3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_le2_nat
% 0.35/0.75 thf(fact_9832_gcd__le1__nat,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat] :
% 0.35/0.75 ( ( A2 != zero_zero_nat )
% 0.35/0.75 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ A2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_le1_nat
% 0.35/0.75 thf(fact_9833_gcd__diff1__nat,axiom,
% 0.35/0.75 ! [N: nat,M: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ N @ M )
% 0.35/0.75 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 0.35/0.75 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_diff1_nat
% 0.35/0.75 thf(fact_9834_gcd__diff2__nat,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 0.35/0.75 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_diff2_nat
% 0.35/0.75 thf(fact_9835_bezout__nat,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat] :
% 0.35/0.75 ( ( A2 != zero_zero_nat )
% 0.35/0.75 => ? [X5: nat,Y4: nat] :
% 0.35/0.75 ( ( times_times_nat @ A2 @ X5 )
% 0.35/0.75 = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezout_nat
% 0.35/0.75 thf(fact_9836_bezout__gcd__nat_H,axiom,
% 0.35/0.75 ! [B3: nat,A2: nat] :
% 0.35/0.75 ? [X5: nat,Y4: nat] :
% 0.35/0.75 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( times_times_nat @ A2 @ X5 ) )
% 0.35/0.75 & ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X5 ) @ ( times_times_nat @ B3 @ Y4 ) )
% 0.35/0.75 = ( gcd_gcd_nat @ A2 @ B3 ) ) )
% 0.35/0.75 | ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y4 ) @ ( times_times_nat @ B3 @ X5 ) )
% 0.35/0.75 & ( ( minus_minus_nat @ ( times_times_nat @ B3 @ X5 ) @ ( times_times_nat @ A2 @ Y4 ) )
% 0.35/0.75 = ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bezout_gcd_nat'
% 0.35/0.75 thf(fact_9837_Gcd__nat__set__eq__fold,axiom,
% 0.35/0.75 ! [Xs: list_nat] :
% 0.35/0.75 ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 0.35/0.75 = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % Gcd_nat_set_eq_fold
% 0.35/0.75 thf(fact_9838_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.35/0.75 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.35/0.75 @ ^ [M2: nat,N2: nat] :
% 0.35/0.75 ( ( dvd_dvd_nat @ M2 @ N2 )
% 0.35/0.75 & ( M2 != N2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.semilattice_neutr_order_axioms
% 0.35/0.75 thf(fact_9839_gcd__is__Max__divisors__nat,axiom,
% 0.35/0.75 ! [N: nat,M: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( ( gcd_gcd_nat @ M @ N )
% 0.35/0.75 = ( lattic8265883725875713057ax_nat
% 0.35/0.75 @ ( collect_nat
% 0.35/0.75 @ ^ [D5: nat] :
% 0.35/0.75 ( ( dvd_dvd_nat @ D5 @ M )
% 0.35/0.75 & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_is_Max_divisors_nat
% 0.35/0.75 thf(fact_9840_gcd__nat_Opelims,axiom,
% 0.35/0.75 ! [X2: nat,Xa2: nat,Y3: nat] :
% 0.35/0.75 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 0.35/0.75 = Y3 )
% 0.35/0.75 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 0.35/0.75 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.35/0.75 => ( Y3 = X2 ) )
% 0.35/0.75 & ( ( Xa2 != zero_zero_nat )
% 0.35/0.75 => ( Y3
% 0.35/0.75 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
% 0.35/0.75 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.pelims
% 0.35/0.75 thf(fact_9841_Field__natLeq__on,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( field_nat
% 0.35/0.75 @ ( collec3392354462482085612at_nat
% 0.35/0.75 @ ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( ( ord_less_nat @ X @ N )
% 0.35/0.75 & ( ord_less_nat @ Y @ N )
% 0.35/0.75 & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 0.35/0.75 = ( collect_nat
% 0.35/0.75 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Field_natLeq_on
% 0.35/0.75 thf(fact_9842_natLess__def,axiom,
% 0.35/0.75 ( bNF_Ca8459412986667044542atLess
% 0.35/0.75 = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % natLess_def
% 0.35/0.75 thf(fact_9843_wf__less,axiom,
% 0.35/0.75 wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % wf_less
% 0.35/0.75 thf(fact_9844_cauchyD,axiom,
% 0.35/0.75 ! [X6: nat > rat,R2: rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.75 => ? [K2: nat] :
% 0.35/0.75 ! [M3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ M3 )
% 0.35/0.75 => ! [N5: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchyD
% 0.35/0.75 thf(fact_9845_cauchy__imp__bounded,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ? [B: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.75 & ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ B ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchy_imp_bounded
% 0.35/0.75 thf(fact_9846_cauchy__def,axiom,
% 0.35/0.75 ( cauchy
% 0.35/0.75 = ( ^ [X8: nat > rat] :
% 0.35/0.75 ! [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 => ? [K3: nat] :
% 0.35/0.75 ! [M2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ M2 )
% 0.35/0.75 => ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchy_def
% 0.35/0.75 thf(fact_9847_cauchyI,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ! [R3: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.35/0.75 => ? [K7: nat] :
% 0.35/0.75 ! [M4: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K7 @ M4 )
% 0.35/0.75 => ! [N3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K7 @ N3 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) ) @ R3 ) ) ) )
% 0.35/0.75 => ( cauchy @ X6 ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchyI
% 0.35/0.75 thf(fact_9848_le__Real,axiom,
% 0.35/0.75 ! [X6: nat > rat,Y7: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ( cauchy @ Y7 )
% 0.35/0.75 => ( ( ord_less_eq_real @ ( real2 @ X6 ) @ ( real2 @ Y7 ) )
% 0.35/0.75 = ( ! [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 => ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % le_Real
% 0.35/0.75 thf(fact_9849_cauchy__not__vanishes,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ~ ( vanishes @ X6 )
% 0.35/0.75 => ? [B: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.75 & ? [K2: nat] :
% 0.35/0.75 ! [N5: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75 => ( ord_less_rat @ B @ ( abs_abs_rat @ ( X6 @ N5 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchy_not_vanishes
% 0.35/0.75 thf(fact_9850_vanishes__mult__bounded,axiom,
% 0.35/0.75 ! [X6: nat > rat,Y7: nat > rat] :
% 0.35/0.75 ( ? [A9: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ A9 )
% 0.35/0.75 & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ A9 ) )
% 0.35/0.75 => ( ( vanishes @ Y7 )
% 0.35/0.75 => ( vanishes
% 0.35/0.75 @ ^ [N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % vanishes_mult_bounded
% 0.35/0.75 thf(fact_9851_vanishes__def,axiom,
% 0.35/0.75 ( vanishes
% 0.35/0.75 = ( ^ [X8: nat > rat] :
% 0.35/0.75 ! [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 => ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % vanishes_def
% 0.35/0.75 thf(fact_9852_vanishesI,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ! [R3: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 0.35/0.75 => ? [K7: nat] :
% 0.35/0.75 ! [N3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K7 @ N3 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ R3 ) ) )
% 0.35/0.75 => ( vanishes @ X6 ) ) ).
% 0.35/0.75
% 0.35/0.75 % vanishesI
% 0.35/0.75 thf(fact_9853_vanishesD,axiom,
% 0.35/0.75 ! [X6: nat > rat,R2: rat] :
% 0.35/0.75 ( ( vanishes @ X6 )
% 0.35/0.75 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.35/0.75 => ? [K2: nat] :
% 0.35/0.75 ! [N5: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ R2 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % vanishesD
% 0.35/0.75 thf(fact_9854_cauchy__not__vanishes__cases,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ~ ( vanishes @ X6 )
% 0.35/0.75 => ? [B: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.35/0.75 & ? [K2: nat] :
% 0.35/0.75 ( ! [N5: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75 => ( ord_less_rat @ B @ ( uminus_uminus_rat @ ( X6 @ N5 ) ) ) )
% 0.35/0.75 | ! [N5: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K2 @ N5 )
% 0.35/0.75 => ( ord_less_rat @ B @ ( X6 @ N5 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % cauchy_not_vanishes_cases
% 0.35/0.75 thf(fact_9855_not__positive__Real,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ( ~ ( positive2 @ ( real2 @ X6 ) ) )
% 0.35/0.75 = ( ! [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 => ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % not_positive_Real
% 0.35/0.75 thf(fact_9856_positive__Real,axiom,
% 0.35/0.75 ! [X6: nat > rat] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ( positive2 @ ( real2 @ X6 ) )
% 0.35/0.75 = ( ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % positive_Real
% 0.35/0.75 thf(fact_9857_less__real__def,axiom,
% 0.35/0.75 ( ord_less_real
% 0.35/0.75 = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_real_def
% 0.35/0.75 thf(fact_9858_Real_Opositive_Orep__eq,axiom,
% 0.35/0.75 ( positive2
% 0.35/0.75 = ( ^ [X: real] :
% 0.35/0.75 ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Real.positive.rep_eq
% 0.35/0.75 thf(fact_9859_mask__nat__positive__iff,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 0.35/0.75 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % mask_nat_positive_iff
% 0.35/0.75 thf(fact_9860_not__mask__negative__int,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 0.35/0.75
% 0.35/0.75 % not_mask_negative_int
% 0.35/0.75 thf(fact_9861_mask__nonnegative__int,axiom,
% 0.35/0.75 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % mask_nonnegative_int
% 0.35/0.75 thf(fact_9862_less__eq__mask,axiom,
% 0.35/0.75 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_mask
% 0.35/0.75 thf(fact_9863_less__mask,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 0.35/0.75 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_mask
% 0.35/0.75 thf(fact_9864_mask__nat__less__exp,axiom,
% 0.35/0.75 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % mask_nat_less_exp
% 0.35/0.75 thf(fact_9865_rat__less__eq__code,axiom,
% 0.35/0.75 ( ord_less_eq_rat
% 0.35/0.75 = ( ^ [P5: rat,Q6: rat] :
% 0.35/0.75 ( produc4947309494688390418_int_o
% 0.35/0.75 @ ^ [A4: int,C5: int] :
% 0.35/0.75 ( produc4947309494688390418_int_o
% 0.35/0.75 @ ^ [B4: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
% 0.35/0.75 @ ( quotient_of @ Q6 ) )
% 0.35/0.75 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % rat_less_eq_code
% 0.35/0.75 thf(fact_9866_quotient__of__denom__pos,axiom,
% 0.35/0.75 ! [R2: rat,P6: int,Q3: int] :
% 0.35/0.75 ( ( ( quotient_of @ R2 )
% 0.35/0.75 = ( product_Pair_int_int @ P6 @ Q3 ) )
% 0.35/0.75 => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % quotient_of_denom_pos
% 0.35/0.75 thf(fact_9867_quotient__of__denom__pos_H,axiom,
% 0.35/0.75 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % quotient_of_denom_pos'
% 0.35/0.75 thf(fact_9868_rat__less__code,axiom,
% 0.35/0.75 ( ord_less_rat
% 0.35/0.75 = ( ^ [P5: rat,Q6: rat] :
% 0.35/0.75 ( produc4947309494688390418_int_o
% 0.35/0.75 @ ^ [A4: int,C5: int] :
% 0.35/0.75 ( produc4947309494688390418_int_o
% 0.35/0.75 @ ^ [B4: int,D5: int] : ( ord_less_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
% 0.35/0.75 @ ( quotient_of @ Q6 ) )
% 0.35/0.75 @ ( quotient_of @ P5 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % rat_less_code
% 0.35/0.75 thf(fact_9869_quotient__of__def,axiom,
% 0.35/0.75 ( quotient_of
% 0.35/0.75 = ( ^ [X: rat] :
% 0.35/0.75 ( the_Pr4378521158711661632nt_int
% 0.35/0.75 @ ^ [Pair: product_prod_int_int] :
% 0.35/0.75 ( ( X
% 0.35/0.75 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 0.35/0.75 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 0.35/0.75 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % quotient_of_def
% 0.35/0.75 thf(fact_9870_of__real__sqrt,axiom,
% 0.35/0.75 ! [X2: real] :
% 0.35/0.75 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.75 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
% 0.35/0.75 = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % of_real_sqrt
% 0.35/0.75 thf(fact_9871_normalize__stable,axiom,
% 0.35/0.75 ! [Q3: int,P6: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 0.35/0.75 => ( ( algebr932160517623751201me_int @ P6 @ Q3 )
% 0.35/0.75 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 0.35/0.75 = ( product_Pair_int_int @ P6 @ Q3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % normalize_stable
% 0.35/0.75 thf(fact_9872_Rat__induct,axiom,
% 0.35/0.75 ! [P: rat > $o,Q3: rat] :
% 0.35/0.75 ( ! [A: int,B: int] :
% 0.35/0.75 ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.75 => ( ( algebr932160517623751201me_int @ A @ B )
% 0.35/0.75 => ( P @ ( fract @ A @ B ) ) ) )
% 0.35/0.75 => ( P @ Q3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat_induct
% 0.35/0.75 thf(fact_9873_Rat__cases,axiom,
% 0.35/0.75 ! [Q3: rat] :
% 0.35/0.75 ~ ! [A: int,B: int] :
% 0.35/0.75 ( ( Q3
% 0.35/0.75 = ( fract @ A @ B ) )
% 0.35/0.75 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.75 => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat_cases
% 0.35/0.75 thf(fact_9874_Rat__cases__nonzero,axiom,
% 0.35/0.75 ! [Q3: rat] :
% 0.35/0.75 ( ! [A: int,B: int] :
% 0.35/0.75 ( ( Q3
% 0.35/0.75 = ( fract @ A @ B ) )
% 0.35/0.75 => ( ( ord_less_int @ zero_zero_int @ B )
% 0.35/0.75 => ( ( A != zero_zero_int )
% 0.35/0.75 => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) )
% 0.35/0.75 => ( Q3 = zero_zero_rat ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat_cases_nonzero
% 0.35/0.75 thf(fact_9875_quotient__of__unique,axiom,
% 0.35/0.75 ! [R2: rat] :
% 0.35/0.75 ? [X5: product_prod_int_int] :
% 0.35/0.75 ( ( R2
% 0.35/0.75 = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
% 0.35/0.75 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
% 0.35/0.75 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
% 0.35/0.75 & ! [Y5: product_prod_int_int] :
% 0.35/0.75 ( ( ( R2
% 0.35/0.75 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.35/0.75 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 0.35/0.75 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 0.35/0.75 => ( Y5 = X5 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % quotient_of_unique
% 0.35/0.75 thf(fact_9876_coprime__Suc__0__right,axiom,
% 0.35/0.75 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % coprime_Suc_0_right
% 0.35/0.75 thf(fact_9877_coprime__Suc__0__left,axiom,
% 0.35/0.75 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 0.35/0.75
% 0.35/0.75 % coprime_Suc_0_left
% 0.35/0.75 thf(fact_9878_coprime__diff__one__left__nat,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % coprime_diff_one_left_nat
% 0.35/0.75 thf(fact_9879_coprime__diff__one__right__nat,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.75 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % coprime_diff_one_right_nat
% 0.35/0.75 thf(fact_9880_Rats__abs__nat__div__natE,axiom,
% 0.35/0.75 ! [X2: real] :
% 0.35/0.75 ( ( member_real @ X2 @ field_5140801741446780682s_real )
% 0.35/0.75 => ~ ! [M4: nat,N3: nat] :
% 0.35/0.75 ( ( N3 != zero_zero_nat )
% 0.35/0.75 => ( ( ( abs_abs_real @ X2 )
% 0.35/0.75 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 0.35/0.75 => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rats_abs_nat_div_natE
% 0.35/0.75 thf(fact_9881_atLeastLessThan__nat__numeral,axiom,
% 0.35/0.75 ! [M: nat,K: num] :
% 0.35/0.75 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.35/0.75 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.35/0.75 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.35/0.75 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = bot_bot_set_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % atLeastLessThan_nat_numeral
% 0.35/0.75 thf(fact_9882_pred__numeral__simps_I1_J,axiom,
% 0.35/0.75 ( ( pred_numeral @ one )
% 0.35/0.75 = zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % pred_numeral_simps(1)
% 0.35/0.75 thf(fact_9883_less__Suc__numeral,axiom,
% 0.35/0.75 ! [N: nat,K: num] :
% 0.35/0.75 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_Suc_numeral
% 0.35/0.75 thf(fact_9884_less__numeral__Suc,axiom,
% 0.35/0.75 ! [K: num,N: nat] :
% 0.35/0.75 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.35/0.75 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_numeral_Suc
% 0.35/0.75 thf(fact_9885_le__numeral__Suc,axiom,
% 0.35/0.75 ! [K: num,N: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 0.35/0.75 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % le_numeral_Suc
% 0.35/0.75 thf(fact_9886_le__Suc__numeral,axiom,
% 0.35/0.75 ! [N: nat,K: num] :
% 0.35/0.75 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % le_Suc_numeral
% 0.35/0.75 thf(fact_9887_lessThan__nat__numeral,axiom,
% 0.35/0.75 ! [K: num] :
% 0.35/0.75 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % lessThan_nat_numeral
% 0.35/0.75 thf(fact_9888_atMost__nat__numeral,axiom,
% 0.35/0.75 ! [K: num] :
% 0.35/0.75 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 0.35/0.75 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % atMost_nat_numeral
% 0.35/0.75 thf(fact_9889_last__upt,axiom,
% 0.35/0.75 ! [I: nat,J: nat] :
% 0.35/0.75 ( ( ord_less_nat @ I @ J )
% 0.35/0.75 => ( ( last_nat @ ( upt @ I @ J ) )
% 0.35/0.75 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % last_upt
% 0.35/0.75 thf(fact_9890_Real_Opositive_Oabs__eq,axiom,
% 0.35/0.75 ! [X2: nat > rat] :
% 0.35/0.75 ( ( realrel @ X2 @ X2 )
% 0.35/0.75 => ( ( positive2 @ ( real2 @ X2 ) )
% 0.35/0.75 = ( ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( X2 @ N2 ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Real.positive.abs_eq
% 0.35/0.75 thf(fact_9891_not__negative__int__iff,axiom,
% 0.35/0.75 ! [K: int] :
% 0.35/0.75 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.35/0.75 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.35/0.75
% 0.35/0.75 % not_negative_int_iff
% 0.35/0.75 thf(fact_9892_not__nonnegative__int__iff,axiom,
% 0.35/0.75 ! [K: int] :
% 0.35/0.75 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.35/0.75 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.35/0.75
% 0.35/0.75 % not_nonnegative_int_iff
% 0.35/0.75 thf(fact_9893_Real_Opositive_Orsp,axiom,
% 0.35/0.75 ( bNF_re728719798268516973at_o_o @ realrel
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [X8: nat > rat] :
% 0.35/0.75 ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
% 0.35/0.75 @ ^ [X8: nat > rat] :
% 0.35/0.75 ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Real.positive.rsp
% 0.35/0.75 thf(fact_9894_less__eq__integer_Orsp,axiom,
% 0.35/0.75 ( bNF_re3403563459893282935_int_o
% 0.35/0.75 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 0.35/0.75 @ ( bNF_re5089333283451836215nt_o_o
% 0.35/0.75 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ord_less_eq_int
% 0.35/0.75 @ ord_less_eq_int ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_integer.rsp
% 0.35/0.75 thf(fact_9895_less__eq__natural_Orsp,axiom,
% 0.35/0.75 ( bNF_re578469030762574527_nat_o
% 0.35/0.75 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.75 @ ( bNF_re4705727531993890431at_o_o
% 0.35/0.75 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ord_less_eq_nat
% 0.35/0.75 @ ord_less_eq_nat ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_natural.rsp
% 0.35/0.75 thf(fact_9896_less__natural_Orsp,axiom,
% 0.35/0.75 ( bNF_re578469030762574527_nat_o
% 0.35/0.75 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.75 @ ( bNF_re4705727531993890431at_o_o
% 0.35/0.75 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ord_less_nat
% 0.35/0.75 @ ord_less_nat ) ).
% 0.35/0.75
% 0.35/0.75 % less_natural.rsp
% 0.35/0.75 thf(fact_9897_less__integer_Orsp,axiom,
% 0.35/0.75 ( bNF_re3403563459893282935_int_o
% 0.35/0.75 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 0.35/0.75 @ ( bNF_re5089333283451836215nt_o_o
% 0.35/0.75 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ord_less_int
% 0.35/0.75 @ ord_less_int ) ).
% 0.35/0.75
% 0.35/0.75 % less_integer.rsp
% 0.35/0.75 thf(fact_9898_Real_Opositive_Otransfer,axiom,
% 0.35/0.75 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [X8: nat > rat] :
% 0.35/0.75 ? [R5: rat] :
% 0.35/0.75 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.75 & ? [K3: nat] :
% 0.35/0.75 ! [N2: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.75 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
% 0.35/0.75 @ positive2 ) ).
% 0.35/0.75
% 0.35/0.75 % Real.positive.transfer
% 0.35/0.75 thf(fact_9899_Rat_Opositive_Otransfer,axiom,
% 0.35/0.75 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.35/0.75 @ positive ) ).
% 0.35/0.75
% 0.35/0.75 % Rat.positive.transfer
% 0.35/0.75 thf(fact_9900_less__eq__int_Otransfer,axiom,
% 0.35/0.75 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.35/0.75 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.35/0.75 @ ord_less_eq_int ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_int.transfer
% 0.35/0.75 thf(fact_9901_zero__int_Otransfer,axiom,
% 0.35/0.75 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 0.35/0.75
% 0.35/0.75 % zero_int.transfer
% 0.35/0.75 thf(fact_9902_int__transfer,axiom,
% 0.35/0.75 ( bNF_re6830278522597306478at_int
% 0.35/0.75 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 0.35/0.75 @ pcr_int
% 0.35/0.75 @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 0.35/0.75 @ semiri1314217659103216013at_int ) ).
% 0.35/0.75
% 0.35/0.75 % int_transfer
% 0.35/0.75 thf(fact_9903_one__int_Otransfer,axiom,
% 0.35/0.75 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 0.35/0.75
% 0.35/0.75 % one_int.transfer
% 0.35/0.75 thf(fact_9904_less__int_Otransfer,axiom,
% 0.35/0.75 ( bNF_re717283939379294677_int_o @ pcr_int
% 0.35/0.75 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.35/0.75 @ ord_less_int ) ).
% 0.35/0.75
% 0.35/0.75 % less_int.transfer
% 0.35/0.75 thf(fact_9905_Rat_Opositive_Orsp,axiom,
% 0.35/0.75 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 0.35/0.75 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 0.35/0.75 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat.positive.rsp
% 0.35/0.75 thf(fact_9906_Rat_Opositive_Oabs__eq,axiom,
% 0.35/0.75 ! [X2: product_prod_int_int] :
% 0.35/0.75 ( ( ratrel @ X2 @ X2 )
% 0.35/0.75 => ( ( positive @ ( abs_Rat @ X2 ) )
% 0.35/0.75 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Rat.positive.abs_eq
% 0.35/0.75 thf(fact_9907_vimage__Suc__insert__0,axiom,
% 0.35/0.75 ! [A3: set_nat] :
% 0.35/0.75 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 0.35/0.75 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % vimage_Suc_insert_0
% 0.35/0.75 thf(fact_9908_finite__vimage__Suc__iff,axiom,
% 0.35/0.75 ! [F2: set_nat] :
% 0.35/0.75 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F2 ) )
% 0.35/0.75 = ( finite_finite_nat @ F2 ) ) ).
% 0.35/0.75
% 0.35/0.75 % finite_vimage_Suc_iff
% 0.35/0.75 thf(fact_9909_pairs__le__eq__Sigma,axiom,
% 0.35/0.75 ! [M: nat] :
% 0.35/0.75 ( ( collec3392354462482085612at_nat
% 0.35/0.75 @ ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 0.35/0.75 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.35/0.75 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % pairs_le_eq_Sigma
% 0.35/0.75 thf(fact_9910_Restr__natLeq,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.35/0.75 @ ( produc457027306803732586at_nat
% 0.35/0.75 @ ( collect_nat
% 0.35/0.75 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
% 0.35/0.75 @ ^ [Uu3: nat] :
% 0.35/0.75 ( collect_nat
% 0.35/0.75 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) )
% 0.35/0.75 = ( collec3392354462482085612at_nat
% 0.35/0.75 @ ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( ( ord_less_nat @ X @ N )
% 0.35/0.75 & ( ord_less_nat @ Y @ N )
% 0.35/0.75 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Restr_natLeq
% 0.35/0.75 thf(fact_9911_natLeq__def,axiom,
% 0.35/0.75 ( bNF_Ca8665028551170535155natLeq
% 0.35/0.75 = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % natLeq_def
% 0.35/0.75 thf(fact_9912_Restr__natLeq2,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.35/0.75 @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.35/0.75 @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 0.35/0.75 = ( collec3392354462482085612at_nat
% 0.35/0.75 @ ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( ( ord_less_nat @ X @ N )
% 0.35/0.75 & ( ord_less_nat @ Y @ N )
% 0.35/0.75 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Restr_natLeq2
% 0.35/0.75 thf(fact_9913_natLeq__underS__less,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 0.35/0.75 = ( collect_nat
% 0.35/0.75 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % natLeq_underS_less
% 0.35/0.75 thf(fact_9914_natLeq__on__wo__rel,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( bNF_We3818239936649020644el_nat
% 0.35/0.75 @ ( collec3392354462482085612at_nat
% 0.35/0.75 @ ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( ( ord_less_nat @ X @ N )
% 0.35/0.75 & ( ord_less_nat @ Y @ N )
% 0.35/0.75 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % natLeq_on_wo_rel
% 0.35/0.75 thf(fact_9915_pred__nat__trancl__eq__le,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.35/0.75 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % pred_nat_trancl_eq_le
% 0.35/0.75 thf(fact_9916_Bseq__monoseq__convergent_H__inc,axiom,
% 0.35/0.75 ! [F: nat > real,M5: nat] :
% 0.35/0.75 ( ( bfun_nat_real
% 0.35/0.75 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 0.35/0.75 @ at_top_nat )
% 0.35/0.75 => ( ! [M4: nat,N3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ M5 @ M4 )
% 0.35/0.75 => ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.75 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N3 ) ) ) )
% 0.35/0.75 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Bseq_monoseq_convergent'_inc
% 0.35/0.75 thf(fact_9917_Bseq__mono__convergent,axiom,
% 0.35/0.75 ! [X6: nat > real] :
% 0.35/0.75 ( ( bfun_nat_real @ X6 @ at_top_nat )
% 0.35/0.75 => ( ! [M4: nat,N3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.75 => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 0.35/0.75 => ( topolo7531315842566124627t_real @ X6 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Bseq_mono_convergent
% 0.35/0.75 thf(fact_9918_convergent__realpow,axiom,
% 0.35/0.75 ! [X2: real] :
% 0.35/0.75 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 0.35/0.75 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 0.35/0.75 => ( topolo7531315842566124627t_real @ ( power_power_real @ X2 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % convergent_realpow
% 0.35/0.75 thf(fact_9919_less__eq,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.35/0.75 = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq
% 0.35/0.75 thf(fact_9920_Bseq__monoseq__convergent_H__dec,axiom,
% 0.35/0.75 ! [F: nat > real,M5: nat] :
% 0.35/0.75 ( ( bfun_nat_real
% 0.35/0.75 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 0.35/0.75 @ at_top_nat )
% 0.35/0.75 => ( ! [M4: nat,N3: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ M5 @ M4 )
% 0.35/0.75 => ( ( ord_less_eq_nat @ M4 @ N3 )
% 0.35/0.75 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ M4 ) ) ) )
% 0.35/0.75 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Bseq_monoseq_convergent'_dec
% 0.35/0.75 thf(fact_9921_pair__lessI2,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.75 => ( ( ord_less_nat @ S2 @ T )
% 0.35/0.75 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_lessI2
% 0.35/0.75 thf(fact_9922_trans__pair__less,axiom,
% 0.35/0.75 trans_4347625901269045472at_nat @ fun_pair_less ).
% 0.35/0.75
% 0.35/0.75 % trans_pair_less
% 0.35/0.75 thf(fact_9923_total__pair__less,axiom,
% 0.35/0.75 ! [A3: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A3 @ fun_pair_less ) ).
% 0.35/0.75
% 0.35/0.75 % total_pair_less
% 0.35/0.75 thf(fact_9924_pair__less__iff1,axiom,
% 0.35/0.75 ! [X2: nat,Y3: nat,Z: nat] :
% 0.35/0.75 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( product_Pair_nat_nat @ X2 @ Z ) ) @ fun_pair_less )
% 0.35/0.75 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_less_iff1
% 0.35/0.75 thf(fact_9925_wf__pair__less,axiom,
% 0.35/0.75 wf_Pro7803398752247294826at_nat @ fun_pair_less ).
% 0.35/0.75
% 0.35/0.75 % wf_pair_less
% 0.35/0.75 thf(fact_9926_pair__lessI1,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 0.35/0.75 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.75 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_lessI1
% 0.35/0.75 thf(fact_9927_gcd__nat_Oordering__top__axioms,axiom,
% 0.35/0.75 ( ordering_top_nat @ dvd_dvd_nat
% 0.35/0.75 @ ^ [M2: nat,N2: nat] :
% 0.35/0.75 ( ( dvd_dvd_nat @ M2 @ N2 )
% 0.35/0.75 & ( M2 != N2 ) )
% 0.35/0.75 @ zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % gcd_nat.ordering_top_axioms
% 0.35/0.75 thf(fact_9928_bot__nat__0_Oordering__top__axioms,axiom,
% 0.35/0.75 ( ordering_top_nat
% 0.35/0.75 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.35/0.75 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.75 @ zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % bot_nat_0.ordering_top_axioms
% 0.35/0.75 thf(fact_9929_pair__leqI2,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 0.35/0.75 ( ( ord_less_eq_nat @ A2 @ B3 )
% 0.35/0.75 => ( ( ord_less_eq_nat @ S2 @ T )
% 0.35/0.75 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_leqI2
% 0.35/0.75 thf(fact_9930_pair__leqI1,axiom,
% 0.35/0.75 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 0.35/0.75 ( ( ord_less_nat @ A2 @ B3 )
% 0.35/0.75 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_leqI1
% 0.35/0.75 thf(fact_9931_pair__leq__def,axiom,
% 0.35/0.75 ( fun_pair_leq
% 0.35/0.75 = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % pair_leq_def
% 0.35/0.75 thf(fact_9932_wmin__insertI,axiom,
% 0.35/0.75 ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ X2 @ XS )
% 0.35/0.75 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
% 0.35/0.75 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % wmin_insertI
% 0.35/0.75 thf(fact_9933_wmax__insertI,axiom,
% 0.35/0.75 ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ Y3 @ YS )
% 0.35/0.75 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
% 0.35/0.75 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % wmax_insertI
% 0.35/0.75 thf(fact_9934_wmin__emptyI,axiom,
% 0.35/0.75 ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 0.35/0.75
% 0.35/0.75 % wmin_emptyI
% 0.35/0.75 thf(fact_9935_wmax__emptyI,axiom,
% 0.35/0.75 ! [X6: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( finite6177210948735845034at_nat @ X6 )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).
% 0.35/0.75
% 0.35/0.75 % wmax_emptyI
% 0.35/0.75 thf(fact_9936_max__weak__def,axiom,
% 0.35/0.75 ( fun_max_weak
% 0.35/0.75 = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % max_weak_def
% 0.35/0.75 thf(fact_9937_min__weak__def,axiom,
% 0.35/0.75 ( fun_min_weak
% 0.35/0.75 = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % min_weak_def
% 0.35/0.75 thf(fact_9938_smin__insertI,axiom,
% 0.35/0.75 ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ X2 @ XS )
% 0.35/0.75 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
% 0.35/0.75 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % smin_insertI
% 0.35/0.75 thf(fact_9939_smax__insertI,axiom,
% 0.35/0.75 ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
% 0.35/0.75 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
% 0.35/0.75 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y7 ) @ fun_max_strict )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % smax_insertI
% 0.35/0.75 thf(fact_9940_smax__emptyI,axiom,
% 0.35/0.75 ! [Y7: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( finite6177210948735845034at_nat @ Y7 )
% 0.35/0.75 => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % smax_emptyI
% 0.35/0.75 thf(fact_9941_smin__emptyI,axiom,
% 0.35/0.75 ! [X6: set_Pr1261947904930325089at_nat] :
% 0.35/0.75 ( ( X6 != bot_bo2099793752762293965at_nat )
% 0.35/0.75 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 0.35/0.75
% 0.35/0.75 % smin_emptyI
% 0.35/0.75 thf(fact_9942_min__strict__def,axiom,
% 0.35/0.75 ( fun_min_strict
% 0.35/0.75 = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).
% 0.35/0.75
% 0.35/0.75 % min_strict_def
% 0.35/0.75 thf(fact_9943_max__strict__def,axiom,
% 0.35/0.75 ( fun_max_strict
% 0.35/0.75 = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).
% 0.35/0.75
% 0.35/0.75 % max_strict_def
% 0.35/0.75 thf(fact_9944_min__rpair__set,axiom,
% 0.35/0.75 fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 0.35/0.75
% 0.35/0.75 % min_rpair_set
% 0.35/0.75 thf(fact_9945_max__rpair__set,axiom,
% 0.35/0.75 fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 0.35/0.75
% 0.35/0.75 % max_rpair_set
% 0.35/0.75 thf(fact_9946_bit__concat__bit__iff,axiom,
% 0.35/0.75 ! [M: nat,K: int,L: int,N: nat] :
% 0.35/0.75 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 0.35/0.75 = ( ( ( ord_less_nat @ N @ M )
% 0.35/0.75 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 0.35/0.75 | ( ( ord_less_eq_nat @ M @ N )
% 0.35/0.75 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % bit_concat_bit_iff
% 0.35/0.75 thf(fact_9947_concat__bit__0,axiom,
% 0.35/0.75 ! [K: int,L: int] :
% 0.35/0.75 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.35/0.75 = L ) ).
% 0.35/0.75
% 0.35/0.75 % concat_bit_0
% 0.35/0.75 thf(fact_9948_concat__bit__nonnegative__iff,axiom,
% 0.35/0.75 ! [N: nat,K: int,L: int] :
% 0.35/0.75 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 0.35/0.75 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 0.35/0.75
% 0.35/0.75 % concat_bit_nonnegative_iff
% 0.35/0.75 thf(fact_9949_concat__bit__negative__iff,axiom,
% 0.35/0.75 ! [N: nat,K: int,L: int] :
% 0.35/0.75 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 0.35/0.75 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.35/0.75
% 0.35/0.75 % concat_bit_negative_iff
% 0.35/0.75 thf(fact_9950_less__eq__int_Orsp,axiom,
% 0.35/0.75 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.35/0.75 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_int.rsp
% 0.35/0.75 thf(fact_9951_zero__int_Orsp,axiom,
% 0.35/0.75 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % zero_int.rsp
% 0.35/0.75 thf(fact_9952_one__int_Orsp,axiom,
% 0.35/0.75 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.35/0.75
% 0.35/0.75 % one_int.rsp
% 0.35/0.75 thf(fact_9953_less__int_Orsp,axiom,
% 0.35/0.75 ( bNF_re4202695980764964119_nat_o @ intrel
% 0.35/0.75 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 0.35/0.75 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 0.35/0.75 @ ( produc8739625826339149834_nat_o
% 0.35/0.75 @ ^ [X: nat,Y: nat] :
% 0.35/0.75 ( produc6081775807080527818_nat_o
% 0.35/0.75 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_int.rsp
% 0.35/0.75 thf(fact_9954_division__segment__int__def,axiom,
% 0.35/0.75 ( euclid3395696857347342551nt_int
% 0.35/0.75 = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % division_segment_int_def
% 0.35/0.75 thf(fact_9955_less__eq__enat__def,axiom,
% 0.35/0.75 ( ord_le2932123472753598470d_enat
% 0.35/0.75 = ( ^ [M2: extended_enat] :
% 0.35/0.75 ( extended_case_enat_o
% 0.35/0.75 @ ^ [N1: nat] :
% 0.35/0.75 ( extended_case_enat_o
% 0.35/0.75 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 0.35/0.75 @ $false
% 0.35/0.75 @ M2 )
% 0.35/0.75 @ $true ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_eq_enat_def
% 0.35/0.75 thf(fact_9956_less__enat__def,axiom,
% 0.35/0.75 ( ord_le72135733267957522d_enat
% 0.35/0.75 = ( ^ [M2: extended_enat,N2: extended_enat] :
% 0.35/0.75 ( extended_case_enat_o
% 0.35/0.75 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 0.35/0.75 @ $false
% 0.35/0.75 @ M2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_enat_def
% 0.35/0.75 thf(fact_9957_of__rat__dense,axiom,
% 0.35/0.75 ! [X2: real,Y3: real] :
% 0.35/0.75 ( ( ord_less_real @ X2 @ Y3 )
% 0.35/0.75 => ? [Q4: rat] :
% 0.35/0.75 ( ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ Q4 ) )
% 0.35/0.75 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q4 ) @ Y3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % of_rat_dense
% 0.35/0.75 thf(fact_9958_less__RealD,axiom,
% 0.35/0.75 ! [Y7: nat > rat,X2: real] :
% 0.35/0.75 ( ( cauchy @ Y7 )
% 0.35/0.75 => ( ( ord_less_real @ X2 @ ( real2 @ Y7 ) )
% 0.35/0.75 => ? [N3: nat] : ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_RealD
% 0.35/0.75 thf(fact_9959_Real__leI,axiom,
% 0.35/0.75 ! [X6: nat > rat,Y3: real] :
% 0.35/0.75 ( ( cauchy @ X6 )
% 0.35/0.75 => ( ! [N3: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X6 @ N3 ) ) @ Y3 )
% 0.35/0.75 => ( ord_less_eq_real @ ( real2 @ X6 ) @ Y3 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % Real_leI
% 0.35/0.75 thf(fact_9960_le__RealI,axiom,
% 0.35/0.75 ! [Y7: nat > rat,X2: real] :
% 0.35/0.75 ( ( cauchy @ Y7 )
% 0.35/0.75 => ( ! [N3: nat] : ( ord_less_eq_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) )
% 0.35/0.75 => ( ord_less_eq_real @ X2 @ ( real2 @ Y7 ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % le_RealI
% 0.35/0.75 thf(fact_9961_compute__powr__real,axiom,
% 0.35/0.75 ( powr_real2
% 0.35/0.75 = ( ^ [B4: real,I4: real] :
% 0.35/0.75 ( if_real @ ( ord_less_eq_real @ B4 @ zero_zero_real )
% 0.35/0.75 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.75 @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) )
% 0.35/0.75 @ ( if_real
% 0.35/0.75 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
% 0.35/0.75 = I4 )
% 0.35/0.75 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
% 0.35/0.75 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.35/0.75 @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % compute_powr_real
% 0.35/0.75 thf(fact_9962_UNIV__char__of__nat,axiom,
% 0.35/0.75 ( top_top_set_char
% 0.35/0.75 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % UNIV_char_of_nat
% 0.35/0.75 thf(fact_9963_inj__on__char__of__nat,axiom,
% 0.35/0.75 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % inj_on_char_of_nat
% 0.35/0.75 thf(fact_9964_range__nat__of__char,axiom,
% 0.35/0.75 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 0.35/0.75 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % range_nat_of_char
% 0.35/0.75 thf(fact_9965_nat__of__char__less__256,axiom,
% 0.35/0.75 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % nat_of_char_less_256
% 0.35/0.75 thf(fact_9966_numeral__le__enat__iff,axiom,
% 0.35/0.75 ! [M: num,N: nat] :
% 0.35/0.75 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.35/0.75 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % numeral_le_enat_iff
% 0.35/0.75 thf(fact_9967_enat__ord__simps_I2_J,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.35/0.75 = ( ord_less_nat @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ord_simps(2)
% 0.35/0.75 thf(fact_9968_enat__ord__simps_I1_J,axiom,
% 0.35/0.75 ! [M: nat,N: nat] :
% 0.35/0.75 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 0.35/0.75 = ( ord_less_eq_nat @ M @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ord_simps(1)
% 0.35/0.75 thf(fact_9969_idiff__enat__0__right,axiom,
% 0.35/0.75 ! [N: extended_enat] :
% 0.35/0.75 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 0.35/0.75 = N ) ).
% 0.35/0.75
% 0.35/0.75 % idiff_enat_0_right
% 0.35/0.75 thf(fact_9970_idiff__enat__0,axiom,
% 0.35/0.75 ! [N: extended_enat] :
% 0.35/0.75 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 0.35/0.75 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % idiff_enat_0
% 0.35/0.75 thf(fact_9971_numeral__less__enat__iff,axiom,
% 0.35/0.75 ! [M: num,N: nat] :
% 0.35/0.75 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 0.35/0.75 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % numeral_less_enat_iff
% 0.35/0.75 thf(fact_9972_zero__enat__def,axiom,
% 0.35/0.75 ( zero_z5237406670263579293d_enat
% 0.35/0.75 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % zero_enat_def
% 0.35/0.75 thf(fact_9973_enat__0__iff_I1_J,axiom,
% 0.35/0.75 ! [X2: nat] :
% 0.35/0.75 ( ( ( extended_enat2 @ X2 )
% 0.35/0.75 = zero_z5237406670263579293d_enat )
% 0.35/0.75 = ( X2 = zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_0_iff(1)
% 0.35/0.75 thf(fact_9974_enat__0__iff_I2_J,axiom,
% 0.35/0.75 ! [X2: nat] :
% 0.35/0.75 ( ( zero_z5237406670263579293d_enat
% 0.35/0.75 = ( extended_enat2 @ X2 ) )
% 0.35/0.75 = ( X2 = zero_zero_nat ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_0_iff(2)
% 0.35/0.75 thf(fact_9975_finite__enat__bounded,axiom,
% 0.35/0.75 ! [A3: set_Extended_enat,N: nat] :
% 0.35/0.75 ( ! [Y4: extended_enat] :
% 0.35/0.75 ( ( member_Extended_enat @ Y4 @ A3 )
% 0.35/0.75 => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
% 0.35/0.75 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 0.35/0.75
% 0.35/0.75 % finite_enat_bounded
% 0.35/0.75 thf(fact_9976_enat__ile,axiom,
% 0.35/0.75 ! [N: extended_enat,M: nat] :
% 0.35/0.75 ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
% 0.35/0.75 => ? [K2: nat] :
% 0.35/0.75 ( N
% 0.35/0.75 = ( extended_enat2 @ K2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ile
% 0.35/0.75 thf(fact_9977_enat__iless,axiom,
% 0.35/0.75 ! [N: extended_enat,M: nat] :
% 0.35/0.75 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 0.35/0.75 => ? [K2: nat] :
% 0.35/0.75 ( N
% 0.35/0.75 = ( extended_enat2 @ K2 ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_iless
% 0.35/0.75 thf(fact_9978_less__enatE,axiom,
% 0.35/0.75 ! [N: extended_enat,M: nat] :
% 0.35/0.75 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 0.35/0.75 => ~ ! [K2: nat] :
% 0.35/0.75 ( ( N
% 0.35/0.75 = ( extended_enat2 @ K2 ) )
% 0.35/0.75 => ~ ( ord_less_nat @ K2 @ M ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % less_enatE
% 0.35/0.75 thf(fact_9979_Suc__ile__eq,axiom,
% 0.35/0.75 ! [M: nat,N: extended_enat] :
% 0.35/0.75 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 0.35/0.75 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.35/0.75
% 0.35/0.75 % Suc_ile_eq
% 0.35/0.75 thf(fact_9980_iadd__le__enat__iff,axiom,
% 0.35/0.75 ! [X2: extended_enat,Y3: extended_enat,N: nat] :
% 0.35/0.75 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ ( extended_enat2 @ N ) )
% 0.35/0.75 = ( ? [Y8: nat,X9: nat] :
% 0.35/0.75 ( ( X2
% 0.35/0.75 = ( extended_enat2 @ X9 ) )
% 0.35/0.75 & ( Y3
% 0.35/0.75 = ( extended_enat2 @ Y8 ) )
% 0.35/0.75 & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % iadd_le_enat_iff
% 0.35/0.75 thf(fact_9981_elimnum,axiom,
% 0.35/0.75 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.35/0.75 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.75 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.35/0.75 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % elimnum
% 0.35/0.75 thf(fact_9982_times__enat__simps_I3_J,axiom,
% 0.35/0.75 ! [N: nat] :
% 0.35/0.75 ( ( ( N = zero_zero_nat )
% 0.35/0.75 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.35/0.75 = zero_z5237406670263579293d_enat ) )
% 0.35/0.75 & ( ( N != zero_zero_nat )
% 0.35/0.75 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 0.35/0.75 = extend5688581933313929465d_enat ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % times_enat_simps(3)
% 0.35/0.75 thf(fact_9983_elimcomplete,axiom,
% 0.35/0.75 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 0.35/0.75 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 0.35/0.75 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 0.35/0.75 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% 0.35/0.75
% 0.35/0.75 % elimcomplete
% 0.35/0.75 thf(fact_9984_enat__ord__simps_I4_J,axiom,
% 0.35/0.75 ! [Q3: extended_enat] :
% 0.35/0.75 ( ( ord_le72135733267957522d_enat @ Q3 @ extend5688581933313929465d_enat )
% 0.35/0.75 = ( Q3 != extend5688581933313929465d_enat ) ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ord_simps(4)
% 0.35/0.75 thf(fact_9985_enat__ord__simps_I6_J,axiom,
% 0.35/0.75 ! [Q3: extended_enat] :
% 0.35/0.75 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q3 ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ord_simps(6)
% 0.35/0.75 thf(fact_9986_enat__ord__code_I3_J,axiom,
% 0.35/0.75 ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).
% 0.35/0.75
% 0.35/0.75 % enat_ord_code(3)
% 0.35/0.76 thf(fact_9987_enat__ord__simps_I5_J,axiom,
% 0.35/0.76 ! [Q3: extended_enat] :
% 0.35/0.76 ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q3 )
% 0.35/0.76 = ( Q3 = extend5688581933313929465d_enat ) ) ).
% 0.35/0.76
% 0.35/0.76 % enat_ord_simps(5)
% 0.35/0.76 thf(fact_9988_times__enat__simps_I4_J,axiom,
% 0.35/0.76 ! [M: nat] :
% 0.35/0.76 ( ( ( M = zero_zero_nat )
% 0.35/0.76 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.35/0.76 = zero_z5237406670263579293d_enat ) )
% 0.35/0.76 & ( ( M != zero_zero_nat )
% 0.35/0.76 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 0.35/0.76 = extend5688581933313929465d_enat ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % times_enat_simps(4)
% 0.35/0.76 thf(fact_9989_enat__ord__code_I5_J,axiom,
% 0.35/0.76 ! [N: nat] :
% 0.35/0.76 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % enat_ord_code(5)
% 0.35/0.76 thf(fact_9990_infinity__ileE,axiom,
% 0.35/0.76 ! [M: nat] :
% 0.35/0.76 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % infinity_ileE
% 0.35/0.76 thf(fact_9991_enat__add__left__cancel__less,axiom,
% 0.35/0.76 ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
% 0.35/0.76 = ( ( A2 != extend5688581933313929465d_enat )
% 0.35/0.76 & ( ord_le72135733267957522d_enat @ B3 @ C ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % enat_add_left_cancel_less
% 0.35/0.76 thf(fact_9992_enat__ord__code_I4_J,axiom,
% 0.35/0.76 ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).
% 0.35/0.76
% 0.35/0.76 % enat_ord_code(4)
% 0.35/0.76 thf(fact_9993_less__infinityE,axiom,
% 0.35/0.76 ! [N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
% 0.35/0.76 => ~ ! [K2: nat] :
% 0.35/0.76 ( N
% 0.35/0.76 != ( extended_enat2 @ K2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % less_infinityE
% 0.35/0.76 thf(fact_9994_infinity__ilessE,axiom,
% 0.35/0.76 ! [M: nat] :
% 0.35/0.76 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % infinity_ilessE
% 0.35/0.76 thf(fact_9995_enat__add__left__cancel__le,axiom,
% 0.35/0.76 ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
% 0.35/0.76 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
% 0.35/0.76 = ( ( A2 = extend5688581933313929465d_enat )
% 0.35/0.76 | ( ord_le2932123472753598470d_enat @ B3 @ C ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % enat_add_left_cancel_le
% 0.35/0.76 thf(fact_9996_enat__ord__simps_I3_J,axiom,
% 0.35/0.76 ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).
% 0.35/0.76
% 0.35/0.76 % enat_ord_simps(3)
% 0.35/0.76 thf(fact_9997_Inf__enat__def,axiom,
% 0.35/0.76 ( comple2295165028678016749d_enat
% 0.35/0.76 = ( ^ [A6: set_Extended_enat] :
% 0.35/0.76 ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
% 0.35/0.76 @ ( ord_Le1955565732374568822d_enat
% 0.35/0.76 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Inf_enat_def
% 0.35/0.76 thf(fact_9998_Sup__enat__def,axiom,
% 0.35/0.76 ( comple4398354569131411667d_enat
% 0.35/0.76 = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Sup_enat_def
% 0.35/0.76 thf(fact_9999_imult__infinity,axiom,
% 0.35/0.76 ! [N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.35/0.76 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 0.35/0.76 = extend5688581933313929465d_enat ) ) ).
% 0.35/0.76
% 0.35/0.76 % imult_infinity
% 0.35/0.76 thf(fact_10000_imult__infinity__right,axiom,
% 0.35/0.76 ! [N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 0.35/0.76 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 0.35/0.76 = extend5688581933313929465d_enat ) ) ).
% 0.35/0.76
% 0.35/0.76 % imult_infinity_right
% 0.35/0.76 thf(fact_10001_times__enat__def,axiom,
% 0.35/0.76 ( times_7803423173614009249d_enat
% 0.35/0.76 = ( ^ [M2: extended_enat,N2: extended_enat] :
% 0.35/0.76 ( extend3600170679010898289d_enat
% 0.35/0.76 @ ^ [O: nat] :
% 0.35/0.76 ( extend3600170679010898289d_enat
% 0.35/0.76 @ ^ [P5: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P5 ) )
% 0.35/0.76 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.35/0.76 @ N2 )
% 0.35/0.76 @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 0.35/0.76 @ M2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % times_enat_def
% 0.35/0.76 thf(fact_10002_iless__Suc__eq,axiom,
% 0.35/0.76 ! [M: nat,N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 0.35/0.76 = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % iless_Suc_eq
% 0.35/0.76 thf(fact_10003_eSuc__mono,axiom,
% 0.35/0.76 ! [N: extended_enat,M: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.35/0.76 = ( ord_le72135733267957522d_enat @ N @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % eSuc_mono
% 0.35/0.76 thf(fact_10004_eSuc__ile__mono,axiom,
% 0.35/0.76 ! [N: extended_enat,M: extended_enat] :
% 0.35/0.76 ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 0.35/0.76 = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % eSuc_ile_mono
% 0.35/0.76 thf(fact_10005_iless__eSuc0,axiom,
% 0.35/0.76 ! [N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
% 0.35/0.76 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 0.35/0.76
% 0.35/0.76 % iless_eSuc0
% 0.35/0.76 thf(fact_10006_eSuc__Max,axiom,
% 0.35/0.76 ! [A3: set_Extended_enat] :
% 0.35/0.76 ( ( finite4001608067531595151d_enat @ A3 )
% 0.35/0.76 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.76 => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
% 0.35/0.76 = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % eSuc_Max
% 0.35/0.76 thf(fact_10007_eSuc__Sup,axiom,
% 0.35/0.76 ! [A3: set_Extended_enat] :
% 0.35/0.76 ( ( A3 != bot_bo7653980558646680370d_enat )
% 0.35/0.76 => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
% 0.35/0.76 = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % eSuc_Sup
% 0.35/0.76 thf(fact_10008_not__eSuc__ilei0,axiom,
% 0.35/0.76 ! [N: extended_enat] :
% 0.35/0.76 ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).
% 0.35/0.76
% 0.35/0.76 % not_eSuc_ilei0
% 0.35/0.76 thf(fact_10009_ile__eSuc,axiom,
% 0.35/0.76 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % ile_eSuc
% 0.35/0.76 thf(fact_10010_ileI1,axiom,
% 0.35/0.76 ! [M: extended_enat,N: extended_enat] :
% 0.35/0.76 ( ( ord_le72135733267957522d_enat @ M @ N )
% 0.35/0.76 => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % ileI1
% 0.35/0.76 thf(fact_10011_i0__iless__eSuc,axiom,
% 0.35/0.76 ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % i0_iless_eSuc
% 0.35/0.76 thf(fact_10012_less__than__iff,axiom,
% 0.35/0.76 ! [X2: nat,Y3: nat] :
% 0.35/0.76 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ less_than )
% 0.35/0.76 = ( ord_less_nat @ X2 @ Y3 ) ) ).
% 0.35/0.76
% 0.35/0.76 % less_than_iff
% 0.35/0.76 thf(fact_10013_pair__less__def,axiom,
% 0.35/0.76 ( fun_pair_less
% 0.35/0.76 = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).
% 0.35/0.76
% 0.35/0.76 % pair_less_def
% 0.35/0.76 thf(fact_10014_natLeq__on__well__order__on,axiom,
% 0.35/0.76 ! [N: nat] :
% 0.35/0.76 ( order_2888998067076097458on_nat
% 0.35/0.76 @ ( collect_nat
% 0.35/0.76 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
% 0.35/0.76 @ ( collec3392354462482085612at_nat
% 0.35/0.76 @ ( produc6081775807080527818_nat_o
% 0.35/0.76 @ ^ [X: nat,Y: nat] :
% 0.35/0.76 ( ( ord_less_nat @ X @ N )
% 0.35/0.76 & ( ord_less_nat @ Y @ N )
% 0.35/0.76 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % natLeq_on_well_order_on
% 0.35/0.76 thf(fact_10015_natLeq__on__Well__order,axiom,
% 0.35/0.76 ! [N: nat] :
% 0.35/0.76 ( order_2888998067076097458on_nat
% 0.35/0.76 @ ( field_nat
% 0.35/0.76 @ ( collec3392354462482085612at_nat
% 0.35/0.76 @ ( produc6081775807080527818_nat_o
% 0.35/0.76 @ ^ [X: nat,Y: nat] :
% 0.35/0.76 ( ( ord_less_nat @ X @ N )
% 0.35/0.76 & ( ord_less_nat @ Y @ N )
% 0.35/0.76 & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 0.35/0.76 @ ( collec3392354462482085612at_nat
% 0.35/0.76 @ ( produc6081775807080527818_nat_o
% 0.35/0.76 @ ^ [X: nat,Y: nat] :
% 0.35/0.76 ( ( ord_less_nat @ X @ N )
% 0.35/0.76 & ( ord_less_nat @ Y @ N )
% 0.35/0.76 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % natLeq_on_Well_order
% 0.35/0.76 thf(fact_10016_nat_Odisc__eq__case_I2_J,axiom,
% 0.35/0.76 ! [Nat: nat] :
% 0.35/0.76 ( ( Nat != zero_zero_nat )
% 0.35/0.76 = ( case_nat_o @ $false
% 0.35/0.76 @ ^ [Uu3: nat] : $true
% 0.35/0.76 @ Nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % nat.disc_eq_case(2)
% 0.35/0.76 thf(fact_10017_nat_Odisc__eq__case_I1_J,axiom,
% 0.35/0.76 ! [Nat: nat] :
% 0.35/0.76 ( ( Nat = zero_zero_nat )
% 0.35/0.76 = ( case_nat_o @ $true
% 0.35/0.76 @ ^ [Uu3: nat] : $false
% 0.35/0.76 @ Nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % nat.disc_eq_case(1)
% 0.35/0.76 thf(fact_10018_less__eq__nat_Osimps_I2_J,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 0.35/0.76 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 0.35/0.76
% 0.35/0.76 % less_eq_nat.simps(2)
% 0.35/0.76 thf(fact_10019_max__Suc1,axiom,
% 0.35/0.76 ! [N: nat,M: nat] :
% 0.35/0.76 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 0.35/0.76 = ( case_nat_nat @ ( suc @ N )
% 0.35/0.76 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 0.35/0.76 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % max_Suc1
% 0.35/0.76 thf(fact_10020_max__Suc2,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 0.35/0.76 = ( case_nat_nat @ ( suc @ N )
% 0.35/0.76 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 0.35/0.76 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % max_Suc2
% 0.35/0.76 thf(fact_10021_diff__Suc,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 0.35/0.76 = ( case_nat_nat @ zero_zero_nat
% 0.35/0.76 @ ^ [K3: nat] : K3
% 0.35/0.76 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % diff_Suc
% 0.35/0.76 thf(fact_10022_min__Suc2,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 0.35/0.76 = ( case_nat_nat @ zero_zero_nat
% 0.35/0.76 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 0.35/0.76 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % min_Suc2
% 0.35/0.76 thf(fact_10023_min__Suc1,axiom,
% 0.35/0.76 ! [N: nat,M: nat] :
% 0.35/0.76 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 0.35/0.76 = ( case_nat_nat @ zero_zero_nat
% 0.35/0.76 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 0.35/0.76 @ M ) ) ).
% 0.35/0.76
% 0.35/0.76 % min_Suc1
% 0.35/0.76 thf(fact_10024_pred__def,axiom,
% 0.35/0.76 ( pred
% 0.35/0.76 = ( case_nat_nat @ zero_zero_nat
% 0.35/0.76 @ ^ [X24: nat] : X24 ) ) ).
% 0.35/0.76
% 0.35/0.76 % pred_def
% 0.35/0.76 thf(fact_10025_UNIV__bool,axiom,
% 0.35/0.76 ( top_top_set_o
% 0.35/0.76 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % UNIV_bool
% 0.35/0.76 thf(fact_10026_Rep__unit__induct,axiom,
% 0.35/0.76 ! [Y3: $o,P: $o > $o] :
% 0.35/0.76 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
% 0.35/0.76 => ( P @ Y3 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Rep_unit_induct
% 0.35/0.76 thf(fact_10027_Abs__unit__inject,axiom,
% 0.35/0.76 ! [X2: $o,Y3: $o] :
% 0.35/0.76 ( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ( ( ( product_Abs_unit @ X2 )
% 0.35/0.76 = ( product_Abs_unit @ Y3 ) )
% 0.35/0.76 = ( X2 = Y3 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Abs_unit_inject
% 0.35/0.76 thf(fact_10028_Abs__unit__inverse,axiom,
% 0.35/0.76 ! [Y3: $o] :
% 0.35/0.76 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
% 0.35/0.76 = Y3 ) ) ).
% 0.35/0.76
% 0.35/0.76 % Abs_unit_inverse
% 0.35/0.76 thf(fact_10029_Rep__unit,axiom,
% 0.35/0.76 ! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% 0.35/0.76
% 0.35/0.76 % Rep_unit
% 0.35/0.76 thf(fact_10030_Abs__unit__cases,axiom,
% 0.35/0.76 ! [X2: product_unit] :
% 0.35/0.76 ~ ! [Y4: $o] :
% 0.35/0.76 ( ( X2
% 0.35/0.76 = ( product_Abs_unit @ Y4 ) )
% 0.35/0.76 => ~ ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Abs_unit_cases
% 0.35/0.76 thf(fact_10031_Rep__unit__cases,axiom,
% 0.35/0.76 ! [Y3: $o] :
% 0.35/0.76 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ~ ! [X5: product_unit] :
% 0.35/0.76 ( Y3
% 0.35/0.76 = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Rep_unit_cases
% 0.35/0.76 thf(fact_10032_Abs__unit__induct,axiom,
% 0.35/0.76 ! [P: product_unit > $o,X2: product_unit] :
% 0.35/0.76 ( ! [Y4: $o] :
% 0.35/0.76 ( ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 0.35/0.76 => ( P @ ( product_Abs_unit @ Y4 ) ) )
% 0.35/0.76 => ( P @ X2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % Abs_unit_induct
% 0.35/0.76 thf(fact_10033_type__definition__unit,axiom,
% 0.35/0.76 type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% 0.35/0.76
% 0.35/0.76 % type_definition_unit
% 0.35/0.76 thf(fact_10034_Real_Opositive__def,axiom,
% 0.35/0.76 ( positive2
% 0.35/0.76 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 0.35/0.76 @ ^ [X8: nat > rat] :
% 0.35/0.76 ? [R5: rat] :
% 0.35/0.76 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 0.35/0.76 & ? [K3: nat] :
% 0.35/0.76 ! [N2: nat] :
% 0.35/0.76 ( ( ord_less_eq_nat @ K3 @ N2 )
% 0.35/0.76 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Real.positive_def
% 0.35/0.76 thf(fact_10035_Rat_Opositive__def,axiom,
% 0.35/0.76 ( positive
% 0.35/0.76 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.35/0.76 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Rat.positive_def
% 0.35/0.76 thf(fact_10036_cmod__plus__Re__le__0__iff,axiom,
% 0.35/0.76 ! [Z: complex] :
% 0.35/0.76 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 0.35/0.76 = ( ( re @ Z )
% 0.35/0.76 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % cmod_plus_Re_le_0_iff
% 0.35/0.76 thf(fact_10037_complex__Re__le__cmod,axiom,
% 0.35/0.76 ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % complex_Re_le_cmod
% 0.35/0.76 thf(fact_10038_abs__Re__le__cmod,axiom,
% 0.35/0.76 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % abs_Re_le_cmod
% 0.35/0.76 thf(fact_10039_Re__csqrt,axiom,
% 0.35/0.76 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Re_csqrt
% 0.35/0.76 thf(fact_10040_complex__abs__le__norm,axiom,
% 0.35/0.76 ! [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 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % complex_abs_le_norm
% 0.35/0.76 thf(fact_10041_csqrt__unique,axiom,
% 0.35/0.76 ! [W2: complex,Z: complex] :
% 0.35/0.76 ( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.35/0.76 = Z )
% 0.35/0.76 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
% 0.35/0.76 | ( ( ( re @ W2 )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
% 0.35/0.76 => ( ( csqrt @ Z )
% 0.35/0.76 = W2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_unique
% 0.35/0.76 thf(fact_10042_csqrt__of__real__nonneg,axiom,
% 0.35/0.76 ! [X2: complex] :
% 0.35/0.76 ( ( ( im @ X2 )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
% 0.35/0.76 => ( ( csqrt @ X2 )
% 0.35/0.76 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_of_real_nonneg
% 0.35/0.76 thf(fact_10043_abs__Im__le__cmod,axiom,
% 0.35/0.76 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % abs_Im_le_cmod
% 0.35/0.76 thf(fact_10044_cmod__Im__le__iff,axiom,
% 0.35/0.76 ! [X2: complex,Y3: complex] :
% 0.35/0.76 ( ( ( re @ X2 )
% 0.35/0.76 = ( re @ Y3 ) )
% 0.35/0.76 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.35/0.76 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % cmod_Im_le_iff
% 0.35/0.76 thf(fact_10045_cmod__Re__le__iff,axiom,
% 0.35/0.76 ! [X2: complex,Y3: complex] :
% 0.35/0.76 ( ( ( im @ X2 )
% 0.35/0.76 = ( im @ Y3 ) )
% 0.35/0.76 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 0.35/0.76 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % cmod_Re_le_iff
% 0.35/0.76 thf(fact_10046_csqrt__principal,axiom,
% 0.35/0.76 ! [Z: complex] :
% 0.35/0.76 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 0.35/0.76 | ( ( ( re @ ( csqrt @ Z ) )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_principal
% 0.35/0.76 thf(fact_10047_cmod__le,axiom,
% 0.35/0.76 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % cmod_le
% 0.35/0.76 thf(fact_10048_complex__neq__0,axiom,
% 0.35/0.76 ! [Z: complex] :
% 0.35/0.76 ( ( Z != zero_zero_complex )
% 0.35/0.76 = ( 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 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % complex_neq_0
% 0.35/0.76 thf(fact_10049_csqrt__square,axiom,
% 0.35/0.76 ! [B3: complex] :
% 0.35/0.76 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B3 ) )
% 0.35/0.76 | ( ( ( re @ B3 )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B3 ) ) ) )
% 0.35/0.76 => ( ( csqrt @ ( power_power_complex @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.35/0.76 = B3 ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_square
% 0.35/0.76 thf(fact_10050_csqrt__of__real__nonpos,axiom,
% 0.35/0.76 ! [X2: complex] :
% 0.35/0.76 ( ( ( im @ X2 )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
% 0.35/0.76 => ( ( csqrt @ X2 )
% 0.35/0.76 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_of_real_nonpos
% 0.35/0.76 thf(fact_10051_csqrt__minus,axiom,
% 0.35/0.76 ! [X2: complex] :
% 0.35/0.76 ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
% 0.35/0.76 | ( ( ( im @ X2 )
% 0.35/0.76 = zero_zero_real )
% 0.35/0.76 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
% 0.35/0.76 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
% 0.35/0.76 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % csqrt_minus
% 0.35/0.76 thf(fact_10052_complex__div__gt__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) )
% 0.35/0.76 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % complex_div_gt_0
% 0.35/0.76 thf(fact_10053_Re__complex__div__gt__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Re_complex_div_gt_0
% 0.35/0.76 thf(fact_10054_Re__complex__div__lt__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 0.35/0.76 = ( ord_less_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.76
% 0.35/0.76 % Re_complex_div_lt_0
% 0.35/0.76 thf(fact_10055_Re__complex__div__le__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 0.35/0.76 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.76
% 0.35/0.76 % Re_complex_div_le_0
% 0.35/0.76 thf(fact_10056_Re__complex__div__ge__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Re_complex_div_ge_0
% 0.35/0.76 thf(fact_10057_Im__complex__div__gt__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Im_complex_div_gt_0
% 0.35/0.76 thf(fact_10058_Im__complex__div__lt__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 0.35/0.76 = ( ord_less_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.76
% 0.35/0.76 % Im_complex_div_lt_0
% 0.35/0.76 thf(fact_10059_Im__complex__div__le__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 0.35/0.76 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 0.35/0.76
% 0.35/0.76 % Im_complex_div_le_0
% 0.35/0.76 thf(fact_10060_Im__complex__div__ge__0,axiom,
% 0.35/0.76 ! [A2: complex,B3: complex] :
% 0.35/0.76 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 0.35/0.76 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Im_complex_div_ge_0
% 0.35/0.76 thf(fact_10061_less__eq__int__def,axiom,
% 0.35/0.76 ( ord_less_eq_int
% 0.35/0.76 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.35/0.76 @ ( produc8739625826339149834_nat_o
% 0.35/0.76 @ ^ [X: nat,Y: nat] :
% 0.35/0.76 ( produc6081775807080527818_nat_o
% 0.35/0.76 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % less_eq_int_def
% 0.35/0.76 thf(fact_10062_less__int__def,axiom,
% 0.35/0.76 ( ord_less_int
% 0.35/0.76 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.35/0.76 @ ( produc8739625826339149834_nat_o
% 0.35/0.76 @ ^ [X: nat,Y: nat] :
% 0.35/0.76 ( produc6081775807080527818_nat_o
% 0.35/0.76 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % less_int_def
% 0.35/0.76 thf(fact_10063_MOST__SucD,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( eventually_nat
% 0.35/0.76 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 0.35/0.76 @ cofinite_nat )
% 0.35/0.76 => ( eventually_nat @ P @ cofinite_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_SucD
% 0.35/0.76 thf(fact_10064_MOST__SucI,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( eventually_nat @ P @ cofinite_nat )
% 0.35/0.76 => ( eventually_nat
% 0.35/0.76 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 0.35/0.76 @ cofinite_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_SucI
% 0.35/0.76 thf(fact_10065_MOST__Suc__iff,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( eventually_nat
% 0.35/0.76 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 0.35/0.76 @ cofinite_nat )
% 0.35/0.76 = ( eventually_nat @ P @ cofinite_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_Suc_iff
% 0.35/0.76 thf(fact_10066_MOST__nat__le,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( eventually_nat @ P @ cofinite_nat )
% 0.35/0.76 = ( ? [M2: nat] :
% 0.35/0.76 ! [N2: nat] :
% 0.35/0.76 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.76 => ( P @ N2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_nat_le
% 0.35/0.76 thf(fact_10067_MOST__ge__nat,axiom,
% 0.35/0.76 ! [M: nat] : ( eventually_nat @ ( ord_less_eq_nat @ M ) @ cofinite_nat ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_ge_nat
% 0.35/0.76 thf(fact_10068_MOST__nat,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( eventually_nat @ P @ cofinite_nat )
% 0.35/0.76 = ( ? [M2: nat] :
% 0.35/0.76 ! [N2: nat] :
% 0.35/0.76 ( ( ord_less_nat @ M2 @ N2 )
% 0.35/0.76 => ( P @ N2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % MOST_nat
% 0.35/0.76 thf(fact_10069_infinity__enat__def,axiom,
% 0.35/0.76 ( extend5688581933313929465d_enat
% 0.35/0.76 = ( extended_Abs_enat @ none_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % infinity_enat_def
% 0.35/0.76 thf(fact_10070_enat__def,axiom,
% 0.35/0.76 ( extended_enat2
% 0.35/0.76 = ( ^ [N2: nat] : ( extended_Abs_enat @ ( some_nat @ N2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % enat_def
% 0.35/0.76 thf(fact_10071_INFM__nat__le,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( frequently_nat @ P @ cofinite_nat )
% 0.35/0.76 = ( ! [M2: nat] :
% 0.35/0.76 ? [N2: nat] :
% 0.35/0.76 ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.35/0.76 & ( P @ N2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % INFM_nat_le
% 0.35/0.76 thf(fact_10072_INFM__nat,axiom,
% 0.35/0.76 ! [P: nat > $o] :
% 0.35/0.76 ( ( frequently_nat @ P @ cofinite_nat )
% 0.35/0.76 = ( ! [M2: nat] :
% 0.35/0.76 ? [N2: nat] :
% 0.35/0.76 ( ( ord_less_nat @ M2 @ N2 )
% 0.35/0.76 & ( P @ N2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % INFM_nat
% 0.35/0.76 thf(fact_10073_list__encode_Oelims,axiom,
% 0.35/0.76 ! [X2: list_nat,Y3: nat] :
% 0.35/0.76 ( ( ( nat_list_encode @ X2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( ( X2 = nil_nat )
% 0.35/0.76 => ( Y3 != zero_zero_nat ) )
% 0.35/0.76 => ~ ! [X5: nat,Xs3: list_nat] :
% 0.35/0.76 ( ( X2
% 0.35/0.76 = ( cons_nat @ X5 @ Xs3 ) )
% 0.35/0.76 => ( Y3
% 0.35/0.76 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_encode.elims
% 0.35/0.76 thf(fact_10074_le__prod__encode__1,axiom,
% 0.35/0.76 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % le_prod_encode_1
% 0.35/0.76 thf(fact_10075_le__prod__encode__2,axiom,
% 0.35/0.76 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % le_prod_encode_2
% 0.35/0.76 thf(fact_10076_list__encode_Osimps_I1_J,axiom,
% 0.35/0.76 ( ( nat_list_encode @ nil_nat )
% 0.35/0.76 = zero_zero_nat ) ).
% 0.35/0.76
% 0.35/0.76 % list_encode.simps(1)
% 0.35/0.76 thf(fact_10077_list__encode_Opelims,axiom,
% 0.35/0.76 ! [X2: list_nat,Y3: nat] :
% 0.35/0.76 ( ( ( nat_list_encode @ X2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( accp_list_nat @ nat_list_encode_rel @ X2 )
% 0.35/0.76 => ( ( ( X2 = nil_nat )
% 0.35/0.76 => ( ( Y3 = zero_zero_nat )
% 0.35/0.76 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.35/0.76 => ~ ! [X5: nat,Xs3: list_nat] :
% 0.35/0.76 ( ( X2
% 0.35/0.76 = ( cons_nat @ X5 @ Xs3 ) )
% 0.35/0.76 => ( ( Y3
% 0.35/0.76 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 0.35/0.76 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_encode.pelims
% 0.35/0.76 thf(fact_10078_prod__decode__def,axiom,
% 0.35/0.76 ( nat_prod_decode
% 0.35/0.76 = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % prod_decode_def
% 0.35/0.76 thf(fact_10079_list__decode_Opinduct,axiom,
% 0.35/0.76 ! [A0: nat,P: nat > $o] :
% 0.35/0.76 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 0.35/0.76 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 0.35/0.76 => ( P @ zero_zero_nat ) )
% 0.35/0.76 => ( ! [N3: nat] :
% 0.35/0.76 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
% 0.35/0.76 => ( ! [X4: nat,Y5: nat] :
% 0.35/0.76 ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 0.35/0.76 = ( nat_prod_decode @ N3 ) )
% 0.35/0.76 => ( P @ Y5 ) )
% 0.35/0.76 => ( P @ ( suc @ N3 ) ) ) )
% 0.35/0.76 => ( P @ A0 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_decode.pinduct
% 0.35/0.76 thf(fact_10080_list__decode_Oelims,axiom,
% 0.35/0.76 ! [X2: nat,Y3: list_nat] :
% 0.35/0.76 ( ( ( nat_list_decode @ X2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( ( X2 = zero_zero_nat )
% 0.35/0.76 => ( Y3 != nil_nat ) )
% 0.35/0.76 => ~ ! [N3: nat] :
% 0.35/0.76 ( ( X2
% 0.35/0.76 = ( suc @ N3 ) )
% 0.35/0.76 => ( Y3
% 0.35/0.76 != ( produc2761476792215241774st_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.35/0.76 @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_decode.elims
% 0.35/0.76 thf(fact_10081_list__decode_Opsimps_I1_J,axiom,
% 0.35/0.76 ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 0.35/0.76 => ( ( nat_list_decode @ zero_zero_nat )
% 0.35/0.76 = nil_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_decode.psimps(1)
% 0.35/0.76 thf(fact_10082_list__decode_Osimps_I1_J,axiom,
% 0.35/0.76 ( ( nat_list_decode @ zero_zero_nat )
% 0.35/0.76 = nil_nat ) ).
% 0.35/0.76
% 0.35/0.76 % list_decode.simps(1)
% 0.35/0.76 thf(fact_10083_list__decode_Opelims,axiom,
% 0.35/0.76 ! [X2: nat,Y3: list_nat] :
% 0.35/0.76 ( ( ( nat_list_decode @ X2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( accp_nat @ nat_list_decode_rel @ X2 )
% 0.35/0.76 => ( ( ( X2 = zero_zero_nat )
% 0.35/0.76 => ( ( Y3 = nil_nat )
% 0.35/0.76 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 0.35/0.76 => ~ ! [N3: nat] :
% 0.35/0.76 ( ( X2
% 0.35/0.76 = ( suc @ N3 ) )
% 0.35/0.76 => ( ( Y3
% 0.35/0.76 = ( produc2761476792215241774st_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 0.35/0.76 @ ( nat_prod_decode @ N3 ) ) )
% 0.35/0.76 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % list_decode.pelims
% 0.35/0.76 thf(fact_10084_unit__factor__simps_I1_J,axiom,
% 0.35/0.76 ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
% 0.35/0.76 = zero_zero_nat ) ).
% 0.35/0.76
% 0.35/0.76 % unit_factor_simps(1)
% 0.35/0.76 thf(fact_10085_unit__factor__nat__def,axiom,
% 0.35/0.76 ( unit_f2748546683901255202or_nat
% 0.35/0.76 = ( ^ [N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % unit_factor_nat_def
% 0.35/0.76 thf(fact_10086_Lcm__eq__0__I__nat,axiom,
% 0.35/0.76 ! [A3: set_nat] :
% 0.35/0.76 ( ( member_nat @ zero_zero_nat @ A3 )
% 0.35/0.76 => ( ( gcd_Lcm_nat @ A3 )
% 0.35/0.76 = zero_zero_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_eq_0_I_nat
% 0.35/0.76 thf(fact_10087_Lcm__0__iff__nat,axiom,
% 0.35/0.76 ! [A3: set_nat] :
% 0.35/0.76 ( ( finite_finite_nat @ A3 )
% 0.35/0.76 => ( ( ( gcd_Lcm_nat @ A3 )
% 0.35/0.76 = zero_zero_nat )
% 0.35/0.76 = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_0_iff_nat
% 0.35/0.76 thf(fact_10088_Lcm__nat__infinite,axiom,
% 0.35/0.76 ! [M5: set_nat] :
% 0.35/0.76 ( ~ ( finite_finite_nat @ M5 )
% 0.35/0.76 => ( ( gcd_Lcm_nat @ M5 )
% 0.35/0.76 = zero_zero_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_nat_infinite
% 0.35/0.76 thf(fact_10089_Lcm__int__greater__eq__0,axiom,
% 0.35/0.76 ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K4 ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_int_greater_eq_0
% 0.35/0.76 thf(fact_10090_Lcm__nat__empty,axiom,
% 0.35/0.76 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 0.35/0.76 = one_one_nat ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_nat_empty
% 0.35/0.76 thf(fact_10091_Lcm__eq__Max__nat,axiom,
% 0.35/0.76 ! [M5: set_nat] :
% 0.35/0.76 ( ( finite_finite_nat @ M5 )
% 0.35/0.76 => ( ( M5 != bot_bot_set_nat )
% 0.35/0.76 => ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 0.35/0.76 => ( ! [M4: nat,N3: nat] :
% 0.35/0.76 ( ( member_nat @ M4 @ M5 )
% 0.35/0.76 => ( ( member_nat @ N3 @ M5 )
% 0.35/0.76 => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
% 0.35/0.76 => ( ( gcd_Lcm_nat @ M5 )
% 0.35/0.76 = ( lattic8265883725875713057ax_nat @ M5 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_eq_Max_nat
% 0.35/0.76 thf(fact_10092_lcm__0__iff__nat,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ( gcd_lcm_nat @ M @ N )
% 0.35/0.76 = zero_zero_nat )
% 0.35/0.76 = ( ( M = zero_zero_nat )
% 0.35/0.76 | ( N = zero_zero_nat ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_0_iff_nat
% 0.35/0.76 thf(fact_10093_lcm__1__iff__nat,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ( gcd_lcm_nat @ M @ N )
% 0.35/0.76 = ( suc @ zero_zero_nat ) )
% 0.35/0.76 = ( ( M
% 0.35/0.76 = ( suc @ zero_zero_nat ) )
% 0.35/0.76 & ( N
% 0.35/0.76 = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_1_iff_nat
% 0.35/0.76 thf(fact_10094_Lcm__in__lcm__closed__set__nat,axiom,
% 0.35/0.76 ! [M5: set_nat] :
% 0.35/0.76 ( ( finite_finite_nat @ M5 )
% 0.35/0.76 => ( ( M5 != bot_bot_set_nat )
% 0.35/0.76 => ( ! [M4: nat,N3: nat] :
% 0.35/0.76 ( ( member_nat @ M4 @ M5 )
% 0.35/0.76 => ( ( member_nat @ N3 @ M5 )
% 0.35/0.76 => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
% 0.35/0.76 => ( member_nat @ ( gcd_Lcm_nat @ M5 ) @ M5 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_in_lcm_closed_set_nat
% 0.35/0.76 thf(fact_10095_lcm__pos__int,axiom,
% 0.35/0.76 ! [M: int,N: int] :
% 0.35/0.76 ( ( M != zero_zero_int )
% 0.35/0.76 => ( ( N != zero_zero_int )
% 0.35/0.76 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_pos_int
% 0.35/0.76 thf(fact_10096_lcm__pos__nat,axiom,
% 0.35/0.76 ! [M: nat,N: nat] :
% 0.35/0.76 ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.35/0.76 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 0.35/0.76 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_pos_nat
% 0.35/0.76 thf(fact_10097_lcm__ge__0__int,axiom,
% 0.35/0.76 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X2 @ Y3 ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_ge_0_int
% 0.35/0.76 thf(fact_10098_lcm__unique__int,axiom,
% 0.35/0.76 ! [D: int,A2: int,B3: int] :
% 0.35/0.76 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.35/0.76 & ( dvd_dvd_int @ A2 @ D )
% 0.35/0.76 & ( dvd_dvd_int @ B3 @ D )
% 0.35/0.76 & ! [E3: int] :
% 0.35/0.76 ( ( ( dvd_dvd_int @ A2 @ E3 )
% 0.35/0.76 & ( dvd_dvd_int @ B3 @ E3 ) )
% 0.35/0.76 => ( dvd_dvd_int @ D @ E3 ) ) )
% 0.35/0.76 = ( D
% 0.35/0.76 = ( gcd_lcm_int @ A2 @ B3 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_unique_int
% 0.35/0.76 thf(fact_10099_lcm__cases__int,axiom,
% 0.35/0.76 ! [X2: int,Y3: int,P: int > $o] :
% 0.35/0.76 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.76 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.76 => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) )
% 0.35/0.76 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 0.35/0.76 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.35/0.76 => ( P @ ( gcd_lcm_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.35/0.76 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.35/0.76 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.35/0.76 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 0.35/0.76 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 0.35/0.76 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.35/0.76 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.35/0.76 => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % lcm_cases_int
% 0.35/0.76 thf(fact_10100_Lcm__nat__def,axiom,
% 0.35/0.76 ( gcd_Lcm_nat
% 0.35/0.76 = ( ^ [M8: set_nat] : ( if_nat @ ( finite_finite_nat @ M8 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M8 ) @ zero_zero_nat ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % Lcm_nat_def
% 0.35/0.76 thf(fact_10101_VEBT__internal_Olesseq_Osimps,axiom,
% 0.35/0.76 ( vEBT_VEBT_lesseq
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.simps
% 0.35/0.76 thf(fact_10102_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.elims(1)
% 0.35/0.76 thf(fact_10103_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.elims(2)
% 0.35/0.76 thf(fact_10104_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.elims(3)
% 0.35/0.76 thf(fact_10105_VEBT__internal_Oless_Oelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.elims(3)
% 0.35/0.76 thf(fact_10106_VEBT__internal_Oless_Oelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.elims(2)
% 0.35/0.76 thf(fact_10107_VEBT__internal_Oless_Oelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.elims(1)
% 0.35/0.76 thf(fact_10108_VEBT__internal_Oless_Osimps,axiom,
% 0.35/0.76 ( vEBT_VEBT_less
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.simps
% 0.35/0.76 thf(fact_10109_VEBT__internal_Ogreater_Osimps,axiom,
% 0.35/0.76 ( vEBT_VEBT_greater
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.simps
% 0.35/0.76 thf(fact_10110_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.elims(1)
% 0.35/0.76 thf(fact_10111_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.elims(2)
% 0.35/0.76 thf(fact_10112_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.elims(3)
% 0.35/0.76 thf(fact_10113_VEBT__internal_Ogreater_Opelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) )
% 0.35/0.76 => ~ ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.pelims(1)
% 0.35/0.76 thf(fact_10114_VEBT__internal_Ogreater_Ocases,axiom,
% 0.35/0.76 ! [X2: produc4953844613479565601on_nat] :
% 0.35/0.76 ~ ! [X5: option_nat,Y4: option_nat] :
% 0.35/0.76 ( X2
% 0.35/0.76 != ( produc5098337634421038937on_nat @ X5 @ Y4 ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.cases
% 0.35/0.76 thf(fact_10115_VEBT__internal_Ogreater_Opelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.pelims(3)
% 0.35/0.76 thf(fact_10116_VEBT__internal_Ogreater_Opelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat
% 0.35/0.76 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 0.35/0.76 @ X2
% 0.35/0.76 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.greater.pelims(2)
% 0.35/0.76 thf(fact_10117_VEBT__internal_Olesseq_Opelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.pelims(1)
% 0.35/0.76 thf(fact_10118_VEBT__internal_Olesseq_Opelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.pelims(2)
% 0.35/0.76 thf(fact_10119_VEBT__internal_Olesseq_Opelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.lesseq.pelims(3)
% 0.35/0.76 thf(fact_10120_VEBT__internal_Oless_Opelims_I3_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.pelims(3)
% 0.35/0.76 thf(fact_10121_VEBT__internal_Oless_Opelims_I2_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat] :
% 0.35/0.76 ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.pelims(2)
% 0.35/0.76 thf(fact_10122_VEBT__internal_Oless_Opelims_I1_J,axiom,
% 0.35/0.76 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 0.35/0.76 ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 0.35/0.76 = Y3 )
% 0.35/0.76 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( ( Y3
% 0.35/0.76 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) )
% 0.35/0.76 => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 0.35/0.76
% 0.35/0.76 % VEBT_internal.less.pelims(1)
% 0.35/0.76
% 0.35/0.76 % Helper facts (38)
% 0.35/0.76 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.35/0.76 ! [X2: int,Y3: int] :
% 0.35/0.76 ( ( if_int @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.35/0.76 ! [X2: int,Y3: int] :
% 0.35/0.76 ( ( if_int @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.35/0.76 ! [X2: nat,Y3: nat] :
% 0.35/0.76 ( ( if_nat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.35/0.76 ! [X2: nat,Y3: nat] :
% 0.35/0.76 ( ( if_nat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 0.35/0.76 ! [X2: num,Y3: num] :
% 0.35/0.76 ( ( if_num @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 0.35/0.76 ! [X2: num,Y3: num] :
% 0.35/0.76 ( ( if_num @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.35/0.76 ! [X2: rat,Y3: rat] :
% 0.35/0.76 ( ( if_rat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.35/0.76 ! [X2: rat,Y3: rat] :
% 0.35/0.76 ( ( if_rat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.35/0.76 ! [X2: real,Y3: real] :
% 0.35/0.76 ( ( if_real @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.35/0.76 ! [X2: real,Y3: real] :
% 0.35/0.76 ( ( if_real @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.35/0.76 ! [P: real > $o] :
% 0.35/0.76 ( ( P @ ( fChoice_real @ P ) )
% 0.35/0.76 = ( ? [X8: real] : ( P @ X8 ) ) ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.35/0.76 ! [X2: complex,Y3: complex] :
% 0.35/0.76 ( ( if_complex @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.35/0.76 ! [X2: complex,Y3: complex] :
% 0.35/0.76 ( ( if_complex @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.35/0.76 ! [X2: extended_enat,Y3: extended_enat] :
% 0.35/0.76 ( ( if_Extended_enat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.35/0.76 ! [X2: extended_enat,Y3: extended_enat] :
% 0.35/0.76 ( ( if_Extended_enat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.35/0.76 ! [X2: code_integer,Y3: code_integer] :
% 0.35/0.76 ( ( if_Code_integer @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.35/0.76 ! [X2: code_integer,Y3: code_integer] :
% 0.35/0.76 ( ( if_Code_integer @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: set_int,Y3: set_int] :
% 0.35/0.76 ( ( if_set_int @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: set_int,Y3: set_int] :
% 0.35/0.76 ( ( if_set_int @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.35/0.76 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.35/0.76 ( ( if_VEBT_VEBT @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 0.35/0.76 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 0.35/0.76 ( ( if_VEBT_VEBT @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: list_int,Y3: list_int] :
% 0.35/0.76 ( ( if_list_int @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: list_int,Y3: list_int] :
% 0.35/0.76 ( ( if_list_int @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: list_nat,Y3: list_nat] :
% 0.35/0.76 ( ( if_list_nat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: list_nat,Y3: list_nat] :
% 0.35/0.76 ( ( if_list_nat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: option_nat,Y3: option_nat] :
% 0.35/0.76 ( ( if_option_nat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: option_nat,Y3: option_nat] :
% 0.35/0.76 ( ( if_option_nat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.35/0.76 ! [X2: option_num,Y3: option_num] :
% 0.35/0.76 ( ( if_option_num @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.35/0.76 ! [X2: option_num,Y3: option_num] :
% 0.35/0.76 ( ( if_option_num @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.35/0.76 ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.35/0.76 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 0.35/0.76 ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.35/0.76 ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.35/0.76 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.35/0.76 ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.35/0.76 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.35/0.76 ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.35/0.76 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.35/0.76 ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76 ! [P: $o] :
% 0.35/0.76 ( ( P = $true )
% 0.35/0.76 | ( P = $false ) ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.35/0.76 ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
% 0.35/0.76 = Y3 ) ).
% 0.35/0.76
% 0.35/0.76 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.35/0.76 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.35/0.76 ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
% 0.35/0.76 = X2 ) ).
% 0.35/0.76
% 0.35/0.76 % Conjectures (4)
% 0.35/0.76 thf(conj_0,hypothesis,
% 0.35/0.76 ~ ( ( ( vEBT_vebt_mint @ ta )
% 0.35/0.76 = none_nat )
% 0.35/0.76 & ( ( vEBT_vebt_mint @ k )
% 0.35/0.76 = ( some_nat @ b ) ) ) ).
% 0.35/0.76
% 0.35/0.76 thf(conj_1,hypothesis,
% 0.35/0.76 ~ ( ( ( vEBT_vebt_mint @ ta )
% 0.35/0.76 = ( some_nat @ a ) )
% 0.35/0.76 & ( ( vEBT_vebt_mint @ k )
% 0.35/0.76 = none_nat ) ) ).
% 0.35/0.76
% 0.35/0.76 thf(conj_2,hypothesis,
% 0.35/0.76 ( ( ord_less_nat @ a @ b )
% 0.35/0.76 & ( ( some_nat @ a )
% 0.35/0.76 = ( vEBT_vebt_mint @ ta ) )
% 0.35/0.76 & ( ( some_nat @ b )
% 0.35/0.76 = ( vEBT_vebt_mint @ k ) ) ) ).
% 0.35/0.76
% 0.35/0.76 thf(conj_3,conjecture,
% 0.35/0.76 $false ).
% 0.35/0.76
% 0.35/0.76 %------------------------------------------------------------------------------
% 0.35/0.76 ------------------------
% 6.45/2.52 ---- Embedded in THF ---
% 6.45/2.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.45/2.52 %%% Generated on Fri Feb 27 14:01:56 EST 2026
% 6.45/2.52 %%% using '$$dhol' embedding, version 1.3.0.
% 6.45/2.52 %%% Logic specification used:
% 6.45/2.52 %%% thf(spec, logic, $$dhol).
% 6.45/2.52
% 6.45/2.52 % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2DTF3077.p
% See solution above
% 6.45/2.55 ------------------------
% 8.84/5.01 ---- Cleaned THF ---
% 8.84/5.02 thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_J_J,type,
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% 8.84/5.02 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,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
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% 8.84/5.02 thf(sy_c_Binomial_Obinomial,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Bit__Operations_Oand__int__rel,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Bit__Operations_Oconcat__bit,type,
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% 8.84/5.02 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
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% 8.84/5.02 thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit_001t__Nat__Onat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Bit__Operations_Otake__bit__num,type,
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% 8.84/5.02 thf(sy_c_Code__Numeral_Obit__cut__integer,type,
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% 8.84/5.02 thf(sy_c_Code__Numeral_Odivmod__abs,type,
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% 8.84/5.02 thf(sy_c_Code__Numeral_Odivmod__integer,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Code__Numeral_Onat__of__integer,type,
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% 8.84/5.02 thf(sy_c_Code__Numeral_Onum__of__integer,type,
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% 8.84/5.02 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Extended____Nat__Oenat,type,
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% 8.84/5.02 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Nat__Onat,type,
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% 8.84/5.02 thf(sy_c_Complete__Lattices_OInf__class_OInf_001t__Set__Oset_It__Nat__Onat_J,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Int__Oint,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Nat__Onat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Real__Oreal,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complex_OArg,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complex_Ocis,type,
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% 8.84/5.02 thf(sy_c_Complex_Ocnj,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complex_Ocomplex_OComplex,type,
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% 8.84/5.02
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% 8.84/5.02 thf(sy_c_Complex_Ocomplex_ORe,type,
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% 8.84/5.02 thf(sy_c_Complex_Ocsqrt,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Complex_Oimaginary__unit,type,
% 8.84/5.02 imaginary_unit: complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above_001t__Nat__Onat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Deriv_Ohas__derivative_001t__Real__Oreal_001t__Real__Oreal,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Divides_Odivmod__nat,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Divides_Oeucl__rel__int,type,
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% 8.84/5.02 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
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% 8.84/5.02 thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
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% 8.84/5.02 thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 cons_set_nat: set_nat > list_set_nat > list_set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 8.84/5.02 nil_int: list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 8.84/5.02 nil_nat: list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 8.84/5.02 hd_nat: list_nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.84/5.02 map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001_Eo,type,
% 8.84/5.02 set_o2: list_o > set_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
% 8.84/5.02 set_complex2: list_complex > set_complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Extended____Nat__Oenat,type,
% 8.84/5.02 set_Extended_enat2: list_Extended_enat > set_Extended_enat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
% 8.84/5.02 set_int2: list_int > set_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
% 8.84/5.02 set_list_o2: list_list_o > set_list_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
% 8.84/5.02 set_list_nat2: list_list_nat > set_list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 8.84/5.02 set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 8.84/5.02 set_nat2: list_nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
% 8.84/5.02 set_num2: list_num > set_num ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 8.84/5.02 set_real2: list_real > set_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 set_set_nat2: list_set_nat > set_set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001_Eo,type,
% 8.84/5.02 list_update_o: list_o > nat > $o > list_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 8.84/5.02 list_update_int: list_int > nat > int > list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 8.84/5.02 list_update_nat: list_nat > nat > nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 8.84/5.02 list_update_real: list_real > nat > real > list_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001_Eo,type,
% 8.84/5.02 nth_o: list_o > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Int__Oint,type,
% 8.84/5.02 nth_int: list_int > nat > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 8.84/5.02 nth_nat: list_nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Num__Onum,type,
% 8.84/5.02 nth_num: list_num > nat > num ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 8.84/5.02 nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.84/5.02 nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_M_Eo_J,type,
% 8.84/5.02 nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 8.84/5.02 nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
% 8.84/5.02 nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.84/5.02 nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 8.84/5.02 nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 8.84/5.02 nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.84/5.02 nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 8.84/5.02 nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 8.84/5.02 nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 8.84/5.02 nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 8.84/5.02 nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 8.84/5.02 nth_real: list_real > nat > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 nth_set_nat: list_set_nat > nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 8.84/5.02 product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__Int__Oint_001_Eo,type,
% 8.84/5.02 product_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Int__Oint,type,
% 8.84/5.02 product_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__Nat__Onat,type,
% 8.84/5.02 product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 8.84/5.02 product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 8.84/5.02 produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 8.84/5.02 produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001_Eo,type,
% 8.84/5.02 removeAll_o: $o > list_o > list_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__Int__Oint,type,
% 8.84/5.02 removeAll_int: int > list_int > list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__Nat__Onat,type,
% 8.84/5.02 removeAll_nat: nat > list_nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__Real__Oreal,type,
% 8.84/5.02 removeAll_real: real > list_real > list_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 removeAll_set_nat: set_nat > list_set_nat > list_set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_OremoveAll_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Orotate1_001_Eo,type,
% 8.84/5.02 rotate1_o: list_o > list_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Orotate1_001t__Int__Oint,type,
% 8.84/5.02 rotate1_int: list_int > list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
% 8.84/5.02 rotate1_nat: list_nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 8.84/5.02 sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 8.84/5.02 sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 8.84/5.02 take_nat: nat > list_nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oupt,type,
% 8.84/5.02 upt: nat > nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oupto,type,
% 8.84/5.02 upto: int > int > list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oupto__aux,type,
% 8.84/5.02 upto_aux: int > int > list_int > list_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Oupto__rel,type,
% 8.84/5.02 upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
% 8.84/5.02 zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__Int__Oint_001_Eo,type,
% 8.84/5.02 zip_int_o: list_int > list_o > list_P5087981734274514673_int_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
% 8.84/5.02 zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__Int__Oint_001t__Nat__Onat,type,
% 8.84/5.02 zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 8.84/5.02 zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 8.84/5.02 zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 8.84/5.02 zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_OSuc,type,
% 8.84/5.02 suc: nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 8.84/5.02 case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 8.84/5.02 case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Onat_Opred,type,
% 8.84/5.02 pred: nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 8.84/5.02 semiri4939895301339042750nteger: nat > code_integer ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 8.84/5.02 semiri8010041392384452111omplex: nat > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
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% 8.84/5.02 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
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% 8.84/5.02 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
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% 8.84/5.02 thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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% 8.84/5.02 thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
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% 8.84/5.02 thf(sy_c_Num_Onum_Osize__num,type,
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% 8.84/5.02 thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
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% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Complex__Ocomplex_J_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 8.84/5.02 produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Extended____Nat__Oenat_J_001t__Set__Oset_It__Extended____Nat__Oenat_J,type,
% 8.84/5.02 produc6639060556116774935d_enat: set_Extended_enat > set_Extended_enat > produc1621487020699730983d_enat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
% 8.84/5.02 produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 8.84/5.02 produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
% 8.84/5.02 produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 8.84/5.02 produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 8.84/5.02 produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 8.84/5.02 produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 8.84/5.02 produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.84/5.02 produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
% 8.84/5.02 produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 8.84/5.02 produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 8.84/5.02 produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Num__Onum,type,
% 8.84/5.02 produc7336495610019696514er_num: ( code_integer > code_integer > num ) > produc8923325533196201883nteger > num ).
% 8.84/5.02
% 8.84/5.02 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,
% 8.84/5.02 produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 8.84/5.02
% 8.84/5.02 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,
% 8.84/5.02 produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 8.84/5.02 produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 8.84/5.02 produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 8.84/5.02
% 8.84/5.02 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,
% 8.84/5.02 produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 8.84/5.02
% 8.84/5.02 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,
% 8.84/5.02 produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 8.84/5.02 produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
% 8.84/5.02 produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).
% 8.84/5.02
% 8.84/5.02 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,
% 8.84/5.02 produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 8.84/5.02 produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 8.84/5.02 product_fst_int_int: product_prod_int_int > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.84/5.02 product_fst_nat_nat: product_prod_nat_nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 8.84/5.02 product_snd_int_int: product_prod_int_int > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 8.84/5.02 product_snd_nat_nat: product_prod_nat_nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Ounit_OAbs__unit,type,
% 8.84/5.02 product_Abs_unit: $o > product_unit ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Product__Type_Ounit_ORep__unit,type,
% 8.84/5.02 product_Rep_unit: product_unit > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_OAbs__Rat,type,
% 8.84/5.02 abs_Rat: product_prod_int_int > rat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_OFract,type,
% 8.84/5.02 fract: int > int > rat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_ORep__Rat,type,
% 8.84/5.02 rep_Rat: rat > product_prod_int_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 8.84/5.02 field_5140801741446780682s_real: set_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Ofield__char__0__class_Oof__rat_001t__Real__Oreal,type,
% 8.84/5.02 field_7254667332652039916t_real: rat > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Onormalize,type,
% 8.84/5.02 normalize: product_prod_int_int > product_prod_int_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Opcr__rat,type,
% 8.84/5.02 pcr_rat: product_prod_int_int > rat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Opositive,type,
% 8.84/5.02 positive: rat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Oquotient__of,type,
% 8.84/5.02 quotient_of: rat > product_prod_int_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rat_Oratrel,type,
% 8.84/5.02 ratrel: product_prod_int_int > product_prod_int_int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_OReal,type,
% 8.84/5.02 real2: ( nat > rat ) > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Ocauchy,type,
% 8.84/5.02 cauchy: ( nat > rat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Opcr__real,type,
% 8.84/5.02 pcr_real: ( nat > rat ) > real > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Opositive,type,
% 8.84/5.02 positive2: real > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Orealrel,type,
% 8.84/5.02 realrel: ( nat > rat ) > ( nat > rat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Orep__real,type,
% 8.84/5.02 rep_real: real > nat > rat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real_Ovanishes,type,
% 8.84/5.02 vanishes: ( nat > rat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 8.84/5.02 real_V3694042436643373181omplex: complex > complex > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 8.84/5.02 real_V975177566351809787t_real: real > real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 8.84/5.02 real_V1022390504157884413omplex: complex > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 8.84/5.02 real_V7735802525324610683m_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 8.84/5.02 real_V4546457046886955230omplex: real > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Relation_OField_001t__Nat__Onat,type,
% 8.84/5.02 field_nat: set_Pr1261947904930325089at_nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Relation_OId_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 id_Pro2258643101195443293at_nat: set_Pr8693737435421807431at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Relation_Ototal__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 total_3592101749530773125at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Relation_Otrans_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 8.84/5.02 trans_4347625901269045472at_nat: set_Pr8693737435421807431at_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Int__Oint,type,
% 8.84/5.02 algebr932160517623751201me_int: int > int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime_001t__Nat__Onat,type,
% 8.84/5.02 algebr934650988132801477me_nat: nat > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 8.84/5.02 divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 8.84/5.02 divide1717551699836669952omplex: complex > complex > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 8.84/5.02 divide_divide_int: int > int > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 8.84/5.02 divide_divide_nat: nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 8.84/5.02 divide_divide_rat: rat > rat > rat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 8.84/5.02 divide_divide_real: real > real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 8.84/5.02 dvd_dvd_int: int > int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat,type,
% 8.84/5.02 dvd_dvd_nat: nat > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
% 8.84/5.02 modulo364778990260209775nteger: code_integer > code_integer > code_integer ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Int__Oint,type,
% 8.84/5.02 modulo_modulo_int: int > int > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
% 8.84/5.02 modulo_modulo_nat: nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
% 8.84/5.02 unit_f2748546683901255202or_nat: nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
% 8.84/5.02 zero_n2684676970156552555ol_int: $o > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Nat__Onat,type,
% 8.84/5.02 zero_n2687167440665602831ol_nat: $o > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osuminf_001t__Complex__Ocomplex,type,
% 8.84/5.02 suminf_complex: ( nat > complex ) > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osuminf_001t__Int__Oint,type,
% 8.84/5.02 suminf_int: ( nat > int ) > int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
% 8.84/5.02 suminf_nat: ( nat > nat ) > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
% 8.84/5.02 suminf_real: ( nat > real ) > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
% 8.84/5.02 summable_complex: ( nat > complex ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osummable_001t__Int__Oint,type,
% 8.84/5.02 summable_int: ( nat > int ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
% 8.84/5.02 summable_nat: ( nat > nat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
% 8.84/5.02 summable_real: ( nat > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
% 8.84/5.02 sums_complex: ( nat > complex ) > complex > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osums_001t__Int__Oint,type,
% 8.84/5.02 sums_int: ( nat > int ) > int > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osums_001t__Nat__Onat,type,
% 8.84/5.02 sums_nat: ( nat > nat ) > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Series_Osums_001t__Real__Oreal,type,
% 8.84/5.02 sums_real: ( nat > real ) > real > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001_Eo,type,
% 8.84/5.02 collect_o: ( $o > $o ) > set_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
% 8.84/5.02 collect_complex: ( complex > $o ) > set_complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001t__Extended____Nat__Oenat,type,
% 8.84/5.02 collec4429806609662206161d_enat: ( extended_enat > $o ) > set_Extended_enat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001t__Int__Oint,type,
% 8.84/5.02 collect_int: ( int > $o ) > set_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
% 8.84/5.02 collect_list_o: ( list_o > $o ) > set_list_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
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% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Nat__Onat,type,
% 8.84/5.02 set_ord_atMost_nat: nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 8.84/5.02 set_or6656581121297822940st_int: int > int > set_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 8.84/5.02 set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 8.84/5.02 set_or5832277885323065728an_int: int > int > set_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 8.84/5.02 set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 8.84/5.02 set_or1633881224788618240n_real: real > real > set_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 8.84/5.02 set_or1210151606488870762an_nat: nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 8.84/5.02 set_or5849166863359141190n_real: real > set_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
% 8.84/5.02 set_ord_lessThan_o: $o > set_o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Extended____Nat__Oenat,type,
% 8.84/5.02 set_or8419480210114673929d_enat: extended_enat > set_Extended_enat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 8.84/5.02 set_ord_lessThan_int: int > set_int ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 8.84/5.02 set_ord_lessThan_nat: nat > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 8.84/5.02 set_ord_lessThan_num: num > set_num ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 8.84/5.02 set_ord_lessThan_rat: rat > set_rat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 8.84/5.02 set_or5984915006950818249n_real: real > set_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 8.84/5.02 set_or890127255671739683et_nat: set_nat > set_set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_String_OCode_Oabort_001t__Real__Oreal,type,
% 8.84/5.02 abort_real: literal > ( product_unit > real ) > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_String_OLiteral,type,
% 8.84/5.02 literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_String_Ocomm__semiring__1__class_Oof__char_001t__Nat__Onat,type,
% 8.84/5.02 comm_s629917340098488124ar_nat: char > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of_001t__Nat__Onat,type,
% 8.84/5.02 unique3096191561947761185of_nat: nat > char ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.84/5.02 topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 8.84/5.02 topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 8.84/5.02 topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 8.84/5.02 topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 8.84/5.02 topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 8.84/5.02 topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 8.84/5.02 topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 8.84/5.02 topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 8.84/5.02 topolo2177554685111907308n_real: real > set_real > filter_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Otopological__space__class_Oconvergent_001t__Real__Oreal,type,
% 8.84/5.02 topolo7531315842566124627t_real: ( nat > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 8.84/5.02 topolo2815343760600316023s_real: real > filter_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 8.84/5.02 topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 8.84/5.02 topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 8.84/5.02 topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oarccos,type,
% 8.84/5.02 arccos: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 8.84/5.02 arcosh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oarcsin,type,
% 8.84/5.02 arcsin: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oarctan,type,
% 8.84/5.02 arctan: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 8.84/5.02 arsinh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 8.84/5.02 artanh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 8.84/5.02 cos_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Ocos__coeff,type,
% 8.84/5.02 cos_coeff: nat > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 8.84/5.02 cosh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 8.84/5.02 cot_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 8.84/5.02 diffs_complex: ( nat > complex ) > nat > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 8.84/5.02 diffs_real: ( nat > real ) > nat > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 8.84/5.02 exp_complex: complex > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 8.84/5.02 exp_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 8.84/5.02 ln_ln_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Olog,type,
% 8.84/5.02 log: real > real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Opi,type,
% 8.84/5.02 pi: real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 8.84/5.02 powr_real: real > real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Opowr__real,type,
% 8.84/5.02 powr_real2: real > real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 8.84/5.02 sin_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Osin__coeff,type,
% 8.84/5.02 sin_coeff: nat > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Osinh_001t__Complex__Ocomplex,type,
% 8.84/5.02 sinh_complex: complex > complex ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 8.84/5.02 sinh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 8.84/5.02 tan_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 8.84/5.02 tanh_real: real > real ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 8.84/5.02 transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 8.84/5.02 transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
% 8.84/5.02 type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 8.84/5.02 vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 8.84/5.02 vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 8.84/5.02 vEBT_size_VEBT: vEBT_VEBT > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 8.84/5.02 vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
% 8.84/5.02 vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 8.84/5.02 vEBT_VEBT_high: nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 8.84/5.02 vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 8.84/5.02 vEBT_VEBT_low: nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 8.84/5.02 vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 8.84/5.02 vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 8.84/5.02 vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 8.84/5.02 vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 8.84/5.02 vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 8.84/5.02 vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 8.84/5.02 vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 8.84/5.02 vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 8.84/5.02 vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 8.84/5.02 vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Delete_Ovebt__delete,type,
% 8.84/5.02 vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
% 8.84/5.02 vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H,type,
% 8.84/5.02 vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__InsertCorrectness_OVEBT__internal_Oinsert_H__rel,type,
% 8.84/5.02 vEBT_VEBT_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Insert_Ovebt__insert,type,
% 8.84/5.02 vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
% 8.84/5.02 vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 8.84/5.02 vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 8.84/5.02 vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 8.84/5.02 vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 8.84/5.02 vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_Ovebt__member,type,
% 8.84/5.02 vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 8.84/5.02 vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 8.84/5.02 vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 8.84/5.02 vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater__rel,type,
% 8.84/5.02 vEBT_V5711637165310795180er_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 8.84/5.02 vEBT_VEBT_less: option_nat > option_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless__rel,type,
% 8.84/5.02 vEBT_VEBT_less_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 8.84/5.02 vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq__rel,type,
% 8.84/5.02 vEBT_VEBT_lesseq_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 8.84/5.02 vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 8.84/5.02 vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 8.84/5.02 vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__comp__shift_001t__Nat__Onat,type,
% 8.84/5.02 vEBT_V2881884560877996034ft_nat: ( nat > nat > $o ) > option_nat > option_nat > $o ).
% 8.84/5.02
% 8.84/5.02 thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 8.84/5.02 vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 8.84/5.02
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% 8.84/5.02
% 8.84/5.02 thf(sy_v_sa____,type,
% 8.84/5.02 sa: vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_v_summary_H____,type,
% 8.84/5.02 summary: vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_v_summary____,type,
% 8.84/5.02 summary2: vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_v_ta____,type,
% 8.84/5.02 ta: vEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_v_treeList_H____,type,
% 8.84/5.02 treeList: list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(sy_v_treeList____,type,
% 8.84/5.02 treeList2: list_VEBT_VEBT ).
% 8.84/5.02
% 8.84/5.02 thf(fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062,axiom,
% 8.84/5.02 ( ( vEBT_vebt_mint @ ta )
% 8.84/5.02 != ( vEBT_vebt_mint @ k ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_1_abdef,axiom,
% 8.84/5.02 ( ( ( ( vEBT_vebt_mint @ ta )
% 8.84/5.02 = none_nat )
% 8.84/5.02 & ( ( vEBT_vebt_mint @ k )
% 8.84/5.02 = ( some_nat @ b ) ) )
% 8.84/5.02 | ( ( ( vEBT_vebt_mint @ ta )
% 8.84/5.02 = ( some_nat @ a ) )
% 8.84/5.02 & ( ( vEBT_vebt_mint @ k )
% 8.84/5.02 = none_nat ) )
% 8.84/5.02 | ( ( ord_less_nat @ a @ b )
% 8.84/5.02 & ( ( some_nat @ a )
% 8.84/5.02 = ( vEBT_vebt_mint @ ta ) )
% 8.84/5.02 & ( ( some_nat @ b )
% 8.84/5.02 = ( vEBT_vebt_mint @ k ) ) )
% 8.84/5.02 | ( ( ord_less_nat @ b @ a )
% 8.84/5.02 & ( ( some_nat @ a )
% 8.84/5.02 = ( vEBT_vebt_mint @ ta ) )
% 8.84/5.02 & ( ( some_nat @ b )
% 8.84/5.02 = ( vEBT_vebt_mint @ k ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_2_assms_I3_J,axiom,
% 8.84/5.02 ( ( vEBT_VEBT_set_vebt @ ta )
% 8.84/5.02 = ( vEBT_VEBT_set_vebt @ k ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_3__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062a_Ab_O_Avebt__mint_At_A_061_ANone_A_092_060and_062_Avebt__mint_Ak_A_061_ASome_Ab_A_092_060or_062_Avebt__mint_At_A_061_ASome_Aa_A_092_060and_062_Avebt__mint_Ak_A_061_ANone_A_092_060or_062_Aa_A_060_Ab_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060or_062_Ab_A_060_Aa_A_092_060and_062_ASome_Aa_A_061_Avebt__mint_At_A_092_060and_062_ASome_Ab_A_061_Avebt__mint_Ak_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 8.84/5.02 ~ ! [A: nat,B: nat] :
% 8.84/5.02 ~ ( ( ( ( vEBT_vebt_mint @ ta )
% 8.84/5.02 = none_nat )
% 8.84/5.02 & ( ( vEBT_vebt_mint @ k )
% 8.84/5.02 = ( some_nat @ B ) ) )
% 8.84/5.02 | ( ( ( vEBT_vebt_mint @ ta )
% 8.84/5.02 = ( some_nat @ A ) )
% 8.84/5.02 & ( ( vEBT_vebt_mint @ k )
% 8.84/5.02 = none_nat ) )
% 8.84/5.02 | ( ( ord_less_nat @ A @ B )
% 8.84/5.02 & ( ( some_nat @ A )
% 8.84/5.02 = ( vEBT_vebt_mint @ ta ) )
% 8.84/5.02 & ( ( some_nat @ B )
% 8.84/5.02 = ( vEBT_vebt_mint @ k ) ) )
% 8.84/5.02 | ( ( ord_less_nat @ B @ A )
% 8.84/5.02 & ( ( some_nat @ A )
% 8.84/5.02 = ( vEBT_vebt_mint @ ta ) )
% 8.84/5.02 & ( ( some_nat @ B )
% 8.84/5.02 = ( vEBT_vebt_mint @ k ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_4_greater__shift,axiom,
% 8.84/5.02 ( ord_less_nat
% 8.84/5.02 = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_5_less__shift,axiom,
% 8.84/5.02 ( ord_less_nat
% 8.84/5.02 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_6_not__None__eq,axiom,
% 8.84/5.02 ! [X2: option_nat] :
% 8.84/5.02 ( ( X2 != none_nat )
% 8.84/5.02 = ( ? [Y: nat] :
% 8.84/5.02 ( X2
% 8.84/5.02 = ( some_nat @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_7_not__None__eq,axiom,
% 8.84/5.02 ! [X2: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( X2 != none_P5556105721700978146at_nat )
% 8.84/5.02 = ( ? [Y: product_prod_nat_nat] :
% 8.84/5.02 ( X2
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_8_not__None__eq,axiom,
% 8.84/5.02 ! [X2: option_num] :
% 8.84/5.02 ( ( X2 != none_num )
% 8.84/5.02 = ( ? [Y: num] :
% 8.84/5.02 ( X2
% 8.84/5.02 = ( some_num @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_9_not__Some__eq,axiom,
% 8.84/5.02 ! [X2: option_nat] :
% 8.84/5.02 ( ( ! [Y: nat] :
% 8.84/5.02 ( X2
% 8.84/5.02 != ( some_nat @ Y ) ) )
% 8.84/5.02 = ( X2 = none_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_10_not__Some__eq,axiom,
% 8.84/5.02 ! [X2: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( ! [Y: product_prod_nat_nat] :
% 8.84/5.02 ( X2
% 8.84/5.02 != ( some_P7363390416028606310at_nat @ Y ) ) )
% 8.84/5.02 = ( X2 = none_P5556105721700978146at_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_11_not__Some__eq,axiom,
% 8.84/5.02 ! [X2: option_num] :
% 8.84/5.02 ( ( ! [Y: num] :
% 8.84/5.02 ( X2
% 8.84/5.02 != ( some_num @ Y ) ) )
% 8.84/5.02 = ( X2 = none_num ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_12_minNullmin,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT] :
% 8.84/5.02 ( ( vEBT_VEBT_minNull @ T )
% 8.84/5.02 => ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = none_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_13_minminNull,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT] :
% 8.84/5.02 ( ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = none_nat )
% 8.84/5.02 => ( vEBT_VEBT_minNull @ T ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_14_option_Oinject,axiom,
% 8.84/5.02 ! [X22: nat,Y2: nat] :
% 8.84/5.02 ( ( ( some_nat @ X22 )
% 8.84/5.02 = ( some_nat @ Y2 ) )
% 8.84/5.02 = ( X22 = Y2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_15_option_Oinject,axiom,
% 8.84/5.02 ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 8.84/5.02 ( ( ( some_P7363390416028606310at_nat @ X22 )
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ Y2 ) )
% 8.84/5.02 = ( X22 = Y2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_16_option_Oinject,axiom,
% 8.84/5.02 ! [X22: num,Y2: num] :
% 8.84/5.02 ( ( ( some_num @ X22 )
% 8.84/5.02 = ( some_num @ Y2 ) )
% 8.84/5.02 = ( X22 = Y2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_17_assms_I2_J,axiom,
% 8.84/5.02 vEBT_invar_vebt @ k @ h ).
% 8.84/5.02
% 8.84/5.02 thf(fact_18_assms_I1_J,axiom,
% 8.84/5.02 vEBT_invar_vebt @ ta @ h ).
% 8.84/5.02
% 8.84/5.02 thf(fact_19_option_Odistinct_I1_J,axiom,
% 8.84/5.02 ! [X22: nat] :
% 8.84/5.02 ( none_nat
% 8.84/5.02 != ( some_nat @ X22 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_20_option_Odistinct_I1_J,axiom,
% 8.84/5.02 ! [X22: product_prod_nat_nat] :
% 8.84/5.02 ( none_P5556105721700978146at_nat
% 8.84/5.02 != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_21_option_Odistinct_I1_J,axiom,
% 8.84/5.02 ! [X22: num] :
% 8.84/5.02 ( none_num
% 8.84/5.02 != ( some_num @ X22 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_22_option_OdiscI,axiom,
% 8.84/5.02 ! [Option: option_nat,X22: nat] :
% 8.84/5.02 ( ( Option
% 8.84/5.02 = ( some_nat @ X22 ) )
% 8.84/5.02 => ( Option != none_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_23_option_OdiscI,axiom,
% 8.84/5.02 ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 8.84/5.02 ( ( Option
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ X22 ) )
% 8.84/5.02 => ( Option != none_P5556105721700978146at_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_24_option_OdiscI,axiom,
% 8.84/5.02 ! [Option: option_num,X22: num] :
% 8.84/5.02 ( ( Option
% 8.84/5.02 = ( some_num @ X22 ) )
% 8.84/5.02 => ( Option != none_num ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_25_option_Oexhaust,axiom,
% 8.84/5.02 ! [Y3: option_nat] :
% 8.84/5.02 ( ( Y3 != none_nat )
% 8.84/5.02 => ~ ! [X23: nat] :
% 8.84/5.02 ( Y3
% 8.84/5.02 != ( some_nat @ X23 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_26_option_Oexhaust,axiom,
% 8.84/5.02 ! [Y3: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( Y3 != none_P5556105721700978146at_nat )
% 8.84/5.02 => ~ ! [X23: product_prod_nat_nat] :
% 8.84/5.02 ( Y3
% 8.84/5.02 != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_27_option_Oexhaust,axiom,
% 8.84/5.02 ! [Y3: option_num] :
% 8.84/5.02 ( ( Y3 != none_num )
% 8.84/5.02 => ~ ! [X23: num] :
% 8.84/5.02 ( Y3
% 8.84/5.02 != ( some_num @ X23 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_28_insert_H__pres__valid,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_29_mint__sound,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 8.84/5.02 => ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = ( some_nat @ X2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_30_mint__corr,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = ( some_nat @ X2 ) )
% 8.84/5.02 => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_31_set__vebt__set__vebt_H__valid,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( vEBT_set_vebt @ T )
% 8.84/5.02 = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_32_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 8.84/5.02 ( ( ( X2 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: nat,B: nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_33_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( ( X2 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: nat,B: product_prod_nat_nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_34_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 8.84/5.02 ( ( ( X2 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: nat,B: num] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_num @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_35_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 8.84/5.02 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: product_prod_nat_nat,B: nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_36_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_37_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 8.84/5.02 ( ( ( X2 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: product_prod_nat_nat,B: num] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_num @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_38_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 8.84/5.02 ( ( ( X2 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: num,B: nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_num @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_39_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 8.84/5.02 ( ( ( X2 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: num,B: product_prod_nat_nat] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_num @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_P7363390416028606310at_nat @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_40_combine__options__cases,axiom,
% 8.84/5.02 ! [X2: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 8.84/5.02 ( ( ( X2 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ( ( Y3 = none_num )
% 8.84/5.02 => ( P @ X2 @ Y3 ) )
% 8.84/5.02 => ( ! [A: num,B: num] :
% 8.84/5.02 ( ( X2
% 8.84/5.02 = ( some_num @ A ) )
% 8.84/5.02 => ( ( Y3
% 8.84/5.02 = ( some_num @ B ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) )
% 8.84/5.02 => ( P @ X2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_41_split__option__all,axiom,
% 8.84/5.02 ( ( ^ [P2: option_nat > $o] :
% 8.84/5.02 ! [X3: option_nat] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option_nat > $o] :
% 8.84/5.02 ( ( P3 @ none_nat )
% 8.84/5.02 & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_42_split__option__all,axiom,
% 8.84/5.02 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 8.84/5.02 ! [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 8.84/5.02 ( ( P3 @ none_P5556105721700978146at_nat )
% 8.84/5.02 & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_43_split__option__all,axiom,
% 8.84/5.02 ( ( ^ [P2: option_num > $o] :
% 8.84/5.02 ! [X3: option_num] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option_num > $o] :
% 8.84/5.02 ( ( P3 @ none_num )
% 8.84/5.02 & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_44_split__option__ex,axiom,
% 8.84/5.02 ( ( ^ [P2: option_nat > $o] :
% 8.84/5.02 ? [X3: option_nat] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option_nat > $o] :
% 8.84/5.02 ( ( P3 @ none_nat )
% 8.84/5.02 | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_45_split__option__ex,axiom,
% 8.84/5.02 ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 8.84/5.02 ? [X3: option4927543243414619207at_nat] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 8.84/5.02 ( ( P3 @ none_P5556105721700978146at_nat )
% 8.84/5.02 | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_46_split__option__ex,axiom,
% 8.84/5.02 ( ( ^ [P2: option_num > $o] :
% 8.84/5.02 ? [X3: option_num] : ( P2 @ X3 ) )
% 8.84/5.02 = ( ^ [P3: option_num > $o] :
% 8.84/5.02 ( ( P3 @ none_num )
% 8.84/5.02 | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_47_mint__corr__help__empty,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = none_nat )
% 8.84/5.02 => ( ( vEBT_VEBT_set_vebt @ T )
% 8.84/5.02 = bot_bot_set_nat ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_48_mint__member,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = ( some_nat @ Maxi ) )
% 8.84/5.02 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_49_valid__eq,axiom,
% 8.84/5.02 vEBT_VEBT_valid = vEBT_invar_vebt ).
% 8.84/5.02
% 8.84/5.02 thf(fact_50_valid__eq1,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,D: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ D )
% 8.84/5.02 => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_51_valid__eq2,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,D: nat] :
% 8.84/5.02 ( ( vEBT_VEBT_valid @ T @ D )
% 8.84/5.02 => ( vEBT_invar_vebt @ T @ D ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_52_deg__not__0,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_53_set__vebt__finite,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_54_case4_I12_J,axiom,
% 8.84/5.02 vEBT_invar_vebt @ sa @ deg ).
% 8.84/5.02
% 8.84/5.02 thf(fact_55_succ__corr,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.84/5.02 = ( some_nat @ Sx ) )
% 8.84/5.02 = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_56_pred__corr,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat,Px: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.84/5.02 = ( some_nat @ Px ) )
% 8.84/5.02 = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Px ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_57_valid__tree__deg__neq__0,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT] :
% 8.84/5.02 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_58_valid__0__not,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT] :
% 8.84/5.02 ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_59_min__Null__member,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,X2: nat] :
% 8.84/5.02 ( ( vEBT_VEBT_minNull @ T )
% 8.84/5.02 => ~ ( vEBT_vebt_member @ T @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_60_pred__none__empty,axiom,
% 8.84/5.02 ! [Xs: set_nat,A2: nat] :
% 8.84/5.02 ( ~ ? [X_1: nat] : ( vEBT_is_pred_in_set @ Xs @ A2 @ X_1 )
% 8.84/5.02 => ( ( finite_finite_nat @ Xs )
% 8.84/5.02 => ~ ? [X4: nat] :
% 8.84/5.02 ( ( member_nat @ X4 @ Xs )
% 8.84/5.02 & ( ord_less_nat @ X4 @ A2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_61_succ__none__empty,axiom,
% 8.84/5.02 ! [Xs: set_nat,A2: nat] :
% 8.84/5.02 ( ~ ? [X_1: nat] : ( vEBT_is_succ_in_set @ Xs @ A2 @ X_1 )
% 8.84/5.02 => ( ( finite_finite_nat @ Xs )
% 8.84/5.02 => ~ ? [X4: nat] :
% 8.84/5.02 ( ( member_nat @ X4 @ Xs )
% 8.84/5.02 & ( ord_less_nat @ A2 @ X4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_62_member__correct,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.84/5.02 = ( member_nat @ X2 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_63_obtain__set__pred,axiom,
% 8.84/5.02 ! [Z: nat,X2: nat,A3: set_nat] :
% 8.84/5.02 ( ( ord_less_nat @ Z @ X2 )
% 8.84/5.02 => ( ( vEBT_VEBT_min_in_set @ A3 @ Z )
% 8.84/5.02 => ( ( finite_finite_nat @ A3 )
% 8.84/5.02 => ? [X_1: nat] : ( vEBT_is_pred_in_set @ A3 @ X2 @ X_1 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_64_pred__correct,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 8.84/5.02 = ( some_nat @ Sx ) )
% 8.84/5.02 = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_65_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: $o,P: $o > $o] :
% 8.84/5.02 ( ( member_o @ A2 @ ( collect_o @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_66_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: real,P: real > $o] :
% 8.84/5.02 ( ( member_real @ A2 @ ( collect_real @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_67_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: list_nat,P: list_nat > $o] :
% 8.84/5.02 ( ( member_list_nat @ A2 @ ( collect_list_nat @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_68_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: set_nat,P: set_nat > $o] :
% 8.84/5.02 ( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_69_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: nat,P: nat > $o] :
% 8.84/5.02 ( ( member_nat @ A2 @ ( collect_nat @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_70_mem__Collect__eq,axiom,
% 8.84/5.02 ! [A2: int,P: int > $o] :
% 8.84/5.02 ( ( member_int @ A2 @ ( collect_int @ P ) )
% 8.84/5.02 = ( P @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_71_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_o] :
% 8.84/5.02 ( ( collect_o
% 8.84/5.02 @ ^ [X: $o] : ( member_o @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_72_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_real] :
% 8.84/5.02 ( ( collect_real
% 8.84/5.02 @ ^ [X: real] : ( member_real @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_73_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_list_nat] :
% 8.84/5.02 ( ( collect_list_nat
% 8.84/5.02 @ ^ [X: list_nat] : ( member_list_nat @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_74_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_set_nat] :
% 8.84/5.02 ( ( collect_set_nat
% 8.84/5.02 @ ^ [X: set_nat] : ( member_set_nat @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_75_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_nat] :
% 8.84/5.02 ( ( collect_nat
% 8.84/5.02 @ ^ [X: nat] : ( member_nat @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_76_Collect__mem__eq,axiom,
% 8.84/5.02 ! [A3: set_int] :
% 8.84/5.02 ( ( collect_int
% 8.84/5.02 @ ^ [X: int] : ( member_int @ X @ A3 ) )
% 8.84/5.02 = A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_77_Collect__cong,axiom,
% 8.84/5.02 ! [P: real > $o,Q: real > $o] :
% 8.84/5.02 ( ! [X5: real] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 = ( Q @ X5 ) )
% 8.84/5.02 => ( ( collect_real @ P )
% 8.84/5.02 = ( collect_real @ Q ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_78_Collect__cong,axiom,
% 8.84/5.02 ! [P: list_nat > $o,Q: list_nat > $o] :
% 8.84/5.02 ( ! [X5: list_nat] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 = ( Q @ X5 ) )
% 8.84/5.02 => ( ( collect_list_nat @ P )
% 8.84/5.02 = ( collect_list_nat @ Q ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_79_Collect__cong,axiom,
% 8.84/5.02 ! [P: set_nat > $o,Q: set_nat > $o] :
% 8.84/5.02 ( ! [X5: set_nat] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 = ( Q @ X5 ) )
% 8.84/5.02 => ( ( collect_set_nat @ P )
% 8.84/5.02 = ( collect_set_nat @ Q ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_80_Collect__cong,axiom,
% 8.84/5.02 ! [P: nat > $o,Q: nat > $o] :
% 8.84/5.02 ( ! [X5: nat] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 = ( Q @ X5 ) )
% 8.84/5.02 => ( ( collect_nat @ P )
% 8.84/5.02 = ( collect_nat @ Q ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_81_Collect__cong,axiom,
% 8.84/5.02 ! [P: int > $o,Q: int > $o] :
% 8.84/5.02 ( ! [X5: int] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 = ( Q @ X5 ) )
% 8.84/5.02 => ( ( collect_int @ P )
% 8.84/5.02 = ( collect_int @ Q ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_82_succ__correct,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat,Sx: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 8.84/5.02 = ( some_nat @ Sx ) )
% 8.84/5.02 = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X2 @ Sx ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_83_obtain__set__succ,axiom,
% 8.84/5.02 ! [X2: nat,Z: nat,A3: set_nat,B2: set_nat] :
% 8.84/5.02 ( ( ord_less_nat @ X2 @ Z )
% 8.84/5.02 => ( ( vEBT_VEBT_max_in_set @ A3 @ Z )
% 8.84/5.02 => ( ( finite_finite_nat @ B2 )
% 8.84/5.02 => ( ( A3 = B2 )
% 8.84/5.02 => ? [X_1: nat] : ( vEBT_is_succ_in_set @ A3 @ X2 @ X_1 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_84_buildup__gives__valid,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.84/5.02 => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_85_buildup__gives__empty,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
% 8.84/5.02 = bot_bot_set_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_86_bot__nat__0_Onot__eq__extremum,axiom,
% 8.84/5.02 ! [A2: nat] :
% 8.84/5.02 ( ( A2 != zero_zero_nat )
% 8.84/5.02 = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_87_neq0__conv,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( N != zero_zero_nat )
% 8.84/5.02 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_88_less__nat__zero__code,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_89_not__gr__zero,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.84/5.02 = ( N = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_90_succ__member,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 8.84/5.02 ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 8.84/5.02 = ( ( vEBT_vebt_member @ T @ Y3 )
% 8.84/5.02 & ( ord_less_nat @ X2 @ Y3 )
% 8.84/5.02 & ! [Z2: nat] :
% 8.84/5.02 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 8.84/5.02 & ( ord_less_nat @ X2 @ Z2 ) )
% 8.84/5.02 => ( ord_less_eq_nat @ Y3 @ Z2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_91_pred__member,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,X2: nat,Y3: nat] :
% 8.84/5.02 ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 @ Y3 )
% 8.84/5.02 = ( ( vEBT_vebt_member @ T @ Y3 )
% 8.84/5.02 & ( ord_less_nat @ Y3 @ X2 )
% 8.84/5.02 & ! [Z2: nat] :
% 8.84/5.02 ( ( ( vEBT_vebt_member @ T @ Z2 )
% 8.84/5.02 & ( ord_less_nat @ Z2 @ X2 ) )
% 8.84/5.02 => ( ord_less_eq_nat @ Z2 @ Y3 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_92_mint__corr__help,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,Mini: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_mint @ T )
% 8.84/5.02 = ( some_nat @ Mini ) )
% 8.84/5.02 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.84/5.02 => ( ord_less_eq_nat @ Mini @ X2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_93_maxt__corr__help__empty,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_maxt @ T )
% 8.84/5.02 = none_nat )
% 8.84/5.02 => ( ( vEBT_VEBT_set_vebt @ T )
% 8.84/5.02 = bot_bot_set_nat ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_94_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_Extended_enat] :
% 8.84/5.02 ( ( finite4001608067531595151d_enat @ S )
% 8.84/5.02 => ( ( S != bot_bo7653980558646680370d_enat )
% 8.84/5.02 => ? [X5: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ Xa @ S )
% 8.84/5.02 & ( ord_le72135733267957522d_enat @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_95_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_o] :
% 8.84/5.02 ( ( finite_finite_o @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_o )
% 8.84/5.02 => ? [X5: $o] :
% 8.84/5.02 ( ( member_o @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: $o] :
% 8.84/5.02 ( ( member_o @ Xa @ S )
% 8.84/5.02 & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_96_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_real] :
% 8.84/5.02 ( ( finite_finite_real @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_real )
% 8.84/5.02 => ? [X5: real] :
% 8.84/5.02 ( ( member_real @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: real] :
% 8.84/5.02 ( ( member_real @ Xa @ S )
% 8.84/5.02 & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_97_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_rat] :
% 8.84/5.02 ( ( finite_finite_rat @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_rat )
% 8.84/5.02 => ? [X5: rat] :
% 8.84/5.02 ( ( member_rat @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: rat] :
% 8.84/5.02 ( ( member_rat @ Xa @ S )
% 8.84/5.02 & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_98_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_num] :
% 8.84/5.02 ( ( finite_finite_num @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_num )
% 8.84/5.02 => ? [X5: num] :
% 8.84/5.02 ( ( member_num @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: num] :
% 8.84/5.02 ( ( member_num @ Xa @ S )
% 8.84/5.02 & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_99_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_nat] :
% 8.84/5.02 ( ( finite_finite_nat @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_nat )
% 8.84/5.02 => ? [X5: nat] :
% 8.84/5.02 ( ( member_nat @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: nat] :
% 8.84/5.02 ( ( member_nat @ Xa @ S )
% 8.84/5.02 & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_100_ex__min__if__finite,axiom,
% 8.84/5.02 ! [S: set_int] :
% 8.84/5.02 ( ( finite_finite_int @ S )
% 8.84/5.02 => ( ( S != bot_bot_set_int )
% 8.84/5.02 => ? [X5: int] :
% 8.84/5.02 ( ( member_int @ X5 @ S )
% 8.84/5.02 & ~ ? [Xa: int] :
% 8.84/5.02 ( ( member_int @ Xa @ S )
% 8.84/5.02 & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_101_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_Extended_enat] :
% 8.84/5.02 ( ( X6 != bot_bo7653980558646680370d_enat )
% 8.84/5.02 => ( ! [X5: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ Xa @ X6 )
% 8.84/5.02 & ( ord_le72135733267957522d_enat @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite4001608067531595151d_enat @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_102_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_o] :
% 8.84/5.02 ( ( X6 != bot_bot_set_o )
% 8.84/5.02 => ( ! [X5: $o] :
% 8.84/5.02 ( ( member_o @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: $o] :
% 8.84/5.02 ( ( member_o @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_o @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_o @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_103_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_real] :
% 8.84/5.02 ( ( X6 != bot_bot_set_real )
% 8.84/5.02 => ( ! [X5: real] :
% 8.84/5.02 ( ( member_real @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: real] :
% 8.84/5.02 ( ( member_real @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_real @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_real @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_104_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_rat] :
% 8.84/5.02 ( ( X6 != bot_bot_set_rat )
% 8.84/5.02 => ( ! [X5: rat] :
% 8.84/5.02 ( ( member_rat @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: rat] :
% 8.84/5.02 ( ( member_rat @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_rat @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_rat @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_105_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_num] :
% 8.84/5.02 ( ( X6 != bot_bot_set_num )
% 8.84/5.02 => ( ! [X5: num] :
% 8.84/5.02 ( ( member_num @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: num] :
% 8.84/5.02 ( ( member_num @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_num @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_num @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_106_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_nat] :
% 8.84/5.02 ( ( X6 != bot_bot_set_nat )
% 8.84/5.02 => ( ! [X5: nat] :
% 8.84/5.02 ( ( member_nat @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: nat] :
% 8.84/5.02 ( ( member_nat @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_nat @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_nat @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_107_infinite__growing,axiom,
% 8.84/5.02 ! [X6: set_int] :
% 8.84/5.02 ( ( X6 != bot_bot_set_int )
% 8.84/5.02 => ( ! [X5: int] :
% 8.84/5.02 ( ( member_int @ X5 @ X6 )
% 8.84/5.02 => ? [Xa: int] :
% 8.84/5.02 ( ( member_int @ Xa @ X6 )
% 8.84/5.02 & ( ord_less_int @ X5 @ Xa ) ) )
% 8.84/5.02 => ~ ( finite_finite_int @ X6 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_108_max__in__set__def,axiom,
% 8.84/5.02 ( vEBT_VEBT_max_in_set
% 8.84/5.02 = ( ^ [Xs2: set_nat,X: nat] :
% 8.84/5.02 ( ( member_nat @ X @ Xs2 )
% 8.84/5.02 & ! [Y: nat] :
% 8.84/5.02 ( ( member_nat @ Y @ Xs2 )
% 8.84/5.02 => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_109_min__in__set__def,axiom,
% 8.84/5.02 ( vEBT_VEBT_min_in_set
% 8.84/5.02 = ( ^ [Xs2: set_nat,X: nat] :
% 8.84/5.02 ( ( member_nat @ X @ Xs2 )
% 8.84/5.02 & ! [Y: nat] :
% 8.84/5.02 ( ( member_nat @ Y @ Xs2 )
% 8.84/5.02 => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_110_maxt__member,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,Maxi: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_maxt @ T )
% 8.84/5.02 = ( some_nat @ Maxi ) )
% 8.84/5.02 => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_111_maxt__corr__help,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,Maxi: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_maxt @ T )
% 8.84/5.02 = ( some_nat @ Maxi ) )
% 8.84/5.02 => ( ( vEBT_vebt_member @ T @ X2 )
% 8.84/5.02 => ( ord_less_eq_nat @ X2 @ Maxi ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_112_maxt__corr,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( ( vEBT_vebt_maxt @ T )
% 8.84/5.02 = ( some_nat @ X2 ) )
% 8.84/5.02 => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_113_maxt__sound,axiom,
% 8.84/5.02 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 8.84/5.02 ( ( vEBT_invar_vebt @ T @ N )
% 8.84/5.02 => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X2 )
% 8.84/5.02 => ( ( vEBT_vebt_maxt @ T )
% 8.84/5.02 = ( some_nat @ X2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_114_le__zero__eq,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.84/5.02 = ( N = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_115_bot__nat__0_Oextremum,axiom,
% 8.84/5.02 ! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_116_le0,axiom,
% 8.84/5.02 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_117_lesseq__shift,axiom,
% 8.84/5.02 ( ord_less_eq_nat
% 8.84/5.02 = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_118_le__refl,axiom,
% 8.84/5.02 ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_119_le__trans,axiom,
% 8.84/5.02 ! [I: nat,J: nat,K: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ I @ J )
% 8.84/5.02 => ( ( ord_less_eq_nat @ J @ K )
% 8.84/5.02 => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_120_eq__imp__le,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( M = N )
% 8.84/5.02 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_121_le__antisym,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ M @ N )
% 8.84/5.02 => ( ( ord_less_eq_nat @ N @ M )
% 8.84/5.02 => ( M = N ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_122_nat__le__linear,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ M @ N )
% 8.84/5.02 | ( ord_less_eq_nat @ N @ M ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_123_Nat_Oex__has__greatest__nat,axiom,
% 8.84/5.02 ! [P: nat > $o,K: nat,B3: nat] :
% 8.84/5.02 ( ( P @ K )
% 8.84/5.02 => ( ! [Y4: nat] :
% 8.84/5.02 ( ( P @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 8.84/5.02 => ? [X5: nat] :
% 8.84/5.02 ( ( P @ X5 )
% 8.84/5.02 & ! [Y5: nat] :
% 8.84/5.02 ( ( P @ Y5 )
% 8.84/5.02 => ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_124_zero__le,axiom,
% 8.84/5.02 ! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_125_less__eq__nat_Osimps_I1_J,axiom,
% 8.84/5.02 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_126_bot__nat__0_Oextremum__unique,axiom,
% 8.84/5.02 ! [A2: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 8.84/5.02 = ( A2 = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_127_bot__nat__0_Oextremum__uniqueI,axiom,
% 8.84/5.02 ! [A2: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 8.84/5.02 => ( A2 = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_128_le__0__eq,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ N @ zero_zero_nat )
% 8.84/5.02 = ( N = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_129_less__mono__imp__le__mono,axiom,
% 8.84/5.02 ! [F: nat > nat,I: nat,J: nat] :
% 8.84/5.02 ( ! [I2: nat,J2: nat] :
% 8.84/5.02 ( ( ord_less_nat @ I2 @ J2 )
% 8.84/5.02 => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
% 8.84/5.02 => ( ( ord_less_eq_nat @ I @ J )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_130_le__neq__implies__less,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ M @ N )
% 8.84/5.02 => ( ( M != N )
% 8.84/5.02 => ( ord_less_nat @ M @ N ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_131_less__or__eq__imp__le,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ( ord_less_nat @ M @ N )
% 8.84/5.02 | ( M = N ) )
% 8.84/5.02 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_132_le__eq__less__or__eq,axiom,
% 8.84/5.02 ( ord_less_eq_nat
% 8.84/5.02 = ( ^ [M2: nat,N2: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M2 @ N2 )
% 8.84/5.02 | ( M2 = N2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_133_less__imp__le__nat,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M @ N )
% 8.84/5.02 => ( ord_less_eq_nat @ M @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_134_nat__less__le,axiom,
% 8.84/5.02 ( ord_less_nat
% 8.84/5.02 = ( ^ [M2: nat,N2: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ M2 @ N2 )
% 8.84/5.02 & ( M2 != N2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_135_ex__least__nat__le,axiom,
% 8.84/5.02 ! [P: nat > $o,N: nat] :
% 8.84/5.02 ( ( P @ N )
% 8.84/5.02 => ( ~ ( P @ zero_zero_nat )
% 8.84/5.02 => ? [K2: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ K2 @ N )
% 8.84/5.02 & ! [I3: nat] :
% 8.84/5.02 ( ( ord_less_nat @ I3 @ K2 )
% 8.84/5.02 => ~ ( P @ I3 ) )
% 8.84/5.02 & ( P @ K2 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_136_zero__reorient,axiom,
% 8.84/5.02 ! [X2: literal] :
% 8.84/5.02 ( ( zero_zero_literal = X2 )
% 8.84/5.02 = ( X2 = zero_zero_literal ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_137_zero__reorient,axiom,
% 8.84/5.02 ! [X2: real] :
% 8.84/5.02 ( ( zero_zero_real = X2 )
% 8.84/5.02 = ( X2 = zero_zero_real ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_138_zero__reorient,axiom,
% 8.84/5.02 ! [X2: rat] :
% 8.84/5.02 ( ( zero_zero_rat = X2 )
% 8.84/5.02 = ( X2 = zero_zero_rat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_139_zero__reorient,axiom,
% 8.84/5.02 ! [X2: nat] :
% 8.84/5.02 ( ( zero_zero_nat = X2 )
% 8.84/5.02 = ( X2 = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_140_zero__reorient,axiom,
% 8.84/5.02 ! [X2: int] :
% 8.84/5.02 ( ( zero_zero_int = X2 )
% 8.84/5.02 = ( X2 = zero_zero_int ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_141_linorder__neqE__nat,axiom,
% 8.84/5.02 ! [X2: nat,Y3: nat] :
% 8.84/5.02 ( ( X2 != Y3 )
% 8.84/5.02 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 8.84/5.02 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_142_infinite__descent,axiom,
% 8.84/5.02 ! [P: nat > $o,N: nat] :
% 8.84/5.02 ( ! [N3: nat] :
% 8.84/5.02 ( ~ ( P @ N3 )
% 8.84/5.02 => ? [M3: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M3 @ N3 )
% 8.84/5.02 & ~ ( P @ M3 ) ) )
% 8.84/5.02 => ( P @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_143_nat__less__induct,axiom,
% 8.84/5.02 ! [P: nat > $o,N: nat] :
% 8.84/5.02 ( ! [N3: nat] :
% 8.84/5.02 ( ! [M3: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M3 @ N3 )
% 8.84/5.02 => ( P @ M3 ) )
% 8.84/5.02 => ( P @ N3 ) )
% 8.84/5.02 => ( P @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_144_less__irrefl__nat,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ N ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_145_less__not__refl3,axiom,
% 8.84/5.02 ! [S2: nat,T: nat] :
% 8.84/5.02 ( ( ord_less_nat @ S2 @ T )
% 8.84/5.02 => ( S2 != T ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_146_less__not__refl2,axiom,
% 8.84/5.02 ! [N: nat,M: nat] :
% 8.84/5.02 ( ( ord_less_nat @ N @ M )
% 8.84/5.02 => ( M != N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_147_less__not__refl,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ N ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_148_nat__neq__iff,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( M != N )
% 8.84/5.02 = ( ( ord_less_nat @ M @ N )
% 8.84/5.02 | ( ord_less_nat @ N @ M ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_149_zero__less__iff__neq__zero,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ord_less_nat @ zero_zero_nat @ N )
% 8.84/5.02 = ( N != zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_150_gr__implies__not__zero,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M @ N )
% 8.84/5.02 => ( N != zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_151_not__less__zero,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_152_gr__zeroI,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( N != zero_zero_nat )
% 8.84/5.02 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_153_infinite__descent0,axiom,
% 8.84/5.02 ! [P: nat > $o,N: nat] :
% 8.84/5.02 ( ( P @ zero_zero_nat )
% 8.84/5.02 => ( ! [N3: nat] :
% 8.84/5.02 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 8.84/5.02 => ( ~ ( P @ N3 )
% 8.84/5.02 => ? [M3: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M3 @ N3 )
% 8.84/5.02 & ~ ( P @ M3 ) ) ) )
% 8.84/5.02 => ( P @ N ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_154_gr__implies__not0,axiom,
% 8.84/5.02 ! [M: nat,N: nat] :
% 8.84/5.02 ( ( ord_less_nat @ M @ N )
% 8.84/5.02 => ( N != zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_155_less__zeroE,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_156_not__less0,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_157_not__gr0,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
% 8.84/5.02 = ( N = zero_zero_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_158_gr0I,axiom,
% 8.84/5.02 ! [N: nat] :
% 8.84/5.02 ( ( N != zero_zero_nat )
% 8.84/5.02 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_159_bot__nat__0_Oextremum__strict,axiom,
% 8.84/5.02 ! [A2: nat] :
% 8.84/5.02 ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_160_buildup__nothing__in__leaf,axiom,
% 8.84/5.02 ! [N: nat,X2: nat] :
% 8.84/5.02 ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_161_is__succ__in__set__def,axiom,
% 8.84/5.02 ( vEBT_is_succ_in_set
% 8.84/5.02 = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
% 8.84/5.02 ( ( member_nat @ Y @ Xs2 )
% 8.84/5.02 & ( ord_less_nat @ X @ Y )
% 8.84/5.02 & ! [Z2: nat] :
% 8.84/5.02 ( ( member_nat @ Z2 @ Xs2 )
% 8.84/5.02 => ( ( ord_less_nat @ X @ Z2 )
% 8.84/5.02 => ( ord_less_eq_nat @ Y @ Z2 ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_162_is__pred__in__set__def,axiom,
% 8.84/5.02 ( vEBT_is_pred_in_set
% 8.84/5.02 = ( ^ [Xs2: set_nat,X: nat,Y: nat] :
% 8.84/5.02 ( ( member_nat @ Y @ Xs2 )
% 8.84/5.02 & ( ord_less_nat @ Y @ X )
% 8.84/5.02 & ! [Z2: nat] :
% 8.84/5.02 ( ( member_nat @ Z2 @ Xs2 )
% 8.84/5.02 => ( ( ord_less_nat @ Z2 @ X )
% 8.84/5.02 => ( ord_less_eq_nat @ Z2 @ Y ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_163_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_Extended_enat] :
% 8.84/5.02 ( ( finite4001608067531595151d_enat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 8.84/5.02 => ? [X5: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_164_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_real] :
% 8.84/5.02 ( ( finite_finite_real @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_real )
% 8.84/5.02 => ? [X5: real] :
% 8.84/5.02 ( ( member_real @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: real] :
% 8.84/5.02 ( ( member_real @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_real @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_165_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_o] :
% 8.84/5.02 ( ( finite_finite_o @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_o )
% 8.84/5.02 => ? [X5: $o] :
% 8.84/5.02 ( ( member_o @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: $o] :
% 8.84/5.02 ( ( member_o @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_o @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_166_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_set_int] :
% 8.84/5.02 ( ( finite6197958912794628473et_int @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_set_int )
% 8.84/5.02 => ? [X5: set_int] :
% 8.84/5.02 ( ( member_set_int @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: set_int] :
% 8.84/5.02 ( ( member_set_int @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_167_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_rat] :
% 8.84/5.02 ( ( finite_finite_rat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_rat )
% 8.84/5.02 => ? [X5: rat] :
% 8.84/5.02 ( ( member_rat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: rat] :
% 8.84/5.02 ( ( member_rat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_168_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_num] :
% 8.84/5.02 ( ( finite_finite_num @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_num )
% 8.84/5.02 => ? [X5: num] :
% 8.84/5.02 ( ( member_num @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: num] :
% 8.84/5.02 ( ( member_num @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_169_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_nat] :
% 8.84/5.02 ( ( finite_finite_nat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_nat )
% 8.84/5.02 => ? [X5: nat] :
% 8.84/5.02 ( ( member_nat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: nat] :
% 8.84/5.02 ( ( member_nat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_170_finite__has__maximal,axiom,
% 8.84/5.02 ! [A3: set_int] :
% 8.84/5.02 ( ( finite_finite_int @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_int )
% 8.84/5.02 => ? [X5: int] :
% 8.84/5.02 ( ( member_int @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: int] :
% 8.84/5.02 ( ( member_int @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_int @ X5 @ Xa )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_171_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_Extended_enat] :
% 8.84/5.02 ( ( finite4001608067531595151d_enat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 8.84/5.02 => ? [X5: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: extended_enat] :
% 8.84/5.02 ( ( member_Extended_enat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_172_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_real] :
% 8.84/5.02 ( ( finite_finite_real @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_real )
% 8.84/5.02 => ? [X5: real] :
% 8.84/5.02 ( ( member_real @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: real] :
% 8.84/5.02 ( ( member_real @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_real @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_173_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_o] :
% 8.84/5.02 ( ( finite_finite_o @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_o )
% 8.84/5.02 => ? [X5: $o] :
% 8.84/5.02 ( ( member_o @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: $o] :
% 8.84/5.02 ( ( member_o @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_o @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_174_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_set_int] :
% 8.84/5.02 ( ( finite6197958912794628473et_int @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_set_int )
% 8.84/5.02 => ? [X5: set_int] :
% 8.84/5.02 ( ( member_set_int @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: set_int] :
% 8.84/5.02 ( ( member_set_int @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_175_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_rat] :
% 8.84/5.02 ( ( finite_finite_rat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_rat )
% 8.84/5.02 => ? [X5: rat] :
% 8.84/5.02 ( ( member_rat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: rat] :
% 8.84/5.02 ( ( member_rat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_176_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_num] :
% 8.84/5.02 ( ( finite_finite_num @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_num )
% 8.84/5.02 => ? [X5: num] :
% 8.84/5.02 ( ( member_num @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: num] :
% 8.84/5.02 ( ( member_num @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_177_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_nat] :
% 8.84/5.02 ( ( finite_finite_nat @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_nat )
% 8.84/5.02 => ? [X5: nat] :
% 8.84/5.02 ( ( member_nat @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: nat] :
% 8.84/5.02 ( ( member_nat @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_178_finite__has__minimal,axiom,
% 8.84/5.02 ! [A3: set_int] :
% 8.84/5.02 ( ( finite_finite_int @ A3 )
% 8.84/5.02 => ( ( A3 != bot_bot_set_int )
% 8.84/5.02 => ? [X5: int] :
% 8.84/5.02 ( ( member_int @ X5 @ A3 )
% 8.84/5.02 & ! [Xa: int] :
% 8.84/5.02 ( ( member_int @ Xa @ A3 )
% 8.84/5.02 => ( ( ord_less_eq_int @ Xa @ X5 )
% 8.84/5.02 => ( X5 = Xa ) ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_179_empty__iff,axiom,
% 8.84/5.02 ! [C: set_nat] :
% 8.84/5.02 ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_180_empty__iff,axiom,
% 8.84/5.02 ! [C: real] :
% 8.84/5.02 ~ ( member_real @ C @ bot_bot_set_real ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_181_empty__iff,axiom,
% 8.84/5.02 ! [C: $o] :
% 8.84/5.02 ~ ( member_o @ C @ bot_bot_set_o ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_182_empty__iff,axiom,
% 8.84/5.02 ! [C: nat] :
% 8.84/5.02 ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_183_empty__iff,axiom,
% 8.84/5.02 ! [C: int] :
% 8.84/5.02 ~ ( member_int @ C @ bot_bot_set_int ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_184_all__not__in__conv,axiom,
% 8.84/5.02 ! [A3: set_set_nat] :
% 8.84/5.02 ( ( ! [X: set_nat] :
% 8.84/5.02 ~ ( member_set_nat @ X @ A3 ) )
% 8.84/5.02 = ( A3 = bot_bot_set_set_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_185_all__not__in__conv,axiom,
% 8.84/5.02 ! [A3: set_real] :
% 8.84/5.02 ( ( ! [X: real] :
% 8.84/5.02 ~ ( member_real @ X @ A3 ) )
% 8.84/5.02 = ( A3 = bot_bot_set_real ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_186_all__not__in__conv,axiom,
% 8.84/5.02 ! [A3: set_o] :
% 8.84/5.02 ( ( ! [X: $o] :
% 8.84/5.02 ~ ( member_o @ X @ A3 ) )
% 8.84/5.02 = ( A3 = bot_bot_set_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_187_all__not__in__conv,axiom,
% 8.84/5.02 ! [A3: set_nat] :
% 8.84/5.02 ( ( ! [X: nat] :
% 8.84/5.02 ~ ( member_nat @ X @ A3 ) )
% 8.84/5.02 = ( A3 = bot_bot_set_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_188_all__not__in__conv,axiom,
% 8.84/5.02 ! [A3: set_int] :
% 8.84/5.02 ( ( ! [X: int] :
% 8.84/5.02 ~ ( member_int @ X @ A3 ) )
% 8.84/5.02 = ( A3 = bot_bot_set_int ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_189_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: list_nat > $o] :
% 8.84/5.02 ( ( ( collect_list_nat @ P )
% 8.84/5.02 = bot_bot_set_list_nat )
% 8.84/5.02 = ( ! [X: list_nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_190_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: set_nat > $o] :
% 8.84/5.02 ( ( ( collect_set_nat @ P )
% 8.84/5.02 = bot_bot_set_set_nat )
% 8.84/5.02 = ( ! [X: set_nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_191_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: real > $o] :
% 8.84/5.02 ( ( ( collect_real @ P )
% 8.84/5.02 = bot_bot_set_real )
% 8.84/5.02 = ( ! [X: real] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_192_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: $o > $o] :
% 8.84/5.02 ( ( ( collect_o @ P )
% 8.84/5.02 = bot_bot_set_o )
% 8.84/5.02 = ( ! [X: $o] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_193_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: nat > $o] :
% 8.84/5.02 ( ( ( collect_nat @ P )
% 8.84/5.02 = bot_bot_set_nat )
% 8.84/5.02 = ( ! [X: nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_194_Collect__empty__eq,axiom,
% 8.84/5.02 ! [P: int > $o] :
% 8.84/5.02 ( ( ( collect_int @ P )
% 8.84/5.02 = bot_bot_set_int )
% 8.84/5.02 = ( ! [X: int] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_195_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: list_nat > $o] :
% 8.84/5.02 ( ( bot_bot_set_list_nat
% 8.84/5.02 = ( collect_list_nat @ P ) )
% 8.84/5.02 = ( ! [X: list_nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_196_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: set_nat > $o] :
% 8.84/5.02 ( ( bot_bot_set_set_nat
% 8.84/5.02 = ( collect_set_nat @ P ) )
% 8.84/5.02 = ( ! [X: set_nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_197_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: real > $o] :
% 8.84/5.02 ( ( bot_bot_set_real
% 8.84/5.02 = ( collect_real @ P ) )
% 8.84/5.02 = ( ! [X: real] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_198_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: $o > $o] :
% 8.84/5.02 ( ( bot_bot_set_o
% 8.84/5.02 = ( collect_o @ P ) )
% 8.84/5.02 = ( ! [X: $o] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_199_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: nat > $o] :
% 8.84/5.02 ( ( bot_bot_set_nat
% 8.84/5.02 = ( collect_nat @ P ) )
% 8.84/5.02 = ( ! [X: nat] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_200_empty__Collect__eq,axiom,
% 8.84/5.02 ! [P: int > $o] :
% 8.84/5.02 ( ( bot_bot_set_int
% 8.84/5.02 = ( collect_int @ P ) )
% 8.84/5.02 = ( ! [X: int] :
% 8.84/5.02 ~ ( P @ X ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_201_order__refl,axiom,
% 8.84/5.02 ! [X2: set_int] : ( ord_less_eq_set_int @ X2 @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_202_order__refl,axiom,
% 8.84/5.02 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_203_order__refl,axiom,
% 8.84/5.02 ! [X2: num] : ( ord_less_eq_num @ X2 @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_204_order__refl,axiom,
% 8.84/5.02 ! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_205_order__refl,axiom,
% 8.84/5.02 ! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_206_dual__order_Orefl,axiom,
% 8.84/5.02 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_207_dual__order_Orefl,axiom,
% 8.84/5.02 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_208_dual__order_Orefl,axiom,
% 8.84/5.02 ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_209_dual__order_Orefl,axiom,
% 8.84/5.02 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_210_dual__order_Orefl,axiom,
% 8.84/5.02 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_211_empty__subsetI,axiom,
% 8.84/5.02 ! [A3: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_212_empty__subsetI,axiom,
% 8.84/5.02 ! [A3: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_213_empty__subsetI,axiom,
% 8.84/5.02 ! [A3: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_214_empty__subsetI,axiom,
% 8.84/5.02 ! [A3: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A3 ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_215_subset__empty,axiom,
% 8.84/5.02 ! [A3: set_real] :
% 8.84/5.02 ( ( ord_less_eq_set_real @ A3 @ bot_bot_set_real )
% 8.84/5.02 = ( A3 = bot_bot_set_real ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_216_subset__empty,axiom,
% 8.84/5.02 ! [A3: set_o] :
% 8.84/5.02 ( ( ord_less_eq_set_o @ A3 @ bot_bot_set_o )
% 8.84/5.02 = ( A3 = bot_bot_set_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_217_subset__empty,axiom,
% 8.84/5.02 ! [A3: set_nat] :
% 8.84/5.02 ( ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat )
% 8.84/5.02 = ( A3 = bot_bot_set_nat ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_218_subset__empty,axiom,
% 8.84/5.02 ! [A3: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ A3 @ bot_bot_set_int )
% 8.84/5.02 = ( A3 = bot_bot_set_int ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_219_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_list_nat
% 8.84/5.02 = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_220_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_set_nat
% 8.84/5.02 = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_221_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_real
% 8.84/5.02 = ( collect_real @ bot_bot_real_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_222_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_o
% 8.84/5.02 = ( collect_o @ bot_bot_o_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_223_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_nat
% 8.84/5.02 = ( collect_nat @ bot_bot_nat_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_224_bot__set__def,axiom,
% 8.84/5.02 ( bot_bot_set_int
% 8.84/5.02 = ( collect_int @ bot_bot_int_o ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_225_rev__finite__subset,axiom,
% 8.84/5.02 ! [B2: set_nat,A3: set_nat] :
% 8.84/5.02 ( ( finite_finite_nat @ B2 )
% 8.84/5.02 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 8.84/5.02 => ( finite_finite_nat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_226_rev__finite__subset,axiom,
% 8.84/5.02 ! [B2: set_complex,A3: set_complex] :
% 8.84/5.02 ( ( finite3207457112153483333omplex @ B2 )
% 8.84/5.02 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 8.84/5.02 => ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_227_rev__finite__subset,axiom,
% 8.84/5.02 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 8.84/5.02 ( ( finite6177210948735845034at_nat @ B2 )
% 8.84/5.02 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 8.84/5.02 => ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_228_rev__finite__subset,axiom,
% 8.84/5.02 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 8.84/5.02 ( ( finite4001608067531595151d_enat @ B2 )
% 8.84/5.02 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 8.84/5.02 => ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_229_rev__finite__subset,axiom,
% 8.84/5.02 ! [B2: set_int,A3: set_int] :
% 8.84/5.02 ( ( finite_finite_int @ B2 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 8.84/5.02 => ( finite_finite_int @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_230_infinite__super,axiom,
% 8.84/5.02 ! [S: set_nat,T2: set_nat] :
% 8.84/5.02 ( ( ord_less_eq_set_nat @ S @ T2 )
% 8.84/5.02 => ( ~ ( finite_finite_nat @ S )
% 8.84/5.02 => ~ ( finite_finite_nat @ T2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_231_infinite__super,axiom,
% 8.84/5.02 ! [S: set_complex,T2: set_complex] :
% 8.84/5.02 ( ( ord_le211207098394363844omplex @ S @ T2 )
% 8.84/5.02 => ( ~ ( finite3207457112153483333omplex @ S )
% 8.84/5.02 => ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_232_infinite__super,axiom,
% 8.84/5.02 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 8.84/5.02 ( ( ord_le3146513528884898305at_nat @ S @ T2 )
% 8.84/5.02 => ( ~ ( finite6177210948735845034at_nat @ S )
% 8.84/5.02 => ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_233_infinite__super,axiom,
% 8.84/5.02 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 8.84/5.02 ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 8.84/5.02 => ( ~ ( finite4001608067531595151d_enat @ S )
% 8.84/5.02 => ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_234_infinite__super,axiom,
% 8.84/5.02 ! [S: set_int,T2: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ S @ T2 )
% 8.84/5.02 => ( ~ ( finite_finite_int @ S )
% 8.84/5.02 => ~ ( finite_finite_int @ T2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_235_finite__subset,axiom,
% 8.84/5.02 ! [A3: set_nat,B2: set_nat] :
% 8.84/5.02 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 8.84/5.02 => ( ( finite_finite_nat @ B2 )
% 8.84/5.02 => ( finite_finite_nat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_236_finite__subset,axiom,
% 8.84/5.02 ! [A3: set_complex,B2: set_complex] :
% 8.84/5.02 ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 8.84/5.02 => ( ( finite3207457112153483333omplex @ B2 )
% 8.84/5.02 => ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_237_finite__subset,axiom,
% 8.84/5.02 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 8.84/5.02 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 8.84/5.02 => ( ( finite6177210948735845034at_nat @ B2 )
% 8.84/5.02 => ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_238_finite__subset,axiom,
% 8.84/5.02 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 8.84/5.02 ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 8.84/5.02 => ( ( finite4001608067531595151d_enat @ B2 )
% 8.84/5.02 => ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_239_finite__subset,axiom,
% 8.84/5.02 ! [A3: set_int,B2: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 8.84/5.02 => ( ( finite_finite_int @ B2 )
% 8.84/5.02 => ( finite_finite_int @ A3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_240_bot__nat__def,axiom,
% 8.84/5.02 bot_bot_nat = zero_zero_nat ).
% 8.84/5.02
% 8.84/5.02 thf(fact_241_order__antisym__conv,axiom,
% 8.84/5.02 ! [Y3: set_int,X2: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 8.84/5.02 = ( X2 = Y3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_242_order__antisym__conv,axiom,
% 8.84/5.02 ! [Y3: rat,X2: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.84/5.02 = ( X2 = Y3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_243_order__antisym__conv,axiom,
% 8.84/5.02 ! [Y3: num,X2: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ Y3 @ X2 )
% 8.84/5.02 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.84/5.02 = ( X2 = Y3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_244_order__antisym__conv,axiom,
% 8.84/5.02 ! [Y3: nat,X2: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.84/5.02 = ( X2 = Y3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_245_order__antisym__conv,axiom,
% 8.84/5.02 ! [Y3: int,X2: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ Y3 @ X2 )
% 8.84/5.02 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.84/5.02 = ( X2 = Y3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_246_linorder__le__cases,axiom,
% 8.84/5.02 ! [X2: rat,Y3: rat] :
% 8.84/5.02 ( ~ ( ord_less_eq_rat @ X2 @ Y3 )
% 8.84/5.02 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_247_linorder__le__cases,axiom,
% 8.84/5.02 ! [X2: num,Y3: num] :
% 8.84/5.02 ( ~ ( ord_less_eq_num @ X2 @ Y3 )
% 8.84/5.02 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_248_linorder__le__cases,axiom,
% 8.84/5.02 ! [X2: nat,Y3: nat] :
% 8.84/5.02 ( ~ ( ord_less_eq_nat @ X2 @ Y3 )
% 8.84/5.02 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_249_linorder__le__cases,axiom,
% 8.84/5.02 ! [X2: int,Y3: int] :
% 8.84/5.02 ( ~ ( ord_less_eq_int @ X2 @ Y3 )
% 8.84/5.02 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_250_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_251_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_252_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_253_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_254_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_255_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_256_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_257_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_258_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_259_ord__le__eq__subst,axiom,
% 8.84/5.02 ! [A2: nat,B3: nat,F: nat > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ B3 )
% 8.84/5.02 => ( ( ( F @ B3 )
% 8.84/5.02 = C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_260_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_261_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_262_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_263_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_264_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_265_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: num,F: num > num,B3: num,C: num] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_266_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_267_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: int,F: num > int,B3: num,C: num] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_268_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_nat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_269_ord__eq__le__subst,axiom,
% 8.84/5.02 ! [A2: num,F: nat > num,B3: nat,C: nat] :
% 8.84/5.02 ( ( A2
% 8.84/5.02 = ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_nat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_270_linorder__linear,axiom,
% 8.84/5.02 ! [X2: rat,Y3: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 8.84/5.02 | ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_271_linorder__linear,axiom,
% 8.84/5.02 ! [X2: num,Y3: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X2 @ Y3 )
% 8.84/5.02 | ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_272_linorder__linear,axiom,
% 8.84/5.02 ! [X2: nat,Y3: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 8.84/5.02 | ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_273_linorder__linear,axiom,
% 8.84/5.02 ! [X2: int,Y3: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ X2 @ Y3 )
% 8.84/5.02 | ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_274_order__eq__refl,axiom,
% 8.84/5.02 ! [X2: set_int,Y3: set_int] :
% 8.84/5.02 ( ( X2 = Y3 )
% 8.84/5.02 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_275_order__eq__refl,axiom,
% 8.84/5.02 ! [X2: rat,Y3: rat] :
% 8.84/5.02 ( ( X2 = Y3 )
% 8.84/5.02 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_276_order__eq__refl,axiom,
% 8.84/5.02 ! [X2: num,Y3: num] :
% 8.84/5.02 ( ( X2 = Y3 )
% 8.84/5.02 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_277_order__eq__refl,axiom,
% 8.84/5.02 ! [X2: nat,Y3: nat] :
% 8.84/5.02 ( ( X2 = Y3 )
% 8.84/5.02 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_278_order__eq__refl,axiom,
% 8.84/5.02 ! [X2: int,Y3: int] :
% 8.84/5.02 ( ( X2 = Y3 )
% 8.84/5.02 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_279_order__subst2,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_280_order__subst2,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_281_order__subst2,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_282_order__subst2,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_283_order__subst2,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_284_order__subst2,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_285_order__subst2,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_286_order__subst2,axiom,
% 8.84/5.02 ! [A2: num,B3: num,F: num > int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_int @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_287_order__subst2,axiom,
% 8.84/5.02 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_288_order__subst2,axiom,
% 8.84/5.02 ! [A2: nat,B3: nat,F: nat > num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ ( F @ B3 ) @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_289_order__subst1,axiom,
% 8.84/5.02 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_290_order__subst1,axiom,
% 8.84/5.02 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_291_order__subst1,axiom,
% 8.84/5.02 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_nat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_292_order__subst1,axiom,
% 8.84/5.02 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_int @ B3 @ C )
% 8.84/5.02 => ( ! [X5: int,Y4: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_293_order__subst1,axiom,
% 8.84/5.02 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_294_order__subst1,axiom,
% 8.84/5.02 ! [A2: num,F: num > num,B3: num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_295_order__subst1,axiom,
% 8.84/5.02 ! [A2: num,F: nat > num,B3: nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_nat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: nat,Y4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_296_order__subst1,axiom,
% 8.84/5.02 ! [A2: num,F: int > num,B3: int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_int @ B3 @ C )
% 8.84/5.02 => ( ! [X5: int,Y4: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_297_order__subst1,axiom,
% 8.84/5.02 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ C )
% 8.84/5.02 => ( ! [X5: rat,Y4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_298_order__subst1,axiom,
% 8.84/5.02 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ ( F @ B3 ) )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ C )
% 8.84/5.02 => ( ! [X5: num,Y4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ X5 @ Y4 )
% 8.84/5.02 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 8.84/5.02 => ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_299_Orderings_Oorder__eq__iff,axiom,
% 8.84/5.02 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 8.84/5.02 = ( ^ [A4: set_int,B4: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 8.84/5.02 & ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_300_Orderings_Oorder__eq__iff,axiom,
% 8.84/5.02 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 8.84/5.02 = ( ^ [A4: rat,B4: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A4 @ B4 )
% 8.84/5.02 & ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_301_Orderings_Oorder__eq__iff,axiom,
% 8.84/5.02 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 8.84/5.02 = ( ^ [A4: num,B4: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ A4 @ B4 )
% 8.84/5.02 & ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_302_Orderings_Oorder__eq__iff,axiom,
% 8.84/5.02 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 8.84/5.02 = ( ^ [A4: nat,B4: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A4 @ B4 )
% 8.84/5.02 & ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_303_Orderings_Oorder__eq__iff,axiom,
% 8.84/5.02 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 8.84/5.02 = ( ^ [A4: int,B4: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ A4 @ B4 )
% 8.84/5.02 & ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_304_antisym,axiom,
% 8.84/5.02 ! [A2: set_int,B3: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.84/5.02 => ( A2 = B3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_305_antisym,axiom,
% 8.84/5.02 ! [A2: rat,B3: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ B3 @ A2 )
% 8.84/5.02 => ( A2 = B3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_306_antisym,axiom,
% 8.84/5.02 ! [A2: num,B3: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_num @ B3 @ A2 )
% 8.84/5.02 => ( A2 = B3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_307_antisym,axiom,
% 8.84/5.02 ! [A2: nat,B3: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 8.84/5.02 => ( A2 = B3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_308_antisym,axiom,
% 8.84/5.02 ! [A2: int,B3: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ A2 @ B3 )
% 8.84/5.02 => ( ( ord_less_eq_int @ B3 @ A2 )
% 8.84/5.02 => ( A2 = B3 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_309_dual__order_Otrans,axiom,
% 8.84/5.02 ! [B3: set_int,A2: set_int,C: set_int] :
% 8.84/5.02 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 8.84/5.02 => ( ( ord_less_eq_set_int @ C @ B3 )
% 8.84/5.02 => ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_310_dual__order_Otrans,axiom,
% 8.84/5.02 ! [B3: rat,A2: rat,C: rat] :
% 8.84/5.02 ( ( ord_less_eq_rat @ B3 @ A2 )
% 8.84/5.02 => ( ( ord_less_eq_rat @ C @ B3 )
% 8.84/5.02 => ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_311_dual__order_Otrans,axiom,
% 8.84/5.02 ! [B3: num,A2: num,C: num] :
% 8.84/5.02 ( ( ord_less_eq_num @ B3 @ A2 )
% 8.84/5.02 => ( ( ord_less_eq_num @ C @ B3 )
% 8.84/5.02 => ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_312_dual__order_Otrans,axiom,
% 8.84/5.02 ! [B3: nat,A2: nat,C: nat] :
% 8.84/5.02 ( ( ord_less_eq_nat @ B3 @ A2 )
% 8.84/5.02 => ( ( ord_less_eq_nat @ C @ B3 )
% 8.84/5.02 => ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 8.84/5.02
% 8.84/5.02 thf(fact_313_dual__order_Otrans,axiom,
% 8.84/5.02 ! [B3: int,A2: int,C: int] :
% 8.84/5.02 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_int @ C @ B3 )
% 9.00/5.02 => ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_314_dual__order_Oantisym,axiom,
% 9.00/5.02 ! [B3: set_int,A2: set_int] :
% 9.00/5.02 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.02 => ( A2 = B3 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_315_dual__order_Oantisym,axiom,
% 9.00/5.02 ! [B3: rat,A2: rat] :
% 9.00/5.02 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.02 => ( A2 = B3 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_316_dual__order_Oantisym,axiom,
% 9.00/5.02 ! [B3: num,A2: num] :
% 9.00/5.02 ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.02 => ( A2 = B3 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_317_dual__order_Oantisym,axiom,
% 9.00/5.02 ! [B3: nat,A2: nat] :
% 9.00/5.02 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.02 => ( A2 = B3 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_318_dual__order_Oantisym,axiom,
% 9.00/5.02 ! [B3: int,A2: int] :
% 9.00/5.02 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.02 => ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.02 => ( A2 = B3 ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_319_dual__order_Oeq__iff,axiom,
% 9.00/5.02 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 9.00/5.02 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.02 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.00/5.02 & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_320_dual__order_Oeq__iff,axiom,
% 9.00/5.02 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 9.00/5.02 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.02 ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.00/5.02 & ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_321_dual__order_Oeq__iff,axiom,
% 9.00/5.02 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 9.00/5.02 = ( ^ [A4: num,B4: num] :
% 9.00/5.02 ( ( ord_less_eq_num @ B4 @ A4 )
% 9.00/5.02 & ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_322_dual__order_Oeq__iff,axiom,
% 9.00/5.02 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 9.00/5.02 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.02 ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.00/5.02 & ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.02
% 9.00/5.02 thf(fact_323_dual__order_Oeq__iff,axiom,
% 9.00/5.02 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 9.00/5.02 = ( ^ [A4: int,B4: int] :
% 9.00/5.02 ( ( ord_less_eq_int @ B4 @ A4 )
% 9.00/5.02 & ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_324_linorder__wlog,axiom,
% 9.00/5.03 ! [P: rat > rat > $o,A2: rat,B3: rat] :
% 9.00/5.03 ( ! [A: rat,B: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: rat,B: rat] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_325_linorder__wlog,axiom,
% 9.00/5.03 ! [P: num > num > $o,A2: num,B3: num] :
% 9.00/5.03 ( ! [A: num,B: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: num,B: num] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_326_linorder__wlog,axiom,
% 9.00/5.03 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 9.00/5.03 ( ! [A: nat,B: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: nat,B: nat] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_327_linorder__wlog,axiom,
% 9.00/5.03 ! [P: int > int > $o,A2: int,B3: int] :
% 9.00/5.03 ( ! [A: int,B: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: int,B: int] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_328_order__trans,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_329_order__trans,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_330_order__trans,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_eq_num @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_331_order__trans,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_332_order__trans,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_333_order_Otrans,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_334_order_Otrans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_335_order_Otrans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_336_order_Otrans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_337_order_Otrans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_338_order__antisym,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.03 => ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_339_order__antisym,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.03 => ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_340_order__antisym,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ Y3 @ X2 )
% 9.00/5.03 => ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_341_order__antisym,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.03 => ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_342_order__antisym,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.03 => ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_343_ord__le__eq__trans,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_344_ord__le__eq__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_345_ord__le__eq__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_346_ord__le__eq__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_347_ord__le__eq__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_348_ord__eq__le__trans,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_349_ord__eq__le__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_350_ord__eq__le__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_351_ord__eq__le__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_352_ord__eq__le__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_353_order__class_Oorder__eq__iff,axiom,
% 9.00/5.03 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X @ Y )
% 9.00/5.03 & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_354_order__class_Oorder__eq__iff,axiom,
% 9.00/5.03 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [X: rat,Y: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X @ Y )
% 9.00/5.03 & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_355_order__class_Oorder__eq__iff,axiom,
% 9.00/5.03 ( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [X: num,Y: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X @ Y )
% 9.00/5.03 & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_356_order__class_Oorder__eq__iff,axiom,
% 9.00/5.03 ( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [X: nat,Y: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X @ Y )
% 9.00/5.03 & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_357_order__class_Oorder__eq__iff,axiom,
% 9.00/5.03 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [X: int,Y: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X @ Y )
% 9.00/5.03 & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_358_le__cases3,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ Y3 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ X2 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_rat @ X2 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ Z @ Y3 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_rat @ Z @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ Y3 @ X2 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ Z @ X2 ) )
% 9.00/5.03 => ~ ( ( ord_less_eq_rat @ Z @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_359_le__cases3,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.03 ( ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ Y3 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_num @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ X2 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_num @ X2 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_num @ Z @ Y3 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_num @ Z @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ Y3 @ X2 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_num @ Y3 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_num @ Z @ X2 ) )
% 9.00/5.03 => ~ ( ( ord_less_eq_num @ Z @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_360_le__cases3,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.03 ( ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ X2 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_nat @ X2 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_nat @ Z @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ Y3 @ X2 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ Z @ X2 ) )
% 9.00/5.03 => ~ ( ( ord_less_eq_nat @ Z @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_361_le__cases3,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.03 ( ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ Y3 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ X2 @ Z ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ X2 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_int @ Z @ Y3 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ Z @ Y3 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ Y3 @ X2 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ Y3 @ Z )
% 9.00/5.03 => ~ ( ord_less_eq_int @ Z @ X2 ) )
% 9.00/5.03 => ~ ( ( ord_less_eq_int @ Z @ X2 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_362_nle__le,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.03 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.03 & ( B3 != A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_363_nle__le,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_num @ A2 @ B3 ) )
% 9.00/5.03 = ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.03 & ( B3 != A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_364_nle__le,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_nat @ A2 @ B3 ) )
% 9.00/5.03 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.03 & ( B3 != A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_365_nle__le,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_int @ A2 @ B3 ) )
% 9.00/5.03 = ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.03 & ( B3 != A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_366_lt__ex,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ? [Y4: real] : ( ord_less_real @ Y4 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_367_lt__ex,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_368_lt__ex,axiom,
% 9.00/5.03 ! [X2: int] :
% 9.00/5.03 ? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_369_gt__ex,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_370_gt__ex,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_371_gt__ex,axiom,
% 9.00/5.03 ! [X2: nat] :
% 9.00/5.03 ? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_372_gt__ex,axiom,
% 9.00/5.03 ! [X2: int] :
% 9.00/5.03 ? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_373_dense,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ? [Z4: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Z4 )
% 9.00/5.03 & ( ord_less_real @ Z4 @ Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_374_dense,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ? [Z4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Z4 )
% 9.00/5.03 & ( ord_less_rat @ Z4 @ Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_375_less__imp__neq,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_376_less__imp__neq,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_377_less__imp__neq,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_378_less__imp__neq,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_379_less__imp__neq,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_380_order_Oasym,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_real @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_381_order_Oasym,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_382_order_Oasym,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_num @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_383_order_Oasym,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_384_order_Oasym,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_int @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_385_ord__eq__less__trans,axiom,
% 9.00/5.03 ! [A2: real,B3: real,C: real] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_386_ord__eq__less__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_387_ord__eq__less__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_388_ord__eq__less__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_389_ord__eq__less__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_390_ord__less__eq__trans,axiom,
% 9.00/5.03 ! [A2: real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_391_ord__less__eq__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_392_ord__less__eq__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_393_ord__less__eq__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_394_ord__less__eq__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( B3 = C )
% 9.00/5.03 => ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_395_less__induct,axiom,
% 9.00/5.03 ! [P: nat > $o,A2: nat] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ( ! [Y5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Y5 @ X5 )
% 9.00/5.03 => ( P @ Y5 ) )
% 9.00/5.03 => ( P @ X5 ) )
% 9.00/5.03 => ( P @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_396_antisym__conv3,axiom,
% 9.00/5.03 ! [Y3: real,X2: real] :
% 9.00/5.03 ( ~ ( ord_less_real @ Y3 @ X2 )
% 9.00/5.03 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_397_antisym__conv3,axiom,
% 9.00/5.03 ! [Y3: rat,X2: rat] :
% 9.00/5.03 ( ~ ( ord_less_rat @ Y3 @ X2 )
% 9.00/5.03 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_398_antisym__conv3,axiom,
% 9.00/5.03 ! [Y3: num,X2: num] :
% 9.00/5.03 ( ~ ( ord_less_num @ Y3 @ X2 )
% 9.00/5.03 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_399_antisym__conv3,axiom,
% 9.00/5.03 ! [Y3: nat,X2: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ Y3 @ X2 )
% 9.00/5.03 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_400_antisym__conv3,axiom,
% 9.00/5.03 ! [Y3: int,X2: int] :
% 9.00/5.03 ( ~ ( ord_less_int @ Y3 @ X2 )
% 9.00/5.03 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_401_linorder__cases,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ~ ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( X2 != Y3 )
% 9.00/5.03 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_402_linorder__cases,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( X2 != Y3 )
% 9.00/5.03 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_403_linorder__cases,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ~ ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( X2 != Y3 )
% 9.00/5.03 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_404_linorder__cases,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( X2 != Y3 )
% 9.00/5.03 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_405_linorder__cases,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ~ ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( X2 != Y3 )
% 9.00/5.03 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_406_dual__order_Oasym,axiom,
% 9.00/5.03 ! [B3: real,A2: real] :
% 9.00/5.03 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.03 => ~ ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_407_dual__order_Oasym,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.03 => ~ ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_408_dual__order_Oasym,axiom,
% 9.00/5.03 ! [B3: num,A2: num] :
% 9.00/5.03 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.03 => ~ ( ord_less_num @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_409_dual__order_Oasym,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.03 => ~ ( ord_less_nat @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_410_dual__order_Oasym,axiom,
% 9.00/5.03 ! [B3: int,A2: int] :
% 9.00/5.03 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.03 => ~ ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_411_dual__order_Oirrefl,axiom,
% 9.00/5.03 ! [A2: real] :
% 9.00/5.03 ~ ( ord_less_real @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_412_dual__order_Oirrefl,axiom,
% 9.00/5.03 ! [A2: rat] :
% 9.00/5.03 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_413_dual__order_Oirrefl,axiom,
% 9.00/5.03 ! [A2: num] :
% 9.00/5.03 ~ ( ord_less_num @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_414_dual__order_Oirrefl,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_415_dual__order_Oirrefl,axiom,
% 9.00/5.03 ! [A2: int] :
% 9.00/5.03 ~ ( ord_less_int @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_416_exists__least__iff,axiom,
% 9.00/5.03 ( ( ^ [P2: nat > $o] :
% 9.00/5.03 ? [X3: nat] : ( P2 @ X3 ) )
% 9.00/5.03 = ( ^ [P3: nat > $o] :
% 9.00/5.03 ? [N2: nat] :
% 9.00/5.03 ( ( P3 @ N2 )
% 9.00/5.03 & ! [M2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M2 @ N2 )
% 9.00/5.03 => ~ ( P3 @ M2 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_417_linorder__less__wlog,axiom,
% 9.00/5.03 ! [P: real > real > $o,A2: real,B3: real] :
% 9.00/5.03 ( ! [A: real,B: real] :
% 9.00/5.03 ( ( ord_less_real @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: real] : ( P @ A @ A )
% 9.00/5.03 => ( ! [A: real,B: real] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_418_linorder__less__wlog,axiom,
% 9.00/5.03 ! [P: rat > rat > $o,A2: rat,B3: rat] :
% 9.00/5.03 ( ! [A: rat,B: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: rat] : ( P @ A @ A )
% 9.00/5.03 => ( ! [A: rat,B: rat] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_419_linorder__less__wlog,axiom,
% 9.00/5.03 ! [P: num > num > $o,A2: num,B3: num] :
% 9.00/5.03 ( ! [A: num,B: num] :
% 9.00/5.03 ( ( ord_less_num @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: num] : ( P @ A @ A )
% 9.00/5.03 => ( ! [A: num,B: num] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_420_linorder__less__wlog,axiom,
% 9.00/5.03 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 9.00/5.03 ( ! [A: nat,B: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: nat] : ( P @ A @ A )
% 9.00/5.03 => ( ! [A: nat,B: nat] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_421_linorder__less__wlog,axiom,
% 9.00/5.03 ! [P: int > int > $o,A2: int,B3: int] :
% 9.00/5.03 ( ! [A: int,B: int] :
% 9.00/5.03 ( ( ord_less_int @ A @ B )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( ! [A: int] : ( P @ A @ A )
% 9.00/5.03 => ( ! [A: int,B: int] :
% 9.00/5.03 ( ( P @ B @ A )
% 9.00/5.03 => ( P @ A @ B ) )
% 9.00/5.03 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_422_order_Ostrict__trans,axiom,
% 9.00/5.03 ! [A2: real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_423_order_Ostrict__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_424_order_Ostrict__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_425_order_Ostrict__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_426_order_Ostrict__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_427_not__less__iff__gr__or__eq,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 9.00/5.03 = ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_428_not__less__iff__gr__or__eq,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ( ord_less_rat @ Y3 @ X2 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_429_not__less__iff__gr__or__eq,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 9.00/5.03 = ( ( ord_less_num @ Y3 @ X2 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_430_not__less__iff__gr__or__eq,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ( ord_less_nat @ Y3 @ X2 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_431_not__less__iff__gr__or__eq,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( ( ord_less_int @ Y3 @ X2 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_432_dual__order_Ostrict__trans,axiom,
% 9.00/5.03 ! [B3: real,A2: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_real @ C @ B3 )
% 9.00/5.03 => ( ord_less_real @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_433_dual__order_Ostrict__trans,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_rat @ C @ B3 )
% 9.00/5.03 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_434_dual__order_Ostrict__trans,axiom,
% 9.00/5.03 ! [B3: num,A2: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_num @ C @ B3 )
% 9.00/5.03 => ( ord_less_num @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_435_dual__order_Ostrict__trans,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_nat @ C @ B3 )
% 9.00/5.03 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_436_dual__order_Ostrict__trans,axiom,
% 9.00/5.03 ! [B3: int,A2: int,C: int] :
% 9.00/5.03 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_int @ C @ B3 )
% 9.00/5.03 => ( ord_less_int @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_437_order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_438_order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_439_order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_440_order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_441_order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_442_dual__order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [B3: real,A2: real] :
% 9.00/5.03 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_443_dual__order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_444_dual__order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [B3: num,A2: num] :
% 9.00/5.03 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_445_dual__order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_446_dual__order_Ostrict__implies__not__eq,axiom,
% 9.00/5.03 ! [B3: int,A2: int] :
% 9.00/5.03 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.03 => ( A2 != B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_447_linorder__neqE,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_448_linorder__neqE,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_449_linorder__neqE,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 => ( ~ ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_450_linorder__neqE,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 => ( ~ ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_451_linorder__neqE,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_452_order__less__asym,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_453_order__less__asym,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_454_order__less__asym,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_455_order__less__asym,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_456_order__less__asym,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_457_linorder__neq__iff,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 = ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_458_linorder__neq__iff,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 = ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_459_linorder__neq__iff,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 = ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_num @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_460_linorder__neq__iff,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 = ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_nat @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_461_linorder__neq__iff,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( X2 != Y3 )
% 9.00/5.03 = ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_462_order__less__asym_H,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_real @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_463_order__less__asym_H,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_rat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_464_order__less__asym_H,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_num @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_465_order__less__asym_H,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_nat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_466_order__less__asym_H,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ~ ( ord_less_int @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_467_order__less__trans,axiom,
% 9.00/5.03 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_real @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_468_order__less__trans,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_rat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_469_order__less__trans,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_num @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_470_order__less__trans,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_nat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_471_order__less__trans,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_472_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: real,F: real > real,B3: real,C: real] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_473_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_474_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: num,F: real > num,B3: real,C: real] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_475_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: nat,F: real > nat,B3: real,C: real] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_476_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: int,F: real > int,B3: real,C: real] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_477_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_478_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_479_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_480_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_481_ord__eq__less__subst,axiom,
% 9.00/5.03 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 = ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_482_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_483_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_484_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > num,C: num] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_485_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_486_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > int,C: int] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_487_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_488_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_489_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_490_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_491_ord__less__eq__subst,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ( F @ B3 )
% 9.00/5.03 = C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_492_order__less__irrefl,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ~ ( ord_less_real @ X2 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_493_order__less__irrefl,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ~ ( ord_less_rat @ X2 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_494_order__less__irrefl,axiom,
% 9.00/5.03 ! [X2: num] :
% 9.00/5.03 ~ ( ord_less_num @ X2 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_495_order__less__irrefl,axiom,
% 9.00/5.03 ! [X2: nat] :
% 9.00/5.03 ~ ( ord_less_nat @ X2 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_496_order__less__irrefl,axiom,
% 9.00/5.03 ! [X2: int] :
% 9.00/5.03 ~ ( ord_less_int @ X2 @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_497_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: real > real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_498_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_499_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: num > real,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_500_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: nat > real,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_501_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: int > real,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_502_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_503_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_504_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_505_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_506_order__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_507_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_508_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_509_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > num,C: num] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_510_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_511_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > int,C: int] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_512_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_513_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_514_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_515_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_516_order__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_517_order__less__not__sym,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_518_order__less__not__sym,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_519_order__less__not__sym,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_520_order__less__not__sym,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_521_order__less__not__sym,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_522_order__less__imp__triv,axiom,
% 9.00/5.03 ! [X2: real,Y3: real,P: $o] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.03 => P ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_523_order__less__imp__triv,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,P: $o] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_rat @ Y3 @ X2 )
% 9.00/5.03 => P ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_524_order__less__imp__triv,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,P: $o] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_num @ Y3 @ X2 )
% 9.00/5.03 => P ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_525_order__less__imp__triv,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,P: $o] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_nat @ Y3 @ X2 )
% 9.00/5.03 => P ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_526_order__less__imp__triv,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,P: $o] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_int @ Y3 @ X2 )
% 9.00/5.03 => P ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_527_linorder__less__linear,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 )
% 9.00/5.03 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_528_linorder__less__linear,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 )
% 9.00/5.03 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_529_linorder__less__linear,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 )
% 9.00/5.03 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_530_linorder__less__linear,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 )
% 9.00/5.03 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_531_linorder__less__linear,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 )
% 9.00/5.03 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_532_order__less__imp__not__eq,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_533_order__less__imp__not__eq,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_534_order__less__imp__not__eq,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_535_order__less__imp__not__eq,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_536_order__less__imp__not__eq,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( X2 != Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_537_order__less__imp__not__eq2,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( Y3 != X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_538_order__less__imp__not__eq2,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( Y3 != X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_539_order__less__imp__not__eq2,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( Y3 != X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_540_order__less__imp__not__eq2,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( Y3 != X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_541_order__less__imp__not__eq2,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( Y3 != X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_542_order__less__imp__not__less,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_543_order__less__imp__not__less,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_544_order__less__imp__not__less,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_545_order__less__imp__not__less,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_546_order__less__imp__not__less,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ~ ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_547_ex__in__conv,axiom,
% 9.00/5.03 ! [A3: set_set_nat] :
% 9.00/5.03 ( ( ? [X: set_nat] : ( member_set_nat @ X @ A3 ) )
% 9.00/5.03 = ( A3 != bot_bot_set_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_548_ex__in__conv,axiom,
% 9.00/5.03 ! [A3: set_real] :
% 9.00/5.03 ( ( ? [X: real] : ( member_real @ X @ A3 ) )
% 9.00/5.03 = ( A3 != bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_549_ex__in__conv,axiom,
% 9.00/5.03 ! [A3: set_o] :
% 9.00/5.03 ( ( ? [X: $o] : ( member_o @ X @ A3 ) )
% 9.00/5.03 = ( A3 != bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_550_ex__in__conv,axiom,
% 9.00/5.03 ! [A3: set_nat] :
% 9.00/5.03 ( ( ? [X: nat] : ( member_nat @ X @ A3 ) )
% 9.00/5.03 = ( A3 != bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_551_ex__in__conv,axiom,
% 9.00/5.03 ! [A3: set_int] :
% 9.00/5.03 ( ( ? [X: int] : ( member_int @ X @ A3 ) )
% 9.00/5.03 = ( A3 != bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_552_equals0I,axiom,
% 9.00/5.03 ! [A3: set_set_nat] :
% 9.00/5.03 ( ! [Y4: set_nat] :
% 9.00/5.03 ~ ( member_set_nat @ Y4 @ A3 )
% 9.00/5.03 => ( A3 = bot_bot_set_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_553_equals0I,axiom,
% 9.00/5.03 ! [A3: set_real] :
% 9.00/5.03 ( ! [Y4: real] :
% 9.00/5.03 ~ ( member_real @ Y4 @ A3 )
% 9.00/5.03 => ( A3 = bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_554_equals0I,axiom,
% 9.00/5.03 ! [A3: set_o] :
% 9.00/5.03 ( ! [Y4: $o] :
% 9.00/5.03 ~ ( member_o @ Y4 @ A3 )
% 9.00/5.03 => ( A3 = bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_555_equals0I,axiom,
% 9.00/5.03 ! [A3: set_nat] :
% 9.00/5.03 ( ! [Y4: nat] :
% 9.00/5.03 ~ ( member_nat @ Y4 @ A3 )
% 9.00/5.03 => ( A3 = bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_556_equals0I,axiom,
% 9.00/5.03 ! [A3: set_int] :
% 9.00/5.03 ( ! [Y4: int] :
% 9.00/5.03 ~ ( member_int @ Y4 @ A3 )
% 9.00/5.03 => ( A3 = bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_557_equals0D,axiom,
% 9.00/5.03 ! [A3: set_set_nat,A2: set_nat] :
% 9.00/5.03 ( ( A3 = bot_bot_set_set_nat )
% 9.00/5.03 => ~ ( member_set_nat @ A2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_558_equals0D,axiom,
% 9.00/5.03 ! [A3: set_real,A2: real] :
% 9.00/5.03 ( ( A3 = bot_bot_set_real )
% 9.00/5.03 => ~ ( member_real @ A2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_559_equals0D,axiom,
% 9.00/5.03 ! [A3: set_o,A2: $o] :
% 9.00/5.03 ( ( A3 = bot_bot_set_o )
% 9.00/5.03 => ~ ( member_o @ A2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_560_equals0D,axiom,
% 9.00/5.03 ! [A3: set_nat,A2: nat] :
% 9.00/5.03 ( ( A3 = bot_bot_set_nat )
% 9.00/5.03 => ~ ( member_nat @ A2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_561_equals0D,axiom,
% 9.00/5.03 ! [A3: set_int,A2: int] :
% 9.00/5.03 ( ( A3 = bot_bot_set_int )
% 9.00/5.03 => ~ ( member_int @ A2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_562_emptyE,axiom,
% 9.00/5.03 ! [A2: set_nat] :
% 9.00/5.03 ~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_563_emptyE,axiom,
% 9.00/5.03 ! [A2: real] :
% 9.00/5.03 ~ ( member_real @ A2 @ bot_bot_set_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_564_emptyE,axiom,
% 9.00/5.03 ! [A2: $o] :
% 9.00/5.03 ~ ( member_o @ A2 @ bot_bot_set_o ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_565_emptyE,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_566_emptyE,axiom,
% 9.00/5.03 ! [A2: int] :
% 9.00/5.03 ~ ( member_int @ A2 @ bot_bot_set_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_567_not__psubset__empty,axiom,
% 9.00/5.03 ! [A3: set_real] :
% 9.00/5.03 ~ ( ord_less_set_real @ A3 @ bot_bot_set_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_568_not__psubset__empty,axiom,
% 9.00/5.03 ! [A3: set_o] :
% 9.00/5.03 ~ ( ord_less_set_o @ A3 @ bot_bot_set_o ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_569_not__psubset__empty,axiom,
% 9.00/5.03 ! [A3: set_nat] :
% 9.00/5.03 ~ ( ord_less_set_nat @ A3 @ bot_bot_set_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_570_not__psubset__empty,axiom,
% 9.00/5.03 ! [A3: set_int] :
% 9.00/5.03 ~ ( ord_less_set_int @ A3 @ bot_bot_set_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_571_finite__psubset__induct,axiom,
% 9.00/5.03 ! [A3: set_nat,P: set_nat > $o] :
% 9.00/5.03 ( ( finite_finite_nat @ A3 )
% 9.00/5.03 => ( ! [A5: set_nat] :
% 9.00/5.03 ( ( finite_finite_nat @ A5 )
% 9.00/5.03 => ( ! [B5: set_nat] :
% 9.00/5.03 ( ( ord_less_set_nat @ B5 @ A5 )
% 9.00/5.03 => ( P @ B5 ) )
% 9.00/5.03 => ( P @ A5 ) ) )
% 9.00/5.03 => ( P @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_572_finite__psubset__induct,axiom,
% 9.00/5.03 ! [A3: set_int,P: set_int > $o] :
% 9.00/5.03 ( ( finite_finite_int @ A3 )
% 9.00/5.03 => ( ! [A5: set_int] :
% 9.00/5.03 ( ( finite_finite_int @ A5 )
% 9.00/5.03 => ( ! [B5: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ B5 @ A5 )
% 9.00/5.03 => ( P @ B5 ) )
% 9.00/5.03 => ( P @ A5 ) ) )
% 9.00/5.03 => ( P @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_573_finite__psubset__induct,axiom,
% 9.00/5.03 ! [A3: set_complex,P: set_complex > $o] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.03 => ( ! [A5: set_complex] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ A5 )
% 9.00/5.03 => ( ! [B5: set_complex] :
% 9.00/5.03 ( ( ord_less_set_complex @ B5 @ A5 )
% 9.00/5.03 => ( P @ B5 ) )
% 9.00/5.03 => ( P @ A5 ) ) )
% 9.00/5.03 => ( P @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_574_finite__psubset__induct,axiom,
% 9.00/5.03 ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.03 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.03 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.03 ( ( finite6177210948735845034at_nat @ A5 )
% 9.00/5.03 => ( ! [B5: set_Pr1261947904930325089at_nat] :
% 9.00/5.03 ( ( ord_le7866589430770878221at_nat @ B5 @ A5 )
% 9.00/5.03 => ( P @ B5 ) )
% 9.00/5.03 => ( P @ A5 ) ) )
% 9.00/5.03 => ( P @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_575_finite__psubset__induct,axiom,
% 9.00/5.03 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.03 => ( ! [A5: set_Extended_enat] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.03 => ( ! [B5: set_Extended_enat] :
% 9.00/5.03 ( ( ord_le2529575680413868914d_enat @ B5 @ A5 )
% 9.00/5.03 => ( P @ B5 ) )
% 9.00/5.03 => ( P @ A5 ) ) )
% 9.00/5.03 => ( P @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_576_leD,axiom,
% 9.00/5.03 ! [Y3: real,X2: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_577_leD,axiom,
% 9.00/5.03 ! [Y3: set_int,X2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_578_leD,axiom,
% 9.00/5.03 ! [Y3: rat,X2: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_rat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_579_leD,axiom,
% 9.00/5.03 ! [Y3: num,X2: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_num @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_580_leD,axiom,
% 9.00/5.03 ! [Y3: nat,X2: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_nat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_581_leD,axiom,
% 9.00/5.03 ! [Y3: int,X2: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.03 => ~ ( ord_less_int @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_582_leI,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ~ ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_583_leI,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_584_leI,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ~ ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_585_leI,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_586_leI,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ~ ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_587_nless__le,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ~ ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_588_nless__le,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int] :
% 9.00/5.03 ( ( ~ ( ord_less_set_int @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_589_nless__le,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_590_nless__le,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ~ ( ord_less_num @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_591_nless__le,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_nat @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_592_nless__le,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ~ ( ord_less_int @ A2 @ B3 ) )
% 9.00/5.03 = ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 | ( A2 = B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_593_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ~ ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_594_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int] :
% 9.00/5.03 ( ~ ( ord_less_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_595_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ~ ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_596_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ~ ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_597_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_598_antisym__conv1,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ~ ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_599_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_600_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_set_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_601_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_602_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_603_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_604_antisym__conv2,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_605_dense__ge,axiom,
% 9.00/5.03 ! [Z: real,Y3: real] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ Z @ X5 )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_606_dense__ge,axiom,
% 9.00/5.03 ! [Z: rat,Y3: rat] :
% 9.00/5.03 ( ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z @ X5 )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_607_dense__le,axiom,
% 9.00/5.03 ! [Y3: real,Z: real] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_real @ X5 @ Z ) )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ Z ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_608_dense__le,axiom,
% 9.00/5.03 ! [Y3: rat,Z: rat] :
% 9.00/5.03 ( ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_rat @ X5 @ Z ) )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ Z ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_609_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [X: real,Y: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_610_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_611_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [X: rat,Y: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_612_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [X: num,Y: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_613_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [X: nat,Y: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_614_less__le__not__le,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [X: int,Y: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X @ Y )
% 9.00/5.03 & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_615_not__le__imp__less,axiom,
% 9.00/5.03 ! [Y3: real,X2: real] :
% 9.00/5.03 ( ~ ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.03 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_616_not__le__imp__less,axiom,
% 9.00/5.03 ! [Y3: rat,X2: rat] :
% 9.00/5.03 ( ~ ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.03 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_617_not__le__imp__less,axiom,
% 9.00/5.03 ! [Y3: num,X2: num] :
% 9.00/5.03 ( ~ ( ord_less_eq_num @ Y3 @ X2 )
% 9.00/5.03 => ( ord_less_num @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_618_not__le__imp__less,axiom,
% 9.00/5.03 ! [Y3: nat,X2: nat] :
% 9.00/5.03 ( ~ ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.03 => ( ord_less_nat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_619_not__le__imp__less,axiom,
% 9.00/5.03 ! [Y3: int,X2: int] :
% 9.00/5.03 ( ~ ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.03 => ( ord_less_int @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_620_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_real
% 9.00/5.03 = ( ^ [A4: real,B4: real] :
% 9.00/5.03 ( ( ord_less_real @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_621_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_622_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_rat
% 9.00/5.03 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_623_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_num
% 9.00/5.03 = ( ^ [A4: num,B4: num] :
% 9.00/5.03 ( ( ord_less_num @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_624_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_nat
% 9.00/5.03 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_625_order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_int
% 9.00/5.03 = ( ^ [A4: int,B4: int] :
% 9.00/5.03 ( ( ord_less_int @ A4 @ B4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_626_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [A4: real,B4: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_627_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_628_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_629_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [A4: num,B4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_630_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_631_order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [A4: int,B4: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A4 @ B4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_632_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_633_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_set_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_set_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_634_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_635_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_636_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_637_order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_638_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ B3 @ C )
% 9.00/5.03 => ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_639_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_set_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_640_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_641_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ord_less_num @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_642_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.03 => ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_643_order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [A2: int,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.03 => ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_644_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [A4: real,B4: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_real @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_645_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_set_int @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_646_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_rat @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_647_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [A4: num,B4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_num @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_648_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_649_order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [A4: int,B4: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A4 @ B4 )
% 9.00/5.03 & ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_650_dense__ge__bounded,axiom,
% 9.00/5.03 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ Z @ X2 )
% 9.00/5.03 => ( ! [W: real] :
% 9.00/5.03 ( ( ord_less_real @ Z @ W )
% 9.00/5.03 => ( ( ord_less_real @ W @ X2 )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ W ) ) )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_651_dense__ge__bounded,axiom,
% 9.00/5.03 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z @ X2 )
% 9.00/5.03 => ( ! [W: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z @ W )
% 9.00/5.03 => ( ( ord_less_rat @ W @ X2 )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ W ) ) )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_652_dense__le__bounded,axiom,
% 9.00/5.03 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ! [W: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ W )
% 9.00/5.03 => ( ( ord_less_real @ W @ Y3 )
% 9.00/5.03 => ( ord_less_eq_real @ W @ Z ) ) )
% 9.00/5.03 => ( ord_less_eq_real @ Y3 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_653_dense__le__bounded,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ! [W: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ W )
% 9.00/5.03 => ( ( ord_less_rat @ W @ Y3 )
% 9.00/5.03 => ( ord_less_eq_rat @ W @ Z ) ) )
% 9.00/5.03 => ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_654_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_real
% 9.00/5.03 = ( ^ [B4: real,A4: real] :
% 9.00/5.03 ( ( ord_less_real @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_655_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_656_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_rat
% 9.00/5.03 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_657_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_num
% 9.00/5.03 = ( ^ [B4: num,A4: num] :
% 9.00/5.03 ( ( ord_less_num @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_658_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_nat
% 9.00/5.03 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_659_dual__order_Oorder__iff__strict,axiom,
% 9.00/5.03 ( ord_less_eq_int
% 9.00/5.03 = ( ^ [B4: int,A4: int] :
% 9.00/5.03 ( ( ord_less_int @ B4 @ A4 )
% 9.00/5.03 | ( A4 = B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_660_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [B4: real,A4: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_661_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_662_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_663_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [B4: num,A4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_664_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_665_dual__order_Ostrict__iff__order,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [B4: int,A4: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ B4 @ A4 )
% 9.00/5.03 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_666_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: real,A2: real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_real @ C @ B3 )
% 9.00/5.03 => ( ord_less_real @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_667_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: set_int,A2: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_set_int @ C @ B3 )
% 9.00/5.03 => ( ord_less_set_int @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_668_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_rat @ C @ B3 )
% 9.00/5.03 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_669_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: num,A2: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_num @ C @ B3 )
% 9.00/5.03 => ( ord_less_num @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_670_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_nat @ C @ B3 )
% 9.00/5.03 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_671_dual__order_Ostrict__trans1,axiom,
% 9.00/5.03 ! [B3: int,A2: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_int @ C @ B3 )
% 9.00/5.03 => ( ord_less_int @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_672_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: real,A2: real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_real @ C @ B3 )
% 9.00/5.03 => ( ord_less_real @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_673_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: set_int,A2: set_int,C: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ C @ B3 )
% 9.00/5.03 => ( ord_less_set_int @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_674_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ C @ B3 )
% 9.00/5.03 => ( ord_less_rat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_675_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: num,A2: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_num @ C @ B3 )
% 9.00/5.03 => ( ord_less_num @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_676_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ C @ B3 )
% 9.00/5.03 => ( ord_less_nat @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_677_dual__order_Ostrict__trans2,axiom,
% 9.00/5.03 ! [B3: int,A2: int,C: int] :
% 9.00/5.03 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.03 => ( ( ord_less_eq_int @ C @ B3 )
% 9.00/5.03 => ( ord_less_int @ C @ A2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_678_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [B4: real,A4: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_real @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_679_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_680_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_rat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_681_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [B4: num,A4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_num @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_682_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_683_dual__order_Ostrict__iff__not,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [B4: int,A4: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ B4 @ A4 )
% 9.00/5.03 & ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_684_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_685_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_686_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_687_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_num @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_688_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_689_order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_690_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: real,A2: real] :
% 9.00/5.03 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_real @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_691_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: set_int,A2: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_692_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: rat,A2: rat] :
% 9.00/5.03 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_693_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: num,A2: num] :
% 9.00/5.03 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_694_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: nat,A2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_695_dual__order_Ostrict__implies__order,axiom,
% 9.00/5.03 ! [B3: int,A2: int] :
% 9.00/5.03 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.03 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_696_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_real
% 9.00/5.03 = ( ^ [X: real,Y: real] :
% 9.00/5.03 ( ( ord_less_real @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_697_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_698_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_rat
% 9.00/5.03 = ( ^ [X: rat,Y: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_699_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_num
% 9.00/5.03 = ( ^ [X: num,Y: num] :
% 9.00/5.03 ( ( ord_less_num @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_700_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_nat
% 9.00/5.03 = ( ^ [X: nat,Y: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_701_order__le__less,axiom,
% 9.00/5.03 ( ord_less_eq_int
% 9.00/5.03 = ( ^ [X: int,Y: int] :
% 9.00/5.03 ( ( ord_less_int @ X @ Y )
% 9.00/5.03 | ( X = Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_702_order__less__le,axiom,
% 9.00/5.03 ( ord_less_real
% 9.00/5.03 = ( ^ [X: real,Y: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_703_order__less__le,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_704_order__less__le,axiom,
% 9.00/5.03 ( ord_less_rat
% 9.00/5.03 = ( ^ [X: rat,Y: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_705_order__less__le,axiom,
% 9.00/5.03 ( ord_less_num
% 9.00/5.03 = ( ^ [X: num,Y: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_706_order__less__le,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [X: nat,Y: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_707_order__less__le,axiom,
% 9.00/5.03 ( ord_less_int
% 9.00/5.03 = ( ^ [X: int,Y: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X @ Y )
% 9.00/5.03 & ( X != Y ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_708_linorder__not__le,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_real @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_709_linorder__not__le,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_rat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_710_linorder__not__le,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_num @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_711_linorder__not__le,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_nat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_712_linorder__not__le,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_713_linorder__not__less,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ~ ( ord_less_real @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_eq_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_714_linorder__not__less,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_rat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_eq_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_715_linorder__not__less,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ~ ( ord_less_num @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_eq_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_716_linorder__not__less,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_nat @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_eq_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_717_linorder__not__less,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ~ ( ord_less_int @ X2 @ Y3 ) )
% 9.00/5.03 = ( ord_less_eq_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_718_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_719_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_720_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_721_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_722_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_723_order__less__imp__le,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_724_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_725_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_726_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_727_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_num @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_728_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_729_order__le__neq__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 => ( ( A2 != B3 )
% 9.00/5.03 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_730_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: real,B3: real] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_731_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: set_int,B3: set_int] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_732_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_733_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_num @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_734_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_735_order__neq__le__trans,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( A2 != B3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_736_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_real @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_737_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_set_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_738_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_rat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_739_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_num @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_740_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_nat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_741_order__le__less__trans,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_742_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.03 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_real @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_743_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_set_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_744_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_rat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_745_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.03 ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_num @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_746_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_nat @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_747_order__less__le__trans,axiom,
% 9.00/5.03 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.03 ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.00/5.03 => ( ord_less_int @ X2 @ Z ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_748_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: real > real,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_749_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_750_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: num > real,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_751_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: nat > real,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_752_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: int > real,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_753_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: real > rat,B3: real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_754_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_755_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_756_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: nat > rat,B3: nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_757_order__le__less__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: int > rat,B3: int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_758_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_759_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_760_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_761_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_762_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_763_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_764_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_765_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > num,C: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_num @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_766_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_nat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_767_order__le__less__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_int @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_768_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: rat > real,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_769_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: rat > rat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_770_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: num,F: rat > num,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_771_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: nat,F: rat > nat,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_772_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: int,F: rat > int,B3: rat,C: rat] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_773_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: real,F: num > real,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_774_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: rat,F: num > rat,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_775_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: num,F: num > num,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_num @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_776_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: nat,F: num > nat,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_777_order__less__le__subst1,axiom,
% 9.00/5.03 ! [A2: int,F: num > int,B3: num,C: num] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ ( F @ B3 ) )
% 9.00/5.03 => ( ( ord_less_eq_num @ B3 @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_778_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > real,C: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_779_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > real,C: real] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_780_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > real,C: real] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_781_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,F: nat > real,C: real] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_782_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: int,B3: int,F: int > real,C: real] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_real @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_783_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: real,B3: real,F: real > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: real,Y4: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_784_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat,F: rat > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: rat,Y4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_785_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: num,B3: num,F: num > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: num,Y4: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_786_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,F: nat > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_787_order__less__le__subst2,axiom,
% 9.00/5.03 ! [A2: int,B3: int,F: int > rat,C: rat] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ ( F @ B3 ) @ C )
% 9.00/5.03 => ( ! [X5: int,Y4: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Y4 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
% 9.00/5.03 => ( ord_less_rat @ ( F @ A2 ) @ C ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_788_linorder__le__less__linear,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_real @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_789_linorder__le__less__linear,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_rat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_790_linorder__le__less__linear,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_num @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_791_linorder__le__less__linear,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_nat @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_792_linorder__le__less__linear,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 | ( ord_less_int @ Y3 @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_793_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: real,Y3: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_794_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: set_int,Y3: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_set_int @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_795_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: rat,Y3: rat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_796_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: num,Y3: num] :
% 9.00/5.03 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_num @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_797_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_798_order__le__imp__less__or__eq,axiom,
% 9.00/5.03 ! [X2: int,Y3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.03 => ( ( ord_less_int @ X2 @ Y3 )
% 9.00/5.03 | ( X2 = Y3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_799_bot_Oextremum__uniqueI,axiom,
% 9.00/5.03 ! [A2: set_real] :
% 9.00/5.03 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 9.00/5.03 => ( A2 = bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_800_bot_Oextremum__uniqueI,axiom,
% 9.00/5.03 ! [A2: set_o] :
% 9.00/5.03 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 9.00/5.03 => ( A2 = bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_801_bot_Oextremum__uniqueI,axiom,
% 9.00/5.03 ! [A2: set_nat] :
% 9.00/5.03 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 9.00/5.03 => ( A2 = bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_802_bot_Oextremum__uniqueI,axiom,
% 9.00/5.03 ! [A2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 9.00/5.03 => ( A2 = bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_803_bot_Oextremum__uniqueI,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 9.00/5.03 => ( A2 = bot_bot_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_804_bot_Oextremum__unique,axiom,
% 9.00/5.03 ! [A2: set_real] :
% 9.00/5.03 ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 9.00/5.03 = ( A2 = bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_805_bot_Oextremum__unique,axiom,
% 9.00/5.03 ! [A2: set_o] :
% 9.00/5.03 ( ( ord_less_eq_set_o @ A2 @ bot_bot_set_o )
% 9.00/5.03 = ( A2 = bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_806_bot_Oextremum__unique,axiom,
% 9.00/5.03 ! [A2: set_nat] :
% 9.00/5.03 ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 9.00/5.03 = ( A2 = bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_807_bot_Oextremum__unique,axiom,
% 9.00/5.03 ! [A2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 9.00/5.03 = ( A2 = bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_808_bot_Oextremum__unique,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
% 9.00/5.03 = ( A2 = bot_bot_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_809_bot_Oextremum,axiom,
% 9.00/5.03 ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_810_bot_Oextremum,axiom,
% 9.00/5.03 ! [A2: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_811_bot_Oextremum,axiom,
% 9.00/5.03 ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_812_bot_Oextremum,axiom,
% 9.00/5.03 ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_813_bot_Oextremum,axiom,
% 9.00/5.03 ! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_814_bot_Oextremum__strict,axiom,
% 9.00/5.03 ! [A2: set_real] :
% 9.00/5.03 ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_815_bot_Oextremum__strict,axiom,
% 9.00/5.03 ! [A2: set_o] :
% 9.00/5.03 ~ ( ord_less_set_o @ A2 @ bot_bot_set_o ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_816_bot_Oextremum__strict,axiom,
% 9.00/5.03 ! [A2: set_nat] :
% 9.00/5.03 ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_817_bot_Oextremum__strict,axiom,
% 9.00/5.03 ! [A2: set_int] :
% 9.00/5.03 ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_818_bot_Oextremum__strict,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_819_bot_Onot__eq__extremum,axiom,
% 9.00/5.03 ! [A2: set_real] :
% 9.00/5.03 ( ( A2 != bot_bot_set_real )
% 9.00/5.03 = ( ord_less_set_real @ bot_bot_set_real @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_820_bot_Onot__eq__extremum,axiom,
% 9.00/5.03 ! [A2: set_o] :
% 9.00/5.03 ( ( A2 != bot_bot_set_o )
% 9.00/5.03 = ( ord_less_set_o @ bot_bot_set_o @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_821_bot_Onot__eq__extremum,axiom,
% 9.00/5.03 ! [A2: set_nat] :
% 9.00/5.03 ( ( A2 != bot_bot_set_nat )
% 9.00/5.03 = ( ord_less_set_nat @ bot_bot_set_nat @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_822_bot_Onot__eq__extremum,axiom,
% 9.00/5.03 ! [A2: set_int] :
% 9.00/5.03 ( ( A2 != bot_bot_set_int )
% 9.00/5.03 = ( ord_less_set_int @ bot_bot_set_int @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_823_bot_Onot__eq__extremum,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ( ( A2 != bot_bot_nat )
% 9.00/5.03 = ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_824_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_real,A2: real] :
% 9.00/5.03 ( ( finite_finite_real @ A3 )
% 9.00/5.03 => ( ( member_real @ A2 @ A3 )
% 9.00/5.03 => ? [X5: real] :
% 9.00/5.03 ( ( member_real @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_real @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: real] :
% 9.00/5.03 ( ( member_real @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_825_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_o,A2: $o] :
% 9.00/5.03 ( ( finite_finite_o @ A3 )
% 9.00/5.03 => ( ( member_o @ A2 @ A3 )
% 9.00/5.03 => ? [X5: $o] :
% 9.00/5.03 ( ( member_o @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_o @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: $o] :
% 9.00/5.03 ( ( member_o @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_o @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_826_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_set_nat,A2: set_nat] :
% 9.00/5.03 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.03 => ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_set_nat @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_set_nat @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_827_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_Extended_enat,A2: extended_enat] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.03 => ( ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.03 & ( ord_le2932123472753598470d_enat @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_le2932123472753598470d_enat @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_828_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_set_int,A2: set_int] :
% 9.00/5.03 ( ( finite6197958912794628473et_int @ A3 )
% 9.00/5.03 => ( ( member_set_int @ A2 @ A3 )
% 9.00/5.03 => ? [X5: set_int] :
% 9.00/5.03 ( ( member_set_int @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_set_int @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: set_int] :
% 9.00/5.03 ( ( member_set_int @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_829_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_rat,A2: rat] :
% 9.00/5.03 ( ( finite_finite_rat @ A3 )
% 9.00/5.03 => ( ( member_rat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: rat] :
% 9.00/5.03 ( ( member_rat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_rat @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: rat] :
% 9.00/5.03 ( ( member_rat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_830_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_num,A2: num] :
% 9.00/5.03 ( ( finite_finite_num @ A3 )
% 9.00/5.03 => ( ( member_num @ A2 @ A3 )
% 9.00/5.03 => ? [X5: num] :
% 9.00/5.03 ( ( member_num @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_num @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: num] :
% 9.00/5.03 ( ( member_num @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_831_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_nat,A2: nat] :
% 9.00/5.03 ( ( finite_finite_nat @ A3 )
% 9.00/5.03 => ( ( member_nat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: nat] :
% 9.00/5.03 ( ( member_nat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_nat @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: nat] :
% 9.00/5.03 ( ( member_nat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_832_finite__has__minimal2,axiom,
% 9.00/5.03 ! [A3: set_int,A2: int] :
% 9.00/5.03 ( ( finite_finite_int @ A3 )
% 9.00/5.03 => ( ( member_int @ A2 @ A3 )
% 9.00/5.03 => ? [X5: int] :
% 9.00/5.03 ( ( member_int @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_int @ X5 @ A2 )
% 9.00/5.03 & ! [Xa: int] :
% 9.00/5.03 ( ( member_int @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ Xa @ X5 )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_833_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_real,A2: real] :
% 9.00/5.03 ( ( finite_finite_real @ A3 )
% 9.00/5.03 => ( ( member_real @ A2 @ A3 )
% 9.00/5.03 => ? [X5: real] :
% 9.00/5.03 ( ( member_real @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: real] :
% 9.00/5.03 ( ( member_real @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_real @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_834_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_o,A2: $o] :
% 9.00/5.03 ( ( finite_finite_o @ A3 )
% 9.00/5.03 => ( ( member_o @ A2 @ A3 )
% 9.00/5.03 => ? [X5: $o] :
% 9.00/5.03 ( ( member_o @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_o @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: $o] :
% 9.00/5.03 ( ( member_o @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_o @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_835_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_set_nat,A2: set_nat] :
% 9.00/5.03 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.03 => ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_set_nat @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_set_nat @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_836_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_Extended_enat,A2: extended_enat] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.03 => ( ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.03 & ( ord_le2932123472753598470d_enat @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_le2932123472753598470d_enat @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_837_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_set_int,A2: set_int] :
% 9.00/5.03 ( ( finite6197958912794628473et_int @ A3 )
% 9.00/5.03 => ( ( member_set_int @ A2 @ A3 )
% 9.00/5.03 => ? [X5: set_int] :
% 9.00/5.03 ( ( member_set_int @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_set_int @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: set_int] :
% 9.00/5.03 ( ( member_set_int @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_838_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_rat,A2: rat] :
% 9.00/5.03 ( ( finite_finite_rat @ A3 )
% 9.00/5.03 => ( ( member_rat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: rat] :
% 9.00/5.03 ( ( member_rat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_rat @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: rat] :
% 9.00/5.03 ( ( member_rat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_rat @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_839_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_num,A2: num] :
% 9.00/5.03 ( ( finite_finite_num @ A3 )
% 9.00/5.03 => ( ( member_num @ A2 @ A3 )
% 9.00/5.03 => ? [X5: num] :
% 9.00/5.03 ( ( member_num @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_num @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: num] :
% 9.00/5.03 ( ( member_num @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_num @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_840_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_nat,A2: nat] :
% 9.00/5.03 ( ( finite_finite_nat @ A3 )
% 9.00/5.03 => ( ( member_nat @ A2 @ A3 )
% 9.00/5.03 => ? [X5: nat] :
% 9.00/5.03 ( ( member_nat @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_nat @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: nat] :
% 9.00/5.03 ( ( member_nat @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_nat @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_841_finite__has__maximal2,axiom,
% 9.00/5.03 ! [A3: set_int,A2: int] :
% 9.00/5.03 ( ( finite_finite_int @ A3 )
% 9.00/5.03 => ( ( member_int @ A2 @ A3 )
% 9.00/5.03 => ? [X5: int] :
% 9.00/5.03 ( ( member_int @ X5 @ A3 )
% 9.00/5.03 & ( ord_less_eq_int @ A2 @ X5 )
% 9.00/5.03 & ! [Xa: int] :
% 9.00/5.03 ( ( member_int @ Xa @ A3 )
% 9.00/5.03 => ( ( ord_less_eq_int @ X5 @ Xa )
% 9.00/5.03 => ( X5 = Xa ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_842_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_complex] :
% 9.00/5.03 ( ~ ( finite3207457112153483333omplex @ S )
% 9.00/5.03 => ( S != bot_bot_set_complex ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_843_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_Pr1261947904930325089at_nat] :
% 9.00/5.03 ( ~ ( finite6177210948735845034at_nat @ S )
% 9.00/5.03 => ( S != bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_844_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_Extended_enat] :
% 9.00/5.03 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( S != bot_bo7653980558646680370d_enat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_845_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_real] :
% 9.00/5.03 ( ~ ( finite_finite_real @ S )
% 9.00/5.03 => ( S != bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_846_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_o] :
% 9.00/5.03 ( ~ ( finite_finite_o @ S )
% 9.00/5.03 => ( S != bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_847_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_nat] :
% 9.00/5.03 ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( S != bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_848_infinite__imp__nonempty,axiom,
% 9.00/5.03 ! [S: set_int] :
% 9.00/5.03 ( ~ ( finite_finite_int @ S )
% 9.00/5.03 => ( S != bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_849_finite_OemptyI,axiom,
% 9.00/5.03 finite3207457112153483333omplex @ bot_bot_set_complex ).
% 9.00/5.03
% 9.00/5.03 thf(fact_850_finite_OemptyI,axiom,
% 9.00/5.03 finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_851_finite_OemptyI,axiom,
% 9.00/5.03 finite4001608067531595151d_enat @ bot_bo7653980558646680370d_enat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_852_finite_OemptyI,axiom,
% 9.00/5.03 finite_finite_real @ bot_bot_set_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_853_finite_OemptyI,axiom,
% 9.00/5.03 finite_finite_o @ bot_bot_set_o ).
% 9.00/5.03
% 9.00/5.03 thf(fact_854_finite_OemptyI,axiom,
% 9.00/5.03 finite_finite_nat @ bot_bot_set_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_855_finite_OemptyI,axiom,
% 9.00/5.03 finite_finite_int @ bot_bot_set_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_856_member__valid__both__member__options,axiom,
% 9.00/5.03 ! [Tree: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ Tree @ N )
% 9.00/5.03 => ( ( vEBT_vebt_member @ Tree @ X2 )
% 9.00/5.03 => ( ( vEBT_V5719532721284313246member @ Tree @ X2 )
% 9.00/5.03 | ( vEBT_VEBT_membermima @ Tree @ X2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_857_buildup__nothing__in__min__max,axiom,
% 9.00/5.03 ! [N: nat,X2: nat] :
% 9.00/5.03 ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_858_dele__member__cont__corr,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
% 9.00/5.03 = ( ( X2 != Y3 )
% 9.00/5.03 & ( vEBT_vebt_member @ T @ Y3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_859_finite__nat__set__iff__bounded__le,axiom,
% 9.00/5.03 ( finite_finite_nat
% 9.00/5.03 = ( ^ [N4: set_nat] :
% 9.00/5.03 ? [M2: nat] :
% 9.00/5.03 ! [X: nat] :
% 9.00/5.03 ( ( member_nat @ X @ N4 )
% 9.00/5.03 => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_860_infinite__nat__iff__unbounded__le,axiom,
% 9.00/5.03 ! [S: set_nat] :
% 9.00/5.03 ( ( ~ ( finite_finite_nat @ S ) )
% 9.00/5.03 = ( ! [M2: nat] :
% 9.00/5.03 ? [N2: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.03 & ( member_nat @ N2 @ S ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_861_unbounded__k__infinite,axiom,
% 9.00/5.03 ! [K: nat,S: set_nat] :
% 9.00/5.03 ( ! [M4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ K @ M4 )
% 9.00/5.03 => ? [N5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M4 @ N5 )
% 9.00/5.03 & ( member_nat @ N5 @ S ) ) )
% 9.00/5.03 => ~ ( finite_finite_nat @ S ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_862_bounded__nat__set__is__finite,axiom,
% 9.00/5.03 ! [N6: set_nat,N: nat] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ( ( member_nat @ X5 @ N6 )
% 9.00/5.03 => ( ord_less_nat @ X5 @ N ) )
% 9.00/5.03 => ( finite_finite_nat @ N6 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_863_infinite__nat__iff__unbounded,axiom,
% 9.00/5.03 ! [S: set_nat] :
% 9.00/5.03 ( ( ~ ( finite_finite_nat @ S ) )
% 9.00/5.03 = ( ! [M2: nat] :
% 9.00/5.03 ? [N2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M2 @ N2 )
% 9.00/5.03 & ( member_nat @ N2 @ S ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_864_finite__nat__set__iff__bounded,axiom,
% 9.00/5.03 ( finite_finite_nat
% 9.00/5.03 = ( ^ [N4: set_nat] :
% 9.00/5.03 ? [M2: nat] :
% 9.00/5.03 ! [X: nat] :
% 9.00/5.03 ( ( member_nat @ X @ N4 )
% 9.00/5.03 => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_865_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_complex,F: complex > real] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_complex )
% 9.00/5.03 => ~ ? [X4: complex] :
% 9.00/5.03 ( ( member_complex @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_866_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ( S != bot_bo7653980558646680370d_enat )
% 9.00/5.03 => ~ ? [X4: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic1189837152898106425t_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_867_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_real,F: real > real] :
% 9.00/5.03 ( ( finite_finite_real @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_real )
% 9.00/5.03 => ~ ? [X4: real] :
% 9.00/5.03 ( ( member_real @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_868_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_o,F: $o > real] :
% 9.00/5.03 ( ( finite_finite_o @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_o )
% 9.00/5.03 => ~ ? [X4: $o] :
% 9.00/5.03 ( ( member_o @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic8697145971487455083o_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_869_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_nat,F: nat > real] :
% 9.00/5.03 ( ( finite_finite_nat @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_nat )
% 9.00/5.03 => ~ ? [X4: nat] :
% 9.00/5.03 ( ( member_nat @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_870_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_int,F: int > real] :
% 9.00/5.03 ( ( finite_finite_int @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_int )
% 9.00/5.03 => ~ ? [X4: int] :
% 9.00/5.03 ( ( member_int @ X4 @ S )
% 9.00/5.03 & ( ord_less_real @ ( F @ X4 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_871_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_complex,F: complex > rat] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_complex )
% 9.00/5.03 => ~ ? [X4: complex] :
% 9.00/5.03 ( ( member_complex @ X4 @ S )
% 9.00/5.03 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_872_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ( S != bot_bo7653980558646680370d_enat )
% 9.00/5.03 => ~ ? [X4: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ X4 @ S )
% 9.00/5.03 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_873_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_real,F: real > rat] :
% 9.00/5.03 ( ( finite_finite_real @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_real )
% 9.00/5.03 => ~ ? [X4: real] :
% 9.00/5.03 ( ( member_real @ X4 @ S )
% 9.00/5.03 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_874_arg__min__if__finite_I2_J,axiom,
% 9.00/5.03 ! [S: set_o,F: $o > rat] :
% 9.00/5.03 ( ( finite_finite_o @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_o )
% 9.00/5.03 => ~ ? [X4: $o] :
% 9.00/5.03 ( ( member_o @ X4 @ S )
% 9.00/5.03 & ( ord_less_rat @ ( F @ X4 ) @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_875_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_complex,Y3: complex,F: complex > rat] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_complex )
% 9.00/5.03 => ( ( member_complex @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_876_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > rat] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ( S != bot_bo7653980558646680370d_enat )
% 9.00/5.03 => ( ( member_Extended_enat @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic3210252021154270693at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_877_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_real,Y3: real,F: real > rat] :
% 9.00/5.03 ( ( finite_finite_real @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_real )
% 9.00/5.03 => ( ( member_real @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_878_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_o,Y3: $o,F: $o > rat] :
% 9.00/5.03 ( ( finite_finite_o @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_o )
% 9.00/5.03 => ( ( member_o @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic2140725968369957399_o_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_879_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_nat,Y3: nat,F: nat > rat] :
% 9.00/5.03 ( ( finite_finite_nat @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_nat )
% 9.00/5.03 => ( ( member_nat @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_880_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_int,Y3: int,F: int > rat] :
% 9.00/5.03 ( ( finite_finite_int @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_int )
% 9.00/5.03 => ( ( member_int @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_881_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_complex,Y3: complex,F: complex > num] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_complex )
% 9.00/5.03 => ( ( member_complex @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_882_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,Y3: extended_enat,F: extended_enat > num] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ( S != bot_bo7653980558646680370d_enat )
% 9.00/5.03 => ( ( member_Extended_enat @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ ( lattic402713867396545063at_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_883_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_real,Y3: real,F: real > num] :
% 9.00/5.03 ( ( finite_finite_real @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_real )
% 9.00/5.03 => ( ( member_real @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_884_arg__min__least,axiom,
% 9.00/5.03 ! [S: set_o,Y3: $o,F: $o > num] :
% 9.00/5.03 ( ( finite_finite_o @ S )
% 9.00/5.03 => ( ( S != bot_bot_set_o )
% 9.00/5.03 => ( ( member_o @ Y3 @ S )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ ( lattic8556559851467007577_o_num @ F @ S ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_885_nat__descend__induct,axiom,
% 9.00/5.03 ! [N: nat,P: nat > $o,M: nat] :
% 9.00/5.03 ( ! [K2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ N @ K2 )
% 9.00/5.03 => ( P @ K2 ) )
% 9.00/5.03 => ( ! [K2: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ K2 @ N )
% 9.00/5.03 => ( ! [I3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ K2 @ I3 )
% 9.00/5.03 => ( P @ I3 ) )
% 9.00/5.03 => ( P @ K2 ) ) )
% 9.00/5.03 => ( P @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_886_delete__pres__valid,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X2 ) @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_887_subsetI,axiom,
% 9.00/5.03 ! [A3: set_real,B2: set_real] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ( ( member_real @ X5 @ A3 )
% 9.00/5.03 => ( member_real @ X5 @ B2 ) )
% 9.00/5.03 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_888_subsetI,axiom,
% 9.00/5.03 ! [A3: set_o,B2: set_o] :
% 9.00/5.03 ( ! [X5: $o] :
% 9.00/5.03 ( ( member_o @ X5 @ A3 )
% 9.00/5.03 => ( member_o @ X5 @ B2 ) )
% 9.00/5.03 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_889_subsetI,axiom,
% 9.00/5.03 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.03 ( ! [X5: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.03 => ( member_set_nat @ X5 @ B2 ) )
% 9.00/5.03 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_890_subsetI,axiom,
% 9.00/5.03 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ( ( member_nat @ X5 @ A3 )
% 9.00/5.03 => ( member_nat @ X5 @ B2 ) )
% 9.00/5.03 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_891_subsetI,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ! [X5: int] :
% 9.00/5.03 ( ( member_int @ X5 @ A3 )
% 9.00/5.03 => ( member_int @ X5 @ B2 ) )
% 9.00/5.03 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_892_subset__antisym,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.03 => ( A3 = B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_893_psubsetI,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( A3 != B2 )
% 9.00/5.03 => ( ord_less_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_894_in__mono,axiom,
% 9.00/5.03 ! [A3: set_real,B2: set_real,X2: real] :
% 9.00/5.03 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.03 => ( ( member_real @ X2 @ A3 )
% 9.00/5.03 => ( member_real @ X2 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_895_in__mono,axiom,
% 9.00/5.03 ! [A3: set_o,B2: set_o,X2: $o] :
% 9.00/5.03 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.03 => ( ( member_o @ X2 @ A3 )
% 9.00/5.03 => ( member_o @ X2 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_896_in__mono,axiom,
% 9.00/5.03 ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat] :
% 9.00/5.03 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.03 => ( member_set_nat @ X2 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_897_in__mono,axiom,
% 9.00/5.03 ! [A3: set_nat,B2: set_nat,X2: nat] :
% 9.00/5.03 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.03 => ( member_nat @ X2 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_898_in__mono,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,X2: int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( member_int @ X2 @ A3 )
% 9.00/5.03 => ( member_int @ X2 @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_899_subsetD,axiom,
% 9.00/5.03 ! [A3: set_real,B2: set_real,C: real] :
% 9.00/5.03 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.03 => ( ( member_real @ C @ A3 )
% 9.00/5.03 => ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_900_subsetD,axiom,
% 9.00/5.03 ! [A3: set_o,B2: set_o,C: $o] :
% 9.00/5.03 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.03 => ( ( member_o @ C @ A3 )
% 9.00/5.03 => ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_901_subsetD,axiom,
% 9.00/5.03 ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.00/5.03 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_set_nat @ C @ A3 )
% 9.00/5.03 => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_902_subsetD,axiom,
% 9.00/5.03 ! [A3: set_nat,B2: set_nat,C: nat] :
% 9.00/5.03 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_nat @ C @ A3 )
% 9.00/5.03 => ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_903_subsetD,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,C: int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( member_int @ C @ A3 )
% 9.00/5.03 => ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_904_psubsetD,axiom,
% 9.00/5.03 ! [A3: set_real,B2: set_real,C: real] :
% 9.00/5.03 ( ( ord_less_set_real @ A3 @ B2 )
% 9.00/5.03 => ( ( member_real @ C @ A3 )
% 9.00/5.03 => ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_905_psubsetD,axiom,
% 9.00/5.03 ! [A3: set_o,B2: set_o,C: $o] :
% 9.00/5.03 ( ( ord_less_set_o @ A3 @ B2 )
% 9.00/5.03 => ( ( member_o @ C @ A3 )
% 9.00/5.03 => ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_906_psubsetD,axiom,
% 9.00/5.03 ! [A3: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.00/5.03 ( ( ord_less_set_set_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_set_nat @ C @ A3 )
% 9.00/5.03 => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_907_psubsetD,axiom,
% 9.00/5.03 ! [A3: set_nat,B2: set_nat,C: nat] :
% 9.00/5.03 ( ( ord_less_set_nat @ A3 @ B2 )
% 9.00/5.03 => ( ( member_nat @ C @ A3 )
% 9.00/5.03 => ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_908_psubsetD,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,C: int] :
% 9.00/5.03 ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( member_int @ C @ A3 )
% 9.00/5.03 => ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_909_psubsetE,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.03 => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_910_equalityE,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( A3 = B2 )
% 9.00/5.03 => ~ ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_911_subset__eq,axiom,
% 9.00/5.03 ( ord_less_eq_set_real
% 9.00/5.03 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.03 ! [X: real] :
% 9.00/5.03 ( ( member_real @ X @ A6 )
% 9.00/5.03 => ( member_real @ X @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_912_subset__eq,axiom,
% 9.00/5.03 ( ord_less_eq_set_o
% 9.00/5.03 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.03 ! [X: $o] :
% 9.00/5.03 ( ( member_o @ X @ A6 )
% 9.00/5.03 => ( member_o @ X @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_913_subset__eq,axiom,
% 9.00/5.03 ( ord_le6893508408891458716et_nat
% 9.00/5.03 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.03 ! [X: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ X @ A6 )
% 9.00/5.03 => ( member_set_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_914_subset__eq,axiom,
% 9.00/5.03 ( ord_less_eq_set_nat
% 9.00/5.03 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.03 ! [X: nat] :
% 9.00/5.03 ( ( member_nat @ X @ A6 )
% 9.00/5.03 => ( member_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_915_subset__eq,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ! [X: int] :
% 9.00/5.03 ( ( member_int @ X @ A6 )
% 9.00/5.03 => ( member_int @ X @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_916_equalityD1,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( A3 = B2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_917_equalityD2,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( A3 = B2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ B2 @ A3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_918_psubset__eq,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.00/5.03 & ( A6 != B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_919_subset__iff,axiom,
% 9.00/5.03 ( ord_less_eq_set_real
% 9.00/5.03 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.03 ! [T3: real] :
% 9.00/5.03 ( ( member_real @ T3 @ A6 )
% 9.00/5.03 => ( member_real @ T3 @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_920_subset__iff,axiom,
% 9.00/5.03 ( ord_less_eq_set_o
% 9.00/5.03 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.03 ! [T3: $o] :
% 9.00/5.03 ( ( member_o @ T3 @ A6 )
% 9.00/5.03 => ( member_o @ T3 @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_921_subset__iff,axiom,
% 9.00/5.03 ( ord_le6893508408891458716et_nat
% 9.00/5.03 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.03 ! [T3: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ T3 @ A6 )
% 9.00/5.03 => ( member_set_nat @ T3 @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_922_subset__iff,axiom,
% 9.00/5.03 ( ord_less_eq_set_nat
% 9.00/5.03 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.03 ! [T3: nat] :
% 9.00/5.03 ( ( member_nat @ T3 @ A6 )
% 9.00/5.03 => ( member_nat @ T3 @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_923_subset__iff,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ! [T3: int] :
% 9.00/5.03 ( ( member_int @ T3 @ A6 )
% 9.00/5.03 => ( member_int @ T3 @ B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_924_subset__refl,axiom,
% 9.00/5.03 ! [A3: set_int] : ( ord_less_eq_set_int @ A3 @ A3 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_925_Collect__mono,axiom,
% 9.00/5.03 ! [P: real > $o,Q: real > $o] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( Q @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_926_Collect__mono,axiom,
% 9.00/5.03 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.03 ( ! [X5: list_nat] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( Q @ X5 ) )
% 9.00/5.03 => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_927_Collect__mono,axiom,
% 9.00/5.03 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.03 ( ! [X5: set_nat] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( Q @ X5 ) )
% 9.00/5.03 => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_928_Collect__mono,axiom,
% 9.00/5.03 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( Q @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_929_Collect__mono,axiom,
% 9.00/5.03 ! [P: int > $o,Q: int > $o] :
% 9.00/5.03 ( ! [X5: int] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( Q @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_930_subset__trans,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ A3 @ C2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_931_set__eq__subset,axiom,
% 9.00/5.03 ( ( ^ [Y6: set_int,Z3: set_int] : ( Y6 = Z3 ) )
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.00/5.03 & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_932_Collect__mono__iff,axiom,
% 9.00/5.03 ! [P: real > $o,Q: real > $o] :
% 9.00/5.03 ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 9.00/5.03 = ( ! [X: real] :
% 9.00/5.03 ( ( P @ X )
% 9.00/5.03 => ( Q @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_933_Collect__mono__iff,axiom,
% 9.00/5.03 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.03 ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 9.00/5.03 = ( ! [X: list_nat] :
% 9.00/5.03 ( ( P @ X )
% 9.00/5.03 => ( Q @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_934_Collect__mono__iff,axiom,
% 9.00/5.03 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.03 ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 9.00/5.03 = ( ! [X: set_nat] :
% 9.00/5.03 ( ( P @ X )
% 9.00/5.03 => ( Q @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_935_Collect__mono__iff,axiom,
% 9.00/5.03 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.03 ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 9.00/5.03 = ( ! [X: nat] :
% 9.00/5.03 ( ( P @ X )
% 9.00/5.03 => ( Q @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_936_Collect__mono__iff,axiom,
% 9.00/5.03 ! [P: int > $o,Q: int > $o] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 9.00/5.03 = ( ! [X: int] :
% 9.00/5.03 ( ( P @ X )
% 9.00/5.03 => ( Q @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_937_psubset__imp__subset,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_938_psubset__subset__trans,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.00/5.03 => ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_939_subset__not__subset__eq,axiom,
% 9.00/5.03 ( ord_less_set_int
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A6 @ B6 )
% 9.00/5.03 & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_940_subset__psubset__trans,axiom,
% 9.00/5.03 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.03 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.03 => ( ( ord_less_set_int @ B2 @ C2 )
% 9.00/5.03 => ( ord_less_set_int @ A3 @ C2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_941_subset__iff__psubset__eq,axiom,
% 9.00/5.03 ( ord_less_eq_set_int
% 9.00/5.03 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.03 ( ( ord_less_set_int @ A6 @ B6 )
% 9.00/5.03 | ( A6 = B6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_942_bounded__Max__nat,axiom,
% 9.00/5.03 ! [P: nat > $o,X2: nat,M5: nat] :
% 9.00/5.03 ( ( P @ X2 )
% 9.00/5.03 => ( ! [X5: nat] :
% 9.00/5.03 ( ( P @ X5 )
% 9.00/5.03 => ( ord_less_eq_nat @ X5 @ M5 ) )
% 9.00/5.03 => ~ ! [M4: nat] :
% 9.00/5.03 ( ( P @ M4 )
% 9.00/5.03 => ~ ! [X4: nat] :
% 9.00/5.03 ( ( P @ X4 )
% 9.00/5.03 => ( ord_less_eq_nat @ X4 @ M4 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_943_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_set_nat,R: set_nat > set_nat > $o] :
% 9.00/5.03 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.03 => ( ! [X5: set_nat] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: set_nat] :
% 9.00/5.03 ( ( member_set_nat @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_set_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_944_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_complex,R: complex > complex > $o] :
% 9.00/5.03 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.03 => ( ! [X5: complex] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: complex,Y4: complex,Z4: complex] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: complex] :
% 9.00/5.03 ( ( member_complex @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: complex] :
% 9.00/5.03 ( ( member_complex @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_complex ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_945_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.00/5.03 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.03 => ( ! [X5: product_prod_nat_nat] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: product_prod_nat_nat] :
% 9.00/5.03 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: product_prod_nat_nat] :
% 9.00/5.03 ( ( member8440522571783428010at_nat @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_946_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_Extended_enat,R: extended_enat > extended_enat > $o] :
% 9.00/5.03 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.03 => ( ! [X5: extended_enat] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: extended_enat,Y4: extended_enat,Z4: extended_enat] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: extended_enat] :
% 9.00/5.03 ( ( member_Extended_enat @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bo7653980558646680370d_enat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_947_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_real,R: real > real > $o] :
% 9.00/5.03 ( ( finite_finite_real @ A3 )
% 9.00/5.03 => ( ! [X5: real] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: real,Y4: real,Z4: real] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: real] :
% 9.00/5.03 ( ( member_real @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: real] :
% 9.00/5.03 ( ( member_real @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_real ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_948_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_o,R: $o > $o > $o] :
% 9.00/5.03 ( ( finite_finite_o @ A3 )
% 9.00/5.03 => ( ! [X5: $o] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: $o,Y4: $o,Z4: $o] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: $o] :
% 9.00/5.03 ( ( member_o @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: $o] :
% 9.00/5.03 ( ( member_o @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_o ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_949_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_nat,R: nat > nat > $o] :
% 9.00/5.03 ( ( finite_finite_nat @ A3 )
% 9.00/5.03 => ( ! [X5: nat] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: nat] :
% 9.00/5.03 ( ( member_nat @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: nat] :
% 9.00/5.03 ( ( member_nat @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_950_finite__transitivity__chain,axiom,
% 9.00/5.03 ! [A3: set_int,R: int > int > $o] :
% 9.00/5.03 ( ( finite_finite_int @ A3 )
% 9.00/5.03 => ( ! [X5: int] :
% 9.00/5.03 ~ ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: int,Y4: int,Z4: int] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [X5: int] :
% 9.00/5.03 ( ( member_int @ X5 @ A3 )
% 9.00/5.03 => ? [Y5: int] :
% 9.00/5.03 ( ( member_int @ Y5 @ A3 )
% 9.00/5.03 & ( R @ X5 @ Y5 ) ) )
% 9.00/5.03 => ( A3 = bot_bot_set_int ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_951_bot__empty__eq,axiom,
% 9.00/5.03 ( bot_bot_set_nat_o
% 9.00/5.03 = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_952_bot__empty__eq,axiom,
% 9.00/5.03 ( bot_bot_real_o
% 9.00/5.03 = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_953_bot__empty__eq,axiom,
% 9.00/5.03 ( bot_bot_o_o
% 9.00/5.03 = ( ^ [X: $o] : ( member_o @ X @ bot_bot_set_o ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_954_bot__empty__eq,axiom,
% 9.00/5.03 ( bot_bot_nat_o
% 9.00/5.03 = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_955_bot__empty__eq,axiom,
% 9.00/5.03 ( bot_bot_int_o
% 9.00/5.03 = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_956_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: list_nat > $o] :
% 9.00/5.03 ( ( ( collect_list_nat @ P )
% 9.00/5.03 = bot_bot_set_list_nat )
% 9.00/5.03 = ( P = bot_bot_list_nat_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_957_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: set_nat > $o] :
% 9.00/5.03 ( ( ( collect_set_nat @ P )
% 9.00/5.03 = bot_bot_set_set_nat )
% 9.00/5.03 = ( P = bot_bot_set_nat_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_958_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: real > $o] :
% 9.00/5.03 ( ( ( collect_real @ P )
% 9.00/5.03 = bot_bot_set_real )
% 9.00/5.03 = ( P = bot_bot_real_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_959_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: $o > $o] :
% 9.00/5.03 ( ( ( collect_o @ P )
% 9.00/5.03 = bot_bot_set_o )
% 9.00/5.03 = ( P = bot_bot_o_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_960_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: nat > $o] :
% 9.00/5.03 ( ( ( collect_nat @ P )
% 9.00/5.03 = bot_bot_set_nat )
% 9.00/5.03 = ( P = bot_bot_nat_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_961_Collect__empty__eq__bot,axiom,
% 9.00/5.03 ! [P: int > $o] :
% 9.00/5.03 ( ( ( collect_int @ P )
% 9.00/5.03 = bot_bot_set_int )
% 9.00/5.03 = ( P = bot_bot_int_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_962_subset__emptyI,axiom,
% 9.00/5.03 ! [A3: set_set_nat] :
% 9.00/5.03 ( ! [X5: set_nat] :
% 9.00/5.03 ~ ( member_set_nat @ X5 @ A3 )
% 9.00/5.03 => ( ord_le6893508408891458716et_nat @ A3 @ bot_bot_set_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_963_subset__emptyI,axiom,
% 9.00/5.03 ! [A3: set_real] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ~ ( member_real @ X5 @ A3 )
% 9.00/5.03 => ( ord_less_eq_set_real @ A3 @ bot_bot_set_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_964_subset__emptyI,axiom,
% 9.00/5.03 ! [A3: set_o] :
% 9.00/5.03 ( ! [X5: $o] :
% 9.00/5.03 ~ ( member_o @ X5 @ A3 )
% 9.00/5.03 => ( ord_less_eq_set_o @ A3 @ bot_bot_set_o ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_965_subset__emptyI,axiom,
% 9.00/5.03 ! [A3: set_nat] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ~ ( member_nat @ X5 @ A3 )
% 9.00/5.03 => ( ord_less_eq_set_nat @ A3 @ bot_bot_set_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_966_subset__emptyI,axiom,
% 9.00/5.03 ! [A3: set_int] :
% 9.00/5.03 ( ! [X5: int] :
% 9.00/5.03 ~ ( member_int @ X5 @ A3 )
% 9.00/5.03 => ( ord_less_eq_set_int @ A3 @ bot_bot_set_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_967_field__lbound__gt__zero,axiom,
% 9.00/5.03 ! [D1: real,D2: real] :
% 9.00/5.03 ( ( ord_less_real @ zero_zero_real @ D1 )
% 9.00/5.03 => ( ( ord_less_real @ zero_zero_real @ D2 )
% 9.00/5.03 => ? [E: real] :
% 9.00/5.03 ( ( ord_less_real @ zero_zero_real @ E )
% 9.00/5.03 & ( ord_less_real @ E @ D1 )
% 9.00/5.03 & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_968_field__lbound__gt__zero,axiom,
% 9.00/5.03 ! [D1: rat,D2: rat] :
% 9.00/5.03 ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 9.00/5.03 => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 9.00/5.03 => ? [E: rat] :
% 9.00/5.03 ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.00/5.03 & ( ord_less_rat @ E @ D1 )
% 9.00/5.03 & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_969_less__numeral__extra_I3_J,axiom,
% 9.00/5.03 ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_970_less__numeral__extra_I3_J,axiom,
% 9.00/5.03 ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_971_less__numeral__extra_I3_J,axiom,
% 9.00/5.03 ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_972_less__numeral__extra_I3_J,axiom,
% 9.00/5.03 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_973_complete__interval,axiom,
% 9.00/5.03 ! [A2: real,B3: real,P: real > $o] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.03 => ( ( P @ A2 )
% 9.00/5.03 => ( ~ ( P @ B3 )
% 9.00/5.03 => ? [C3: real] :
% 9.00/5.03 ( ( ord_less_eq_real @ A2 @ C3 )
% 9.00/5.03 & ( ord_less_eq_real @ C3 @ B3 )
% 9.00/5.03 & ! [X4: real] :
% 9.00/5.03 ( ( ( ord_less_eq_real @ A2 @ X4 )
% 9.00/5.03 & ( ord_less_real @ X4 @ C3 ) )
% 9.00/5.03 => ( P @ X4 ) )
% 9.00/5.03 & ! [D3: real] :
% 9.00/5.03 ( ! [X5: real] :
% 9.00/5.03 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.03 & ( ord_less_real @ X5 @ D3 ) )
% 9.00/5.03 => ( P @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_real @ D3 @ C3 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_974_complete__interval,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat,P: nat > $o] :
% 9.00/5.03 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.03 => ( ( P @ A2 )
% 9.00/5.03 => ( ~ ( P @ B3 )
% 9.00/5.03 => ? [C3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ A2 @ C3 )
% 9.00/5.03 & ( ord_less_eq_nat @ C3 @ B3 )
% 9.00/5.03 & ! [X4: nat] :
% 9.00/5.03 ( ( ( ord_less_eq_nat @ A2 @ X4 )
% 9.00/5.03 & ( ord_less_nat @ X4 @ C3 ) )
% 9.00/5.03 => ( P @ X4 ) )
% 9.00/5.03 & ! [D3: nat] :
% 9.00/5.03 ( ! [X5: nat] :
% 9.00/5.03 ( ( ( ord_less_eq_nat @ A2 @ X5 )
% 9.00/5.03 & ( ord_less_nat @ X5 @ D3 ) )
% 9.00/5.03 => ( P @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_975_complete__interval,axiom,
% 9.00/5.03 ! [A2: int,B3: int,P: int > $o] :
% 9.00/5.03 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.03 => ( ( P @ A2 )
% 9.00/5.03 => ( ~ ( P @ B3 )
% 9.00/5.03 => ? [C3: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ C3 )
% 9.00/5.03 & ( ord_less_eq_int @ C3 @ B3 )
% 9.00/5.03 & ! [X4: int] :
% 9.00/5.03 ( ( ( ord_less_eq_int @ A2 @ X4 )
% 9.00/5.03 & ( ord_less_int @ X4 @ C3 ) )
% 9.00/5.03 => ( P @ X4 ) )
% 9.00/5.03 & ! [D3: int] :
% 9.00/5.03 ( ! [X5: int] :
% 9.00/5.03 ( ( ( ord_less_eq_int @ A2 @ X5 )
% 9.00/5.03 & ( ord_less_int @ X5 @ D3 ) )
% 9.00/5.03 => ( P @ X5 ) )
% 9.00/5.03 => ( ord_less_eq_int @ D3 @ C3 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_976_verit__comp__simplify1_I3_J,axiom,
% 9.00/5.03 ! [B7: real,A7: real] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_real @ B7 @ A7 ) )
% 9.00/5.03 = ( ord_less_real @ A7 @ B7 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_977_verit__comp__simplify1_I3_J,axiom,
% 9.00/5.03 ! [B7: rat,A7: rat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_rat @ B7 @ A7 ) )
% 9.00/5.03 = ( ord_less_rat @ A7 @ B7 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_978_verit__comp__simplify1_I3_J,axiom,
% 9.00/5.03 ! [B7: num,A7: num] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_num @ B7 @ A7 ) )
% 9.00/5.03 = ( ord_less_num @ A7 @ B7 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_979_verit__comp__simplify1_I3_J,axiom,
% 9.00/5.03 ! [B7: nat,A7: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_nat @ B7 @ A7 ) )
% 9.00/5.03 = ( ord_less_nat @ A7 @ B7 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_980_verit__comp__simplify1_I3_J,axiom,
% 9.00/5.03 ! [B7: int,A7: int] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_int @ B7 @ A7 ) )
% 9.00/5.03 = ( ord_less_int @ A7 @ B7 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_981_pinf_I6_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_eq_real @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_982_pinf_I6_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_983_pinf_I6_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_984_pinf_I6_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_985_pinf_I6_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_986_pinf_I8_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_eq_real @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_987_pinf_I8_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_eq_rat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_988_pinf_I8_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_eq_num @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_989_pinf_I8_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_eq_nat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_990_pinf_I8_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_eq_int @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_991_minf_I6_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_eq_real @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_992_minf_I6_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_eq_rat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_993_minf_I6_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_eq_num @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_994_minf_I6_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_eq_nat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_995_minf_I6_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_eq_int @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_996_minf_I8_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_eq_real @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_997_minf_I8_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_eq_rat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_998_minf_I8_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_eq_num @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_999_minf_I8_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1000_minf_I8_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_eq_int @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1001_verit__la__disequality,axiom,
% 9.00/5.03 ! [A2: rat,B3: rat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 | ~ ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.03 | ~ ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1002_verit__la__disequality,axiom,
% 9.00/5.03 ! [A2: num,B3: num] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 | ~ ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.03 | ~ ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1003_verit__la__disequality,axiom,
% 9.00/5.03 ! [A2: nat,B3: nat] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 | ~ ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.03 | ~ ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1004_verit__la__disequality,axiom,
% 9.00/5.03 ! [A2: int,B3: int] :
% 9.00/5.03 ( ( A2 = B3 )
% 9.00/5.03 | ~ ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.03 | ~ ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1005_verit__comp__simplify1_I2_J,axiom,
% 9.00/5.03 ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1006_verit__comp__simplify1_I2_J,axiom,
% 9.00/5.03 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1007_verit__comp__simplify1_I2_J,axiom,
% 9.00/5.03 ! [A2: num] : ( ord_less_eq_num @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1008_verit__comp__simplify1_I2_J,axiom,
% 9.00/5.03 ! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1009_verit__comp__simplify1_I2_J,axiom,
% 9.00/5.03 ! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1010_minf_I7_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_real @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1011_minf_I7_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_rat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1012_minf_I7_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_num @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1013_minf_I7_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_nat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1014_minf_I7_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ~ ( ord_less_int @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1015_minf_I5_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_real @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1016_minf_I5_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_rat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1017_minf_I5_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_num @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1018_minf_I5_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_nat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1019_minf_I5_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( ord_less_int @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1020_minf_I4_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1021_minf_I4_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1022_minf_I4_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1023_minf_I4_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1024_minf_I4_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1025_minf_I3_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1026_minf_I3_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1027_minf_I3_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1028_minf_I3_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1029_minf_I3_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1030_minf_I2_J,axiom,
% 9.00/5.03 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 9.00/5.03 ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1031_minf_I2_J,axiom,
% 9.00/5.03 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.00/5.03 ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1032_minf_I2_J,axiom,
% 9.00/5.03 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 9.00/5.03 ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1033_minf_I2_J,axiom,
% 9.00/5.03 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.00/5.03 ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1034_minf_I2_J,axiom,
% 9.00/5.03 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 9.00/5.03 ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1035_minf_I1_J,axiom,
% 9.00/5.03 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 9.00/5.03 ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1036_minf_I1_J,axiom,
% 9.00/5.03 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.00/5.03 ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1037_minf_I1_J,axiom,
% 9.00/5.03 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 9.00/5.03 ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1038_minf_I1_J,axiom,
% 9.00/5.03 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.00/5.03 ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1039_minf_I1_J,axiom,
% 9.00/5.03 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 9.00/5.03 ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ X4 @ Z4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1040_pinf_I7_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_real @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1041_pinf_I7_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_rat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1042_pinf_I7_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_num @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1043_pinf_I7_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_nat @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1044_pinf_I7_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( ord_less_int @ T @ X4 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1045_pinf_I5_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_real @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1046_pinf_I5_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_rat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1047_pinf_I5_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_num @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1048_pinf_I5_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_nat @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1049_pinf_I5_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ~ ( ord_less_int @ X4 @ T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1050_pinf_I4_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1051_pinf_I4_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1052_pinf_I4_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1053_pinf_I4_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1054_pinf_I4_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1055_pinf_I3_J,axiom,
% 9.00/5.03 ! [T: real] :
% 9.00/5.03 ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1056_pinf_I3_J,axiom,
% 9.00/5.03 ! [T: rat] :
% 9.00/5.03 ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1057_pinf_I3_J,axiom,
% 9.00/5.03 ! [T: num] :
% 9.00/5.03 ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1058_pinf_I3_J,axiom,
% 9.00/5.03 ! [T: nat] :
% 9.00/5.03 ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1059_pinf_I3_J,axiom,
% 9.00/5.03 ! [T: int] :
% 9.00/5.03 ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( X4 != T ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1060_pinf_I2_J,axiom,
% 9.00/5.03 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 9.00/5.03 ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1061_pinf_I2_J,axiom,
% 9.00/5.03 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.00/5.03 ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1062_pinf_I2_J,axiom,
% 9.00/5.03 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 9.00/5.03 ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1063_pinf_I2_J,axiom,
% 9.00/5.03 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.00/5.03 ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1064_pinf_I2_J,axiom,
% 9.00/5.03 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 9.00/5.03 ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 | ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 | ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1065_pinf_I1_J,axiom,
% 9.00/5.03 ! [P: real > $o,P4: real > $o,Q: real > $o,Q2: real > $o] :
% 9.00/5.03 ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: real] :
% 9.00/5.03 ! [X5: real] :
% 9.00/5.03 ( ( ord_less_real @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: real] :
% 9.00/5.03 ! [X4: real] :
% 9.00/5.03 ( ( ord_less_real @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1066_pinf_I1_J,axiom,
% 9.00/5.03 ! [P: rat > $o,P4: rat > $o,Q: rat > $o,Q2: rat > $o] :
% 9.00/5.03 ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: rat] :
% 9.00/5.03 ! [X5: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: rat] :
% 9.00/5.03 ! [X4: rat] :
% 9.00/5.03 ( ( ord_less_rat @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1067_pinf_I1_J,axiom,
% 9.00/5.03 ! [P: num > $o,P4: num > $o,Q: num > $o,Q2: num > $o] :
% 9.00/5.03 ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: num] :
% 9.00/5.03 ! [X5: num] :
% 9.00/5.03 ( ( ord_less_num @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: num] :
% 9.00/5.03 ! [X4: num] :
% 9.00/5.03 ( ( ord_less_num @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1068_pinf_I1_J,axiom,
% 9.00/5.03 ! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
% 9.00/5.03 ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: nat] :
% 9.00/5.03 ! [X5: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: nat] :
% 9.00/5.03 ! [X4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1069_pinf_I1_J,axiom,
% 9.00/5.03 ! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
% 9.00/5.03 ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.03 => ( ( P @ X5 )
% 9.00/5.03 = ( P4 @ X5 ) ) )
% 9.00/5.03 => ( ? [Z5: int] :
% 9.00/5.03 ! [X5: int] :
% 9.00/5.03 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.03 => ( ( Q @ X5 )
% 9.00/5.03 = ( Q2 @ X5 ) ) )
% 9.00/5.03 => ? [Z4: int] :
% 9.00/5.03 ! [X4: int] :
% 9.00/5.03 ( ( ord_less_int @ Z4 @ X4 )
% 9.00/5.03 => ( ( ( P @ X4 )
% 9.00/5.03 & ( Q @ X4 ) )
% 9.00/5.03 = ( ( P4 @ X4 )
% 9.00/5.03 & ( Q2 @ X4 ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1070_verit__comp__simplify1_I1_J,axiom,
% 9.00/5.03 ! [A2: real] :
% 9.00/5.03 ~ ( ord_less_real @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1071_verit__comp__simplify1_I1_J,axiom,
% 9.00/5.03 ! [A2: rat] :
% 9.00/5.03 ~ ( ord_less_rat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1072_verit__comp__simplify1_I1_J,axiom,
% 9.00/5.03 ! [A2: num] :
% 9.00/5.03 ~ ( ord_less_num @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1073_verit__comp__simplify1_I1_J,axiom,
% 9.00/5.03 ! [A2: nat] :
% 9.00/5.03 ~ ( ord_less_nat @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1074_verit__comp__simplify1_I1_J,axiom,
% 9.00/5.03 ! [A2: int] :
% 9.00/5.03 ~ ( ord_less_int @ A2 @ A2 ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1075_ex__gt__or__lt,axiom,
% 9.00/5.03 ! [A2: real] :
% 9.00/5.03 ? [B: real] :
% 9.00/5.03 ( ( ord_less_real @ A2 @ B )
% 9.00/5.03 | ( ord_less_real @ B @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1076_le__numeral__extra_I3_J,axiom,
% 9.00/5.03 ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1077_le__numeral__extra_I3_J,axiom,
% 9.00/5.03 ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1078_le__numeral__extra_I3_J,axiom,
% 9.00/5.03 ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1079_le__numeral__extra_I3_J,axiom,
% 9.00/5.03 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1080_both__member__options__def,axiom,
% 9.00/5.03 ( vEBT_V8194947554948674370ptions
% 9.00/5.03 = ( ^ [T3: vEBT_VEBT,X: nat] :
% 9.00/5.03 ( ( vEBT_V5719532721284313246member @ T3 @ X )
% 9.00/5.03 | ( vEBT_VEBT_membermima @ T3 @ X ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1081_valid__member__both__member__options,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 9.00/5.03 => ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1082_both__member__options__equiv__member,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 9.00/5.03 = ( vEBT_vebt_member @ T @ X2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1083_Leaf__0__not,axiom,
% 9.00/5.03 ! [A2: $o,B3: $o] :
% 9.00/5.03 ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1084_dele__bmo__cont__corr,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X2 ) @ Y3 )
% 9.00/5.03 = ( ( X2 != Y3 )
% 9.00/5.03 & ( vEBT_V8194947554948674370ptions @ T @ Y3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1085_maxbmo,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,X2: nat] :
% 9.00/5.03 ( ( ( vEBT_vebt_maxt @ T )
% 9.00/5.03 = ( some_nat @ X2 ) )
% 9.00/5.03 => ( vEBT_V8194947554948674370ptions @ T @ X2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1086_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.00/5.03 ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 9.00/5.03 ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 9.00/5.03 = none_P5556105721700978146at_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1087_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.00/5.03 ! [Uw: num > num > num,V: num] :
% 9.00/5.03 ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 9.00/5.03 = none_num ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1088_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.00/5.03 ! [Uw: nat > nat > nat,V: nat] :
% 9.00/5.03 ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 9.00/5.03 = none_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1089_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.00/5.03 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 9.00/5.03 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 9.00/5.03 = Y3 )
% 9.00/5.03 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.00/5.03 => ( Y3 != none_P5556105721700978146at_nat ) )
% 9.00/5.03 => ( ( ? [V2: product_prod_nat_nat] :
% 9.00/5.03 ( Xa2
% 9.00/5.03 = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.00/5.03 => ( ( Xb = none_P5556105721700978146at_nat )
% 9.00/5.03 => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 9.00/5.03 => ~ ! [A: product_prod_nat_nat] :
% 9.00/5.03 ( ( Xa2
% 9.00/5.03 = ( some_P7363390416028606310at_nat @ A ) )
% 9.00/5.03 => ! [B: product_prod_nat_nat] :
% 9.00/5.03 ( ( Xb
% 9.00/5.03 = ( some_P7363390416028606310at_nat @ B ) )
% 9.00/5.03 => ( Y3
% 9.00/5.03 != ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1090_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.00/5.03 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 9.00/5.03 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 9.00/5.03 = Y3 )
% 9.00/5.03 => ( ( ( Xa2 = none_num )
% 9.00/5.03 => ( Y3 != none_num ) )
% 9.00/5.03 => ( ( ? [V2: num] :
% 9.00/5.03 ( Xa2
% 9.00/5.03 = ( some_num @ V2 ) )
% 9.00/5.03 => ( ( Xb = none_num )
% 9.00/5.03 => ( Y3 != none_num ) ) )
% 9.00/5.03 => ~ ! [A: num] :
% 9.00/5.03 ( ( Xa2
% 9.00/5.03 = ( some_num @ A ) )
% 9.00/5.03 => ! [B: num] :
% 9.00/5.03 ( ( Xb
% 9.00/5.03 = ( some_num @ B ) )
% 9.00/5.03 => ( Y3
% 9.00/5.03 != ( some_num @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1091_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.00/5.03 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 9.00/5.03 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 9.00/5.03 = Y3 )
% 9.00/5.03 => ( ( ( Xa2 = none_nat )
% 9.00/5.03 => ( Y3 != none_nat ) )
% 9.00/5.03 => ( ( ? [V2: nat] :
% 9.00/5.03 ( Xa2
% 9.00/5.03 = ( some_nat @ V2 ) )
% 9.00/5.03 => ( ( Xb = none_nat )
% 9.00/5.03 => ( Y3 != none_nat ) ) )
% 9.00/5.03 => ~ ! [A: nat] :
% 9.00/5.03 ( ( Xa2
% 9.00/5.03 = ( some_nat @ A ) )
% 9.00/5.03 => ! [B: nat] :
% 9.00/5.03 ( ( Xb
% 9.00/5.03 = ( some_nat @ B ) )
% 9.00/5.03 => ( Y3
% 9.00/5.03 != ( some_nat @ ( X2 @ A @ B ) ) ) ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1092_Set_Ois__empty__def,axiom,
% 9.00/5.03 ( is_empty_real
% 9.00/5.03 = ( ^ [A6: set_real] : ( A6 = bot_bot_set_real ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1093_Set_Ois__empty__def,axiom,
% 9.00/5.03 ( is_empty_o
% 9.00/5.03 = ( ^ [A6: set_o] : ( A6 = bot_bot_set_o ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1094_Set_Ois__empty__def,axiom,
% 9.00/5.03 ( is_empty_nat
% 9.00/5.03 = ( ^ [A6: set_nat] : ( A6 = bot_bot_set_nat ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1095_Set_Ois__empty__def,axiom,
% 9.00/5.03 ( is_empty_int
% 9.00/5.03 = ( ^ [A6: set_int] : ( A6 = bot_bot_set_int ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1096_of__nat__0__less__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1097_of__nat__0__less__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1098_of__nat__0__less__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1099_of__nat__0__less__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1100_enumerate__mono__iff,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,M: nat,N: nat] :
% 9.00/5.03 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1101_enumerate__mono__iff,axiom,
% 9.00/5.03 ! [S: set_nat,M: nat,N: nat] :
% 9.00/5.03 ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1102_not__min__Null__member,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT] :
% 9.00/5.03 ( ~ ( vEBT_VEBT_minNull @ T )
% 9.00/5.03 => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1103_of__nat__eq__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ( semiri1314217659103216013at_int @ M )
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( M = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1104_of__nat__eq__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ( semiri5074537144036343181t_real @ M )
% 9.00/5.03 = ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( M = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1105_of__nat__0,axiom,
% 9.00/5.03 ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 9.00/5.03 = zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1106_of__nat__0,axiom,
% 9.00/5.03 ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 9.00/5.03 = zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1107_of__nat__0,axiom,
% 9.00/5.03 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.00/5.03 = zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1108_of__nat__0,axiom,
% 9.00/5.03 ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 9.00/5.03 = zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1109_of__nat__0__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( zero_zero_rat
% 9.00/5.03 = ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 = ( zero_zero_nat = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1110_of__nat__0__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( zero_zero_nat
% 9.00/5.03 = ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 = ( zero_zero_nat = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1111_of__nat__0__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( zero_zero_int
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( zero_zero_nat = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1112_of__nat__0__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( zero_zero_real
% 9.00/5.03 = ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( zero_zero_nat = N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1113_of__nat__eq__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ( semiri681578069525770553at_rat @ M )
% 9.00/5.03 = zero_zero_rat )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1114_of__nat__eq__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ( semiri1316708129612266289at_nat @ M )
% 9.00/5.03 = zero_zero_nat )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1115_of__nat__eq__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ( semiri1314217659103216013at_int @ M )
% 9.00/5.03 = zero_zero_int )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1116_of__nat__eq__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ( semiri5074537144036343181t_real @ M )
% 9.00/5.03 = zero_zero_real )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1117_of__nat__less__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1118_of__nat__less__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1119_of__nat__less__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1120_of__nat__less__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1121_of__nat__le__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1122_of__nat__le__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1123_of__nat__le__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1124_of__nat__le__iff,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1125_of__nat__le__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1126_of__nat__le__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1127_of__nat__le__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1128_of__nat__le__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 9.00/5.03 = ( M = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1129_int__ops_I1_J,axiom,
% 9.00/5.03 ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.00/5.03 = zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1130_nat__int__comparison_I2_J,axiom,
% 9.00/5.03 ( ord_less_nat
% 9.00/5.03 = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1131_nat__int__comparison_I3_J,axiom,
% 9.00/5.03 ( ord_less_eq_nat
% 9.00/5.03 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1132_enumerate__in__set,axiom,
% 9.00/5.03 ! [S: set_Extended_enat,N: nat] :
% 9.00/5.03 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1133_enumerate__in__set,axiom,
% 9.00/5.03 ! [S: set_nat,N: nat] :
% 9.00/5.03 ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1134_enumerate__Ex,axiom,
% 9.00/5.03 ! [S: set_nat,S2: nat] :
% 9.00/5.03 ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( ( member_nat @ S2 @ S )
% 9.00/5.03 => ? [N3: nat] :
% 9.00/5.03 ( ( infini8530281810654367211te_nat @ S @ N3 )
% 9.00/5.03 = S2 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1135_of__nat__0__le__iff,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1136_of__nat__0__le__iff,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1137_of__nat__0__le__iff,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1138_of__nat__0__le__iff,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1139_of__nat__less__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1140_of__nat__less__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1141_of__nat__less__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1142_of__nat__less__0__iff,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1143_of__nat__less__imp__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1144_of__nat__less__imp__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1145_of__nat__less__imp__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1146_of__nat__less__imp__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1147_less__imp__of__nat__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1148_less__imp__of__nat__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1149_less__imp__of__nat__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1150_less__imp__of__nat__less,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1151_of__nat__mono,axiom,
% 9.00/5.03 ! [I: nat,J: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.03 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1152_of__nat__mono,axiom,
% 9.00/5.03 ! [I: nat,J: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.03 => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1153_of__nat__mono,axiom,
% 9.00/5.03 ! [I: nat,J: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.03 => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1154_of__nat__mono,axiom,
% 9.00/5.03 ! [I: nat,J: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.03 => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1155_le__enumerate,axiom,
% 9.00/5.03 ! [S: set_nat,N: nat] :
% 9.00/5.03 ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1156_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.00/5.03 ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.00/5.03 ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A2 ) @ ( some_P7363390416028606310at_nat @ B3 ) )
% 9.00/5.03 = ( some_P7363390416028606310at_nat @ ( F @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1157_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.00/5.03 ! [F: num > num > num,A2: num,B3: num] :
% 9.00/5.03 ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A2 ) @ ( some_num @ B3 ) )
% 9.00/5.03 = ( some_num @ ( F @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1158_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.00/5.03 ! [F: nat > nat > nat,A2: nat,B3: nat] :
% 9.00/5.03 ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A2 ) @ ( some_nat @ B3 ) )
% 9.00/5.03 = ( some_nat @ ( F @ A2 @ B3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1159_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 9.00/5.03 ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 9.00/5.03 = none_P5556105721700978146at_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1160_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: num > num > num,Uv: option_num] :
% 9.00/5.03 ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 9.00/5.03 = none_num ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1161_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: nat > nat > nat,Uv: option_nat] :
% 9.00/5.03 ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 9.00/5.03 = none_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1162_vebt__pred_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: $o,Uv: $o] :
% 9.00/5.03 ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
% 9.00/5.03 = none_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1163_enumerate__mono,axiom,
% 9.00/5.03 ! [M: nat,N: nat,S: set_Extended_enat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.03 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1164_enumerate__mono,axiom,
% 9.00/5.03 ! [M: nat,N: nat,S: set_nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ~ ( finite_finite_nat @ S )
% 9.00/5.03 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1165_vebt__buildup_Osimps_I1_J,axiom,
% 9.00/5.03 ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 9.00/5.03 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1166_vebt__delete_Osimps_I1_J,axiom,
% 9.00/5.03 ! [A2: $o,B3: $o] :
% 9.00/5.03 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ zero_zero_nat )
% 9.00/5.03 = ( vEBT_Leaf @ $false @ B3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1167_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: $o,Uv: $o,Uw: nat] :
% 9.00/5.03 ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1168_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 9.00/5.03 ! [Uu: $o] :
% 9.00/5.03 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1169_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 9.00/5.03 ! [Uv: $o] :
% 9.00/5.03 ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1170_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 9.00/5.03 vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1171_deg1Leaf,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.00/5.03 = ( ? [A4: $o,B4: $o] :
% 9.00/5.03 ( T
% 9.00/5.03 = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1172_deg__1__Leaf,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.00/5.03 => ? [A: $o,B: $o] :
% 9.00/5.03 ( T
% 9.00/5.03 = ( vEBT_Leaf @ A @ B ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1173_deg__1__Leafy,axiom,
% 9.00/5.03 ! [T: vEBT_VEBT,N: nat] :
% 9.00/5.03 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.03 => ( ( N = one_one_nat )
% 9.00/5.03 => ? [A: $o,B: $o] :
% 9.00/5.03 ( T
% 9.00/5.03 = ( vEBT_Leaf @ A @ B ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1174_pos__int__cases,axiom,
% 9.00/5.03 ! [K: int] :
% 9.00/5.03 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.03 => ~ ! [N3: nat] :
% 9.00/5.03 ( ( K
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N3 ) )
% 9.00/5.03 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1175_zero__less__imp__eq__int,axiom,
% 9.00/5.03 ! [K: int] :
% 9.00/5.03 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.03 => ? [N3: nat] :
% 9.00/5.03 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.03 & ( K
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1176_of__nat__1,axiom,
% 9.00/5.03 ( ( semiri8010041392384452111omplex @ one_one_nat )
% 9.00/5.03 = one_one_complex ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1177_of__nat__1,axiom,
% 9.00/5.03 ( ( semiri681578069525770553at_rat @ one_one_nat )
% 9.00/5.03 = one_one_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1178_of__nat__1,axiom,
% 9.00/5.03 ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 9.00/5.03 = one_one_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1179_of__nat__1,axiom,
% 9.00/5.03 ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.00/5.03 = one_one_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1180_of__nat__1,axiom,
% 9.00/5.03 ( ( semiri5074537144036343181t_real @ one_one_nat )
% 9.00/5.03 = one_one_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1181_of__nat__1__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( one_one_complex
% 9.00/5.03 = ( semiri8010041392384452111omplex @ N ) )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1182_of__nat__1__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( one_one_rat
% 9.00/5.03 = ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1183_of__nat__1__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( one_one_nat
% 9.00/5.03 = ( semiri1316708129612266289at_nat @ N ) )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1184_of__nat__1__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( one_one_int
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1185_of__nat__1__eq__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( one_one_real
% 9.00/5.03 = ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1186_of__nat__eq__1__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ( semiri8010041392384452111omplex @ N )
% 9.00/5.03 = one_one_complex )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1187_of__nat__eq__1__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ( semiri681578069525770553at_rat @ N )
% 9.00/5.03 = one_one_rat )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1188_of__nat__eq__1__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ( semiri1316708129612266289at_nat @ N )
% 9.00/5.03 = one_one_nat )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1189_of__nat__eq__1__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ( semiri1314217659103216013at_int @ N )
% 9.00/5.03 = one_one_int )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1190_of__nat__eq__1__iff,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ( semiri5074537144036343181t_real @ N )
% 9.00/5.03 = one_one_real )
% 9.00/5.03 = ( N = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1191_less__one,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ N @ one_one_nat )
% 9.00/5.03 = ( N = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1192_nonneg__int__cases,axiom,
% 9.00/5.03 ! [K: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.03 => ~ ! [N3: nat] :
% 9.00/5.03 ( K
% 9.00/5.03 != ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1193_zero__le__imp__eq__int,axiom,
% 9.00/5.03 ! [K: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.03 => ? [N3: nat] :
% 9.00/5.03 ( K
% 9.00/5.03 = ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1194_less__eq__int__code_I1_J,axiom,
% 9.00/5.03 ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1195_one__reorient,axiom,
% 9.00/5.03 ! [X2: complex] :
% 9.00/5.03 ( ( one_one_complex = X2 )
% 9.00/5.03 = ( X2 = one_one_complex ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1196_one__reorient,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ( ( one_one_real = X2 )
% 9.00/5.03 = ( X2 = one_one_real ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1197_one__reorient,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ( ( one_one_rat = X2 )
% 9.00/5.03 = ( X2 = one_one_rat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1198_one__reorient,axiom,
% 9.00/5.03 ! [X2: nat] :
% 9.00/5.03 ( ( one_one_nat = X2 )
% 9.00/5.03 = ( X2 = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1199_one__reorient,axiom,
% 9.00/5.03 ! [X2: int] :
% 9.00/5.03 ( ( one_one_int = X2 )
% 9.00/5.03 = ( X2 = one_one_int ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1200_less__int__code_I1_J,axiom,
% 9.00/5.03 ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1201_verit__la__generic,axiom,
% 9.00/5.03 ! [A2: int,X2: int] :
% 9.00/5.03 ( ( ord_less_eq_int @ A2 @ X2 )
% 9.00/5.03 | ( A2 = X2 )
% 9.00/5.03 | ( ord_less_eq_int @ X2 @ A2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1202_imp__le__cong,axiom,
% 9.00/5.03 ! [X2: int,X7: int,P: $o,P4: $o] :
% 9.00/5.03 ( ( X2 = X7 )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.00/5.03 => ( P = P4 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.03 => P )
% 9.00/5.03 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.00/5.03 => P4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1203_conj__le__cong,axiom,
% 9.00/5.03 ! [X2: int,X7: int,P: $o,P4: $o] :
% 9.00/5.03 ( ( X2 = X7 )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.00/5.03 => ( P = P4 ) )
% 9.00/5.03 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.03 & P )
% 9.00/5.03 = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
% 9.00/5.03 & P4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1204_le__numeral__extra_I4_J,axiom,
% 9.00/5.03 ord_less_eq_real @ one_one_real @ one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1205_le__numeral__extra_I4_J,axiom,
% 9.00/5.03 ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1206_le__numeral__extra_I4_J,axiom,
% 9.00/5.03 ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1207_le__numeral__extra_I4_J,axiom,
% 9.00/5.03 ord_less_eq_int @ one_one_int @ one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1208_less__numeral__extra_I4_J,axiom,
% 9.00/5.03 ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1209_less__numeral__extra_I4_J,axiom,
% 9.00/5.03 ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1210_less__numeral__extra_I4_J,axiom,
% 9.00/5.03 ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1211_less__numeral__extra_I4_J,axiom,
% 9.00/5.03 ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1212_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 9.00/5.03 ! [Uu: $o,Uv: $o,D: nat] :
% 9.00/5.03 ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 9.00/5.03 = ( D = one_one_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1213_less__numeral__extra_I1_J,axiom,
% 9.00/5.03 ord_less_real @ zero_zero_real @ one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1214_less__numeral__extra_I1_J,axiom,
% 9.00/5.03 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1215_less__numeral__extra_I1_J,axiom,
% 9.00/5.03 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1216_less__numeral__extra_I1_J,axiom,
% 9.00/5.03 ord_less_int @ zero_zero_int @ one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1217_vebt__member_Osimps_I1_J,axiom,
% 9.00/5.03 ! [A2: $o,B3: $o,X2: nat] :
% 9.00/5.03 ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.03 = ( ( ( X2 = zero_zero_nat )
% 9.00/5.03 => A2 )
% 9.00/5.03 & ( ( X2 != zero_zero_nat )
% 9.00/5.03 => ( ( ( X2 = one_one_nat )
% 9.00/5.03 => B3 )
% 9.00/5.03 & ( X2 = one_one_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1218_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 9.00/5.03 ! [A2: $o,B3: $o,X2: nat] :
% 9.00/5.03 ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.03 = ( ( ( X2 = zero_zero_nat )
% 9.00/5.03 => A2 )
% 9.00/5.03 & ( ( X2 != zero_zero_nat )
% 9.00/5.03 => ( ( ( X2 = one_one_nat )
% 9.00/5.03 => B3 )
% 9.00/5.03 & ( X2 = one_one_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1219_zle__int,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1220_vebt__mint_Osimps_I1_J,axiom,
% 9.00/5.03 ! [A2: $o,B3: $o] :
% 9.00/5.03 ( ( A2
% 9.00/5.03 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.03 & ( ~ A2
% 9.00/5.03 => ( ( B3
% 9.00/5.03 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.03 & ( ~ B3
% 9.00/5.03 => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = none_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1221_vebt__maxt_Osimps_I1_J,axiom,
% 9.00/5.03 ! [B3: $o,A2: $o] :
% 9.00/5.03 ( ( B3
% 9.00/5.03 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.03 & ( ~ B3
% 9.00/5.03 => ( ( A2
% 9.00/5.03 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.03 & ( ~ A2
% 9.00/5.03 => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A2 @ B3 ) )
% 9.00/5.03 = none_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1222_vebt__succ_Osimps_I1_J,axiom,
% 9.00/5.03 ! [B3: $o,Uu: $o] :
% 9.00/5.03 ( ( B3
% 9.00/5.03 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
% 9.00/5.03 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.03 & ( ~ B3
% 9.00/5.03 => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat )
% 9.00/5.03 = none_nat ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1223_not__one__less__zero,axiom,
% 9.00/5.03 ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1224_not__one__less__zero,axiom,
% 9.00/5.03 ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1225_not__one__less__zero,axiom,
% 9.00/5.03 ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1226_not__one__less__zero,axiom,
% 9.00/5.03 ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1227_zero__less__one,axiom,
% 9.00/5.03 ord_less_real @ zero_zero_real @ one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1228_zero__less__one,axiom,
% 9.00/5.03 ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1229_zero__less__one,axiom,
% 9.00/5.03 ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1230_zero__less__one,axiom,
% 9.00/5.03 ord_less_int @ zero_zero_int @ one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1231_zero__less__one__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1232_zero__less__one__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1233_zero__less__one__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1234_zero__less__one__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1235_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1236_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1237_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1238_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.00/5.03 ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1239_not__one__le__zero,axiom,
% 9.00/5.03 ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1240_not__one__le__zero,axiom,
% 9.00/5.03 ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1241_not__one__le__zero,axiom,
% 9.00/5.03 ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1242_not__one__le__zero,axiom,
% 9.00/5.03 ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1243_reals__Archimedean2,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ? [N3: nat] : ( ord_less_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1244_reals__Archimedean2,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ? [N3: nat] : ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1245_real__arch__simple,axiom,
% 9.00/5.03 ! [X2: real] :
% 9.00/5.03 ? [N3: nat] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1246_real__arch__simple,axiom,
% 9.00/5.03 ! [X2: rat] :
% 9.00/5.03 ? [N3: nat] : ( ord_less_eq_rat @ X2 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1247_vebt__pred_Osimps_I3_J,axiom,
% 9.00/5.03 ! [B3: $o,A2: $o,Va: nat] :
% 9.00/5.03 ( ( B3
% 9.00/5.03 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.03 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.03 & ( ~ B3
% 9.00/5.03 => ( ( A2
% 9.00/5.03 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.03 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.03 & ( ~ A2
% 9.00/5.03 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.03 = none_nat ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1248_zero__neq__one,axiom,
% 9.00/5.03 zero_zero_complex != one_one_complex ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1249_zero__neq__one,axiom,
% 9.00/5.03 zero_zero_real != one_one_real ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1250_zero__neq__one,axiom,
% 9.00/5.03 zero_zero_rat != one_one_rat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1251_zero__neq__one,axiom,
% 9.00/5.03 zero_zero_nat != one_one_nat ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1252_zero__neq__one,axiom,
% 9.00/5.03 zero_zero_int != one_one_int ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1253_dbl__inc__simps_I2_J,axiom,
% 9.00/5.03 ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
% 9.00/5.03 = one_one_complex ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1254_dbl__inc__simps_I2_J,axiom,
% 9.00/5.03 ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
% 9.00/5.03 = one_one_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1255_dbl__inc__simps_I2_J,axiom,
% 9.00/5.03 ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
% 9.00/5.03 = one_one_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1256_dbl__inc__simps_I2_J,axiom,
% 9.00/5.03 ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
% 9.00/5.03 = one_one_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1257_VEBT_Osize__gen_I2_J,axiom,
% 9.00/5.03 ! [X21: $o,X222: $o] :
% 9.00/5.03 ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.00/5.03 = zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1258_old_Onat_Oinject,axiom,
% 9.00/5.03 ! [Nat: nat,Nat2: nat] :
% 9.00/5.03 ( ( ( suc @ Nat )
% 9.00/5.03 = ( suc @ Nat2 ) )
% 9.00/5.03 = ( Nat = Nat2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1259_nat_Oinject,axiom,
% 9.00/5.03 ! [X22: nat,Y2: nat] :
% 9.00/5.03 ( ( ( suc @ X22 )
% 9.00/5.03 = ( suc @ Y2 ) )
% 9.00/5.03 = ( X22 = Y2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1260_Suc__less__eq,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.00/5.03 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1261_Suc__mono,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1262_lessI,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1263_Suc__le__mono,axiom,
% 9.00/5.03 ! [N: nat,M: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
% 9.00/5.03 = ( ord_less_eq_nat @ N @ M ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1264_less__Suc0,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.03 = ( N = zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1265_zero__less__Suc,axiom,
% 9.00/5.03 ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1266_n__not__Suc__n,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( N
% 9.00/5.03 != ( suc @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1267_Suc__inject,axiom,
% 9.00/5.03 ! [X2: nat,Y3: nat] :
% 9.00/5.03 ( ( ( suc @ X2 )
% 9.00/5.03 = ( suc @ Y3 ) )
% 9.00/5.03 => ( X2 = Y3 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1268_exists__least__lemma,axiom,
% 9.00/5.03 ! [P: nat > $o] :
% 9.00/5.03 ( ~ ( P @ zero_zero_nat )
% 9.00/5.03 => ( ? [X_12: nat] : ( P @ X_12 )
% 9.00/5.03 => ? [N3: nat] :
% 9.00/5.03 ( ~ ( P @ N3 )
% 9.00/5.03 & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1269_nat_Odistinct_I1_J,axiom,
% 9.00/5.03 ! [X22: nat] :
% 9.00/5.03 ( zero_zero_nat
% 9.00/5.03 != ( suc @ X22 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1270_old_Onat_Odistinct_I2_J,axiom,
% 9.00/5.03 ! [Nat2: nat] :
% 9.00/5.03 ( ( suc @ Nat2 )
% 9.00/5.03 != zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1271_old_Onat_Odistinct_I1_J,axiom,
% 9.00/5.03 ! [Nat2: nat] :
% 9.00/5.03 ( zero_zero_nat
% 9.00/5.03 != ( suc @ Nat2 ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1272_nat_OdiscI,axiom,
% 9.00/5.03 ! [Nat: nat,X22: nat] :
% 9.00/5.03 ( ( Nat
% 9.00/5.03 = ( suc @ X22 ) )
% 9.00/5.03 => ( Nat != zero_zero_nat ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1273_old_Onat_Oexhaust,axiom,
% 9.00/5.03 ! [Y3: nat] :
% 9.00/5.03 ( ( Y3 != zero_zero_nat )
% 9.00/5.03 => ~ ! [Nat3: nat] :
% 9.00/5.03 ( Y3
% 9.00/5.03 != ( suc @ Nat3 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1274_nat__induct,axiom,
% 9.00/5.03 ! [P: nat > $o,N: nat] :
% 9.00/5.03 ( ( P @ zero_zero_nat )
% 9.00/5.03 => ( ! [N3: nat] :
% 9.00/5.03 ( ( P @ N3 )
% 9.00/5.03 => ( P @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( P @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1275_diff__induct,axiom,
% 9.00/5.03 ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.00/5.03 ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
% 9.00/5.03 => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.03 ( ( P @ X5 @ Y4 )
% 9.00/5.03 => ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
% 9.00/5.03 => ( P @ M @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1276_zero__induct,axiom,
% 9.00/5.03 ! [P: nat > $o,K: nat] :
% 9.00/5.03 ( ( P @ K )
% 9.00/5.03 => ( ! [N3: nat] :
% 9.00/5.03 ( ( P @ ( suc @ N3 ) )
% 9.00/5.03 => ( P @ N3 ) )
% 9.00/5.03 => ( P @ zero_zero_nat ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1277_Suc__neq__Zero,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( ( suc @ M )
% 9.00/5.03 != zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1278_Zero__neq__Suc,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( zero_zero_nat
% 9.00/5.03 != ( suc @ M ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1279_Zero__not__Suc,axiom,
% 9.00/5.03 ! [M: nat] :
% 9.00/5.03 ( zero_zero_nat
% 9.00/5.03 != ( suc @ M ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1280_not0__implies__Suc,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( N != zero_zero_nat )
% 9.00/5.03 => ? [M4: nat] :
% 9.00/5.03 ( N
% 9.00/5.03 = ( suc @ M4 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1281_vebt__buildup_Ocases,axiom,
% 9.00/5.03 ! [X2: nat] :
% 9.00/5.03 ( ( X2 != zero_zero_nat )
% 9.00/5.03 => ( ( X2
% 9.00/5.03 != ( suc @ zero_zero_nat ) )
% 9.00/5.03 => ~ ! [Va2: nat] :
% 9.00/5.03 ( X2
% 9.00/5.03 != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1282_not__less__less__Suc__eq,axiom,
% 9.00/5.03 ! [N: nat,M: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ N @ M )
% 9.00/5.03 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.00/5.03 = ( N = M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1283_strict__inc__induct,axiom,
% 9.00/5.03 ! [I: nat,J: nat,P: nat > $o] :
% 9.00/5.03 ( ( ord_less_nat @ I @ J )
% 9.00/5.03 => ( ! [I2: nat] :
% 9.00/5.03 ( ( J
% 9.00/5.03 = ( suc @ I2 ) )
% 9.00/5.03 => ( P @ I2 ) )
% 9.00/5.03 => ( ! [I2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I2 @ J )
% 9.00/5.03 => ( ( P @ ( suc @ I2 ) )
% 9.00/5.03 => ( P @ I2 ) ) )
% 9.00/5.03 => ( P @ I ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1284_less__Suc__induct,axiom,
% 9.00/5.03 ! [I: nat,J: nat,P: nat > nat > $o] :
% 9.00/5.03 ( ( ord_less_nat @ I @ J )
% 9.00/5.03 => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
% 9.00/5.03 => ( ! [I2: nat,J2: nat,K2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I2 @ J2 )
% 9.00/5.03 => ( ( ord_less_nat @ J2 @ K2 )
% 9.00/5.03 => ( ( P @ I2 @ J2 )
% 9.00/5.03 => ( ( P @ J2 @ K2 )
% 9.00/5.03 => ( P @ I2 @ K2 ) ) ) ) )
% 9.00/5.03 => ( P @ I @ J ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1285_less__trans__Suc,axiom,
% 9.00/5.03 ! [I: nat,J: nat,K: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I @ J )
% 9.00/5.03 => ( ( ord_less_nat @ J @ K )
% 9.00/5.03 => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1286_Suc__less__SucD,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1287_less__antisym,axiom,
% 9.00/5.03 ! [N: nat,M: nat] :
% 9.00/5.03 ( ~ ( ord_less_nat @ N @ M )
% 9.00/5.03 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.00/5.03 => ( M = N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1288_Suc__less__eq2,axiom,
% 9.00/5.03 ! [N: nat,M: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.03 = ( ? [M6: nat] :
% 9.00/5.03 ( ( M
% 9.00/5.03 = ( suc @ M6 ) )
% 9.00/5.03 & ( ord_less_nat @ N @ M6 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1289_All__less__Suc,axiom,
% 9.00/5.03 ! [N: nat,P: nat > $o] :
% 9.00/5.03 ( ( ! [I4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.00/5.03 => ( P @ I4 ) ) )
% 9.00/5.03 = ( ( P @ N )
% 9.00/5.03 & ! [I4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I4 @ N )
% 9.00/5.03 => ( P @ I4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1290_not__less__eq,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_nat @ M @ N ) )
% 9.00/5.03 = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1291_less__Suc__eq,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.00/5.03 = ( ( ord_less_nat @ M @ N )
% 9.00/5.03 | ( M = N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1292_Ex__less__Suc,axiom,
% 9.00/5.03 ! [N: nat,P: nat > $o] :
% 9.00/5.03 ( ( ? [I4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.00/5.03 & ( P @ I4 ) ) )
% 9.00/5.03 = ( ( P @ N )
% 9.00/5.03 | ? [I4: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I4 @ N )
% 9.00/5.03 & ( P @ I4 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1293_less__SucI,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1294_less__SucE,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.00/5.03 => ( ~ ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( M = N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1295_Suc__lessI,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ M @ N )
% 9.00/5.03 => ( ( ( suc @ M )
% 9.00/5.03 != N )
% 9.00/5.03 => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1296_Suc__lessE,axiom,
% 9.00/5.03 ! [I: nat,K: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( suc @ I ) @ K )
% 9.00/5.03 => ~ ! [J2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I @ J2 )
% 9.00/5.03 => ( K
% 9.00/5.03 != ( suc @ J2 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1297_Suc__lessD,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_nat @ ( suc @ M ) @ N )
% 9.00/5.03 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1298_Nat_OlessE,axiom,
% 9.00/5.03 ! [I: nat,K: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I @ K )
% 9.00/5.03 => ( ( K
% 9.00/5.03 != ( suc @ I ) )
% 9.00/5.03 => ~ ! [J2: nat] :
% 9.00/5.03 ( ( ord_less_nat @ I @ J2 )
% 9.00/5.03 => ( K
% 9.00/5.03 != ( suc @ J2 ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1299_Suc__leD,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.00/5.03 => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1300_le__SucE,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.03 => ( ~ ( ord_less_eq_nat @ M @ N )
% 9.00/5.03 => ( M
% 9.00/5.03 = ( suc @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1301_le__SucI,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.03 => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1302_Suc__le__D,axiom,
% 9.00/5.03 ! [N: nat,M7: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( suc @ N ) @ M7 )
% 9.00/5.03 => ? [M4: nat] :
% 9.00/5.03 ( M7
% 9.00/5.03 = ( suc @ M4 ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1303_le__Suc__eq,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.03 = ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.03 | ( M
% 9.00/5.03 = ( suc @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1304_Suc__n__not__le__n,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1305_not__less__eq__eq,axiom,
% 9.00/5.03 ! [M: nat,N: nat] :
% 9.00/5.03 ( ( ~ ( ord_less_eq_nat @ M @ N ) )
% 9.00/5.03 = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1306_full__nat__induct,axiom,
% 9.00/5.03 ! [P: nat > $o,N: nat] :
% 9.00/5.03 ( ! [N3: nat] :
% 9.00/5.03 ( ! [M3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
% 9.00/5.03 => ( P @ M3 ) )
% 9.00/5.03 => ( P @ N3 ) )
% 9.00/5.03 => ( P @ N ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1307_nat__induct__at__least,axiom,
% 9.00/5.03 ! [M: nat,N: nat,P: nat > $o] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.03 => ( ( P @ M )
% 9.00/5.03 => ( ! [N3: nat] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ N3 )
% 9.00/5.03 => ( ( P @ N3 )
% 9.00/5.03 => ( P @ ( suc @ N3 ) ) ) )
% 9.00/5.03 => ( P @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1308_transitive__stepwise__le,axiom,
% 9.00/5.03 ! [M: nat,N: nat,R: nat > nat > $o] :
% 9.00/5.03 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.03 => ( ! [X5: nat] : ( R @ X5 @ X5 )
% 9.00/5.03 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 9.00/5.03 ( ( R @ X5 @ Y4 )
% 9.00/5.03 => ( ( R @ Y4 @ Z4 )
% 9.00/5.03 => ( R @ X5 @ Z4 ) ) )
% 9.00/5.03 => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 9.00/5.03 => ( R @ M @ N ) ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1309_lift__Suc__mono__less__iff,axiom,
% 9.00/5.03 ! [F: nat > real,N: nat,M: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_real @ ( F @ N ) @ ( F @ M ) )
% 9.00/5.03 = ( ord_less_nat @ N @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1310_lift__Suc__mono__less__iff,axiom,
% 9.00/5.03 ! [F: nat > rat,N: nat,M: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M ) )
% 9.00/5.03 = ( ord_less_nat @ N @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1311_lift__Suc__mono__less__iff,axiom,
% 9.00/5.03 ! [F: nat > num,N: nat,M: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
% 9.00/5.03 = ( ord_less_nat @ N @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1312_lift__Suc__mono__less__iff,axiom,
% 9.00/5.03 ! [F: nat > nat,N: nat,M: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
% 9.00/5.03 = ( ord_less_nat @ N @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1313_lift__Suc__mono__less__iff,axiom,
% 9.00/5.03 ! [F: nat > int,N: nat,M: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
% 9.00/5.03 = ( ord_less_nat @ N @ M ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1314_lift__Suc__mono__less,axiom,
% 9.00/5.03 ! [F: nat > real,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_real @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1315_lift__Suc__mono__less,axiom,
% 9.00/5.03 ! [F: nat > rat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1316_lift__Suc__mono__less,axiom,
% 9.00/5.03 ! [F: nat > num,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1317_lift__Suc__mono__less,axiom,
% 9.00/5.03 ! [F: nat > nat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1318_lift__Suc__mono__less,axiom,
% 9.00/5.03 ! [F: nat > int,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1319_lift__Suc__mono__le,axiom,
% 9.00/5.03 ! [F: nat > set_int,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_set_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1320_lift__Suc__mono__le,axiom,
% 9.00/5.03 ! [F: nat > rat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1321_lift__Suc__mono__le,axiom,
% 9.00/5.03 ! [F: nat > num,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1322_lift__Suc__mono__le,axiom,
% 9.00/5.03 ! [F: nat > nat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1323_lift__Suc__mono__le,axiom,
% 9.00/5.03 ! [F: nat > int,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N7 ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1324_lift__Suc__antimono__le,axiom,
% 9.00/5.03 ! [F: nat > set_int,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_set_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1325_lift__Suc__antimono__le,axiom,
% 9.00/5.03 ! [F: nat > rat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_rat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1326_lift__Suc__antimono__le,axiom,
% 9.00/5.03 ! [F: nat > num,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_num @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1327_lift__Suc__antimono__le,axiom,
% 9.00/5.03 ! [F: nat > nat,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_nat @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1328_lift__Suc__antimono__le,axiom,
% 9.00/5.03 ! [F: nat > int,N: nat,N7: nat] :
% 9.00/5.03 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.00/5.03 => ( ( ord_less_eq_nat @ N @ N7 )
% 9.00/5.03 => ( ord_less_eq_int @ ( F @ N7 ) @ ( F @ N ) ) ) ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1329_of__nat__neq__0,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
% 9.00/5.03 != zero_zero_rat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1330_of__nat__neq__0,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
% 9.00/5.03 != zero_zero_nat ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1331_of__nat__neq__0,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
% 9.00/5.03 != zero_zero_int ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1332_of__nat__neq__0,axiom,
% 9.00/5.03 ! [N: nat] :
% 9.00/5.03 ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
% 9.00/5.03 != zero_zero_real ) ).
% 9.00/5.03
% 9.00/5.03 thf(fact_1333_Ex__less__Suc2,axiom,
% 9.00/5.03 ! [N: nat,P: nat > $o] :
% 9.00/5.03 ( ( ? [I4: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.00/5.04 & ( P @ I4 ) ) )
% 9.00/5.04 = ( ( P @ zero_zero_nat )
% 9.00/5.04 | ? [I4: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I4 @ N )
% 9.00/5.04 & ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1334_gr0__conv__Suc,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 = ( ? [M2: nat] :
% 9.00/5.04 ( N
% 9.00/5.04 = ( suc @ M2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1335_All__less__Suc2,axiom,
% 9.00/5.04 ! [N: nat,P: nat > $o] :
% 9.00/5.04 ( ( ! [I4: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I4 @ ( suc @ N ) )
% 9.00/5.04 => ( P @ I4 ) ) )
% 9.00/5.04 = ( ( P @ zero_zero_nat )
% 9.00/5.04 & ! [I4: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I4 @ N )
% 9.00/5.04 => ( P @ ( suc @ I4 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1336_gr0__implies__Suc,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ? [M4: nat] :
% 9.00/5.04 ( N
% 9.00/5.04 = ( suc @ M4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1337_less__Suc__eq__0__disj,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.00/5.04 = ( ( M = zero_zero_nat )
% 9.00/5.04 | ? [J3: nat] :
% 9.00/5.04 ( ( M
% 9.00/5.04 = ( suc @ J3 ) )
% 9.00/5.04 & ( ord_less_nat @ J3 @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1338_Suc__leI,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1339_Suc__le__eq,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.00/5.04 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1340_dec__induct,axiom,
% 9.00/5.04 ! [I: nat,J: nat,P: nat > $o] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.04 => ( ( P @ I )
% 9.00/5.04 => ( ! [N3: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I @ N3 )
% 9.00/5.04 => ( ( ord_less_nat @ N3 @ J )
% 9.00/5.04 => ( ( P @ N3 )
% 9.00/5.04 => ( P @ ( suc @ N3 ) ) ) ) )
% 9.00/5.04 => ( P @ J ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1341_inc__induct,axiom,
% 9.00/5.04 ! [I: nat,J: nat,P: nat > $o] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.04 => ( ( P @ J )
% 9.00/5.04 => ( ! [N3: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I @ N3 )
% 9.00/5.04 => ( ( ord_less_nat @ N3 @ J )
% 9.00/5.04 => ( ( P @ ( suc @ N3 ) )
% 9.00/5.04 => ( P @ N3 ) ) ) )
% 9.00/5.04 => ( P @ I ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1342_Suc__le__lessD,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.00/5.04 => ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1343_le__less__Suc__eq,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.00/5.04 = ( N = M ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1344_less__Suc__eq__le,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ ( suc @ N ) )
% 9.00/5.04 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1345_less__eq__Suc__le,axiom,
% 9.00/5.04 ( ord_less_nat
% 9.00/5.04 = ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1346_le__imp__less__Suc,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1347_One__nat__def,axiom,
% 9.00/5.04 ( one_one_nat
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1348_vebt__delete_Osimps_I3_J,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o,N: nat] :
% 9.00/5.04 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ ( suc @ N ) ) )
% 9.00/5.04 = ( vEBT_Leaf @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1349_ex__least__nat__less,axiom,
% 9.00/5.04 ! [P: nat > $o,N: nat] :
% 9.00/5.04 ( ( P @ N )
% 9.00/5.04 => ( ~ ( P @ zero_zero_nat )
% 9.00/5.04 => ? [K2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ K2 @ N )
% 9.00/5.04 & ! [I3: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I3 @ K2 )
% 9.00/5.04 => ~ ( P @ I3 ) )
% 9.00/5.04 & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1350_linorder__neqE__linordered__idom,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( X2 != Y3 )
% 9.00/5.04 => ( ~ ( ord_less_real @ X2 @ Y3 )
% 9.00/5.04 => ( ord_less_real @ Y3 @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1351_linorder__neqE__linordered__idom,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( X2 != Y3 )
% 9.00/5.04 => ( ~ ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.04 => ( ord_less_rat @ Y3 @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1352_linorder__neqE__linordered__idom,axiom,
% 9.00/5.04 ! [X2: int,Y3: int] :
% 9.00/5.04 ( ( X2 != Y3 )
% 9.00/5.04 => ( ~ ( ord_less_int @ X2 @ Y3 )
% 9.00/5.04 => ( ord_less_int @ Y3 @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1353_nat__induct__non__zero,axiom,
% 9.00/5.04 ! [N: nat,P: nat > $o] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( P @ one_one_nat )
% 9.00/5.04 => ( ! [N3: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.04 => ( ( P @ N3 )
% 9.00/5.04 => ( P @ ( suc @ N3 ) ) ) )
% 9.00/5.04 => ( P @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1354_enumerate__step,axiom,
% 9.00/5.04 ! [S: set_Extended_enat,N: nat] :
% 9.00/5.04 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1355_enumerate__step,axiom,
% 9.00/5.04 ! [S: set_nat,N: nat] :
% 9.00/5.04 ( ~ ( finite_finite_nat @ S )
% 9.00/5.04 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1356_invar__vebt_Ointros_I1_J,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1357_vebt__delete_Osimps_I2_J,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o] :
% 9.00/5.04 ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A2 @ B3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( vEBT_Leaf @ A2 @ $false ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1358_int__one__le__iff__zero__less,axiom,
% 9.00/5.04 ! [Z: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ one_one_int @ Z )
% 9.00/5.04 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1359_vebt__buildup_Osimps_I2_J,axiom,
% 9.00/5.04 ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1360_vebt__succ_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Uv: $o,Uw: $o,N: nat] :
% 9.00/5.04 ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N ) )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1361_vebt__pred_Osimps_I2_J,axiom,
% 9.00/5.04 ! [A2: $o,Uw: $o] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.04 & ( ~ A2
% 9.00/5.04 => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A2 @ Uw ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = none_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1362_option_Osize__gen_I1_J,axiom,
% 9.00/5.04 ! [X2: nat > nat] :
% 9.00/5.04 ( ( size_option_nat @ X2 @ none_nat )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1363_option_Osize__gen_I1_J,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat > nat] :
% 9.00/5.04 ( ( size_o8335143837870341156at_nat @ X2 @ none_P5556105721700978146at_nat )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1364_option_Osize__gen_I1_J,axiom,
% 9.00/5.04 ! [X2: num > nat] :
% 9.00/5.04 ( ( size_option_num @ X2 @ none_num )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1365_option_Osize_I3_J,axiom,
% 9.00/5.04 ( ( size_size_option_nat @ none_nat )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1366_option_Osize_I3_J,axiom,
% 9.00/5.04 ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1367_option_Osize_I3_J,axiom,
% 9.00/5.04 ( ( size_size_option_num @ none_num )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1368_option_Osize_I4_J,axiom,
% 9.00/5.04 ! [X22: nat] :
% 9.00/5.04 ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1369_option_Osize_I4_J,axiom,
% 9.00/5.04 ! [X22: product_prod_nat_nat] :
% 9.00/5.04 ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1370_option_Osize_I4_J,axiom,
% 9.00/5.04 ! [X22: num] :
% 9.00/5.04 ( ( size_size_option_num @ ( some_num @ X22 ) )
% 9.00/5.04 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1371_list__decode_Ocases,axiom,
% 9.00/5.04 ! [X2: nat] :
% 9.00/5.04 ( ( X2 != zero_zero_nat )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( suc @ N3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1372_deg__SUcn__Node,axiom,
% 9.00/5.04 ! [Tree: vEBT_VEBT,N: nat] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
% 9.00/5.04 => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.04 ( Tree
% 9.00/5.04 = ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1373_neg__int__cases,axiom,
% 9.00/5.04 ! [K: int] :
% 9.00/5.04 ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ( ( K
% 9.00/5.04 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.00/5.04 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1374_frac__eq,axiom,
% 9.00/5.04 ! [X2: real] :
% 9.00/5.04 ( ( ( archim2898591450579166408c_real @ X2 )
% 9.00/5.04 = X2 )
% 9.00/5.04 = ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 & ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1375_frac__eq,axiom,
% 9.00/5.04 ! [X2: rat] :
% 9.00/5.04 ( ( ( archimedean_frac_rat @ X2 )
% 9.00/5.04 = X2 )
% 9.00/5.04 = ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 & ( ord_less_rat @ X2 @ one_one_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1376_finite__enum__subset,axiom,
% 9.00/5.04 ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
% 9.00/5.04 ( ! [I2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
% 9.00/5.04 => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
% 9.00/5.04 = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ X6 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ Y7 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ X6 ) @ ( finite121521170596916366d_enat @ Y7 ) )
% 9.00/5.04 => ( ord_le7203529160286727270d_enat @ X6 @ Y7 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1377_finite__enum__subset,axiom,
% 9.00/5.04 ! [X6: set_nat,Y7: set_nat] :
% 9.00/5.04 ( ! [I2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
% 9.00/5.04 => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
% 9.00/5.04 = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
% 9.00/5.04 => ( ( finite_finite_nat @ X6 )
% 9.00/5.04 => ( ( finite_finite_nat @ Y7 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_nat @ X6 ) @ ( finite_card_nat @ Y7 ) )
% 9.00/5.04 => ( ord_less_eq_set_nat @ X6 @ Y7 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1378_Suc__diff__1,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.04 = N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1379_inverse__of__nat__le,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( N != zero_zero_nat )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1380_inverse__of__nat__le,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( N != zero_zero_nat )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1381_deg__deg__n,axiom,
% 9.00/5.04 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.04 => ( Deg = N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1382_add_Oinverse__inverse,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1383_add_Oinverse__inverse,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( uminus_uminus_real @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1384_add_Oinverse__inverse,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1385_add_Oinverse__inverse,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1386_add_Oinverse__inverse,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A2 ) )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1387_neg__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ( uminus_uminus_int @ A2 )
% 9.00/5.04 = ( uminus_uminus_int @ B3 ) )
% 9.00/5.04 = ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1388_neg__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( uminus_uminus_real @ A2 )
% 9.00/5.04 = ( uminus_uminus_real @ B3 ) )
% 9.00/5.04 = ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1389_neg__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( uminus_uminus_rat @ A2 )
% 9.00/5.04 = ( uminus_uminus_rat @ B3 ) )
% 9.00/5.04 = ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1390_neg__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.04 = ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.04 = ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1391_neg__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.04 = ( uminus1482373934393186551omplex @ B3 ) )
% 9.00/5.04 = ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1392_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ A2 @ A2 )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1393_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ A2 @ A2 )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1394_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ A2 @ A2 )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1395_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ A2 @ A2 )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1396_diff__zero,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ A2 @ zero_zero_real )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1397_diff__zero,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1398_diff__zero,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ A2 @ zero_zero_nat )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1399_diff__zero,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1400_zero__diff,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ zero_zero_nat @ A2 )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1401_diff__0__right,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ A2 @ zero_zero_real )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1402_diff__0__right,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1403_diff__0__right,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ A2 @ zero_zero_int )
% 9.00/5.04 = A2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1404_diff__self,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ A2 @ A2 )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1405_diff__self,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ A2 @ A2 )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1406_diff__self,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ A2 @ A2 )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1407_div__by__0,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( divide_divide_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1408_div__by__0,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( divide_divide_int @ A2 @ zero_zero_int )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1409_div__by__0,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( divide_divide_nat @ A2 @ zero_zero_nat )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1410_div__by__0,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( divide_divide_real @ A2 @ zero_zero_real )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1411_div__by__0,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1412_div__0,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( divide_divide_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1413_div__0,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( divide_divide_int @ zero_zero_int @ A2 )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1414_div__0,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( divide_divide_nat @ zero_zero_nat @ A2 )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1415_div__0,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( divide_divide_real @ zero_zero_real @ A2 )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1416_div__0,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( divide1717551699836669952omplex @ zero_zero_complex @ A2 )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1417_neg__le__iff__le,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1418_neg__le__iff__le,axiom,
% 9.00/5.04 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1419_neg__le__iff__le,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1420_neg__le__iff__le,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1421_neg__equal__zero,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ( uminus_uminus_int @ A2 )
% 9.00/5.04 = A2 )
% 9.00/5.04 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1422_neg__equal__zero,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ( uminus_uminus_real @ A2 )
% 9.00/5.04 = A2 )
% 9.00/5.04 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1423_neg__equal__zero,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ( uminus_uminus_rat @ A2 )
% 9.00/5.04 = A2 )
% 9.00/5.04 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1424_neg__equal__zero,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.04 = A2 )
% 9.00/5.04 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1425_equal__neg__zero,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1426_equal__neg__zero,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1427_equal__neg__zero,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1428_equal__neg__zero,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1429_neg__equal__0__iff__equal,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ( uminus_uminus_int @ A2 )
% 9.00/5.04 = zero_zero_int )
% 9.00/5.04 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1430_neg__equal__0__iff__equal,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ( uminus_uminus_real @ A2 )
% 9.00/5.04 = zero_zero_real )
% 9.00/5.04 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1431_neg__equal__0__iff__equal,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ( uminus_uminus_rat @ A2 )
% 9.00/5.04 = zero_zero_rat )
% 9.00/5.04 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1432_neg__equal__0__iff__equal,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.04 = zero_z3403309356797280102nteger )
% 9.00/5.04 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1433_neg__equal__0__iff__equal,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.04 = zero_zero_complex )
% 9.00/5.04 = ( A2 = zero_zero_complex ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1434_neg__0__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( zero_zero_int
% 9.00/5.04 = ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( zero_zero_int = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1435_neg__0__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( zero_zero_real
% 9.00/5.04 = ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( zero_zero_real = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1436_neg__0__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( zero_zero_rat
% 9.00/5.04 = ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( zero_zero_rat = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1437_neg__0__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( zero_z3403309356797280102nteger
% 9.00/5.04 = ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( zero_z3403309356797280102nteger = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1438_neg__0__equal__iff__equal,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( zero_zero_complex
% 9.00/5.04 = ( uminus1482373934393186551omplex @ A2 ) )
% 9.00/5.04 = ( zero_zero_complex = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1439_add_Oinverse__neutral,axiom,
% 9.00/5.04 ( ( uminus_uminus_int @ zero_zero_int )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1440_add_Oinverse__neutral,axiom,
% 9.00/5.04 ( ( uminus_uminus_real @ zero_zero_real )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1441_add_Oinverse__neutral,axiom,
% 9.00/5.04 ( ( uminus_uminus_rat @ zero_zero_rat )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1442_add_Oinverse__neutral,axiom,
% 9.00/5.04 ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.00/5.04 = zero_z3403309356797280102nteger ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1443_add_Oinverse__neutral,axiom,
% 9.00/5.04 ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1444_neg__less__iff__less,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1445_neg__less__iff__less,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1446_neg__less__iff__less,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1447_neg__less__iff__less,axiom,
% 9.00/5.04 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ A2 @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1448_minus__diff__eq,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_minus_int @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1449_minus__diff__eq,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( uminus_uminus_real @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_minus_real @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1450_minus__diff__eq,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( uminus_uminus_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_minus_rat @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1451_minus__diff__eq,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1452_minus__diff__eq,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_minus_complex @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1453_diff__Suc__Suc,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.00/5.04 = ( minus_minus_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1454_Suc__diff__diff,axiom,
% 9.00/5.04 ! [M: nat,N: nat,K: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
% 9.00/5.04 = ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1455_diff__self__eq__0,axiom,
% 9.00/5.04 ! [M: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ M @ M )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1456_diff__0__eq__0,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ zero_zero_nat @ N )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1457_diff__diff__cancel,axiom,
% 9.00/5.04 ! [I: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ I @ N )
% 9.00/5.04 => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
% 9.00/5.04 = I ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1458_diff__ge__0__iff__ge,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_real @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1459_diff__ge__0__iff__ge,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1460_diff__ge__0__iff__ge,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1461_diff__gt__0__iff__gt,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_real @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1462_diff__gt__0__iff__gt,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_rat @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1463_diff__gt__0__iff__gt,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_int @ B3 @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1464_neg__0__le__iff__le,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1465_neg__0__le__iff__le,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1466_neg__0__le__iff__le,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1467_neg__0__le__iff__le,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1468_neg__le__0__iff__le,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
% 9.00/5.04 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1469_neg__le__0__iff__le,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1470_neg__le__0__iff__le,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.04 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1471_neg__le__0__iff__le,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 9.00/5.04 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1472_less__eq__neg__nonpos,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1473_less__eq__neg__nonpos,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1474_less__eq__neg__nonpos,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1475_less__eq__neg__nonpos,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1476_neg__less__eq__nonneg,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1477_neg__less__eq__nonneg,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1478_neg__less__eq__nonneg,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1479_neg__less__eq__nonneg,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1480_neg__less__0__iff__less,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
% 9.00/5.04 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1481_neg__less__0__iff__less,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ zero_zero_real )
% 9.00/5.04 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1482_neg__less__0__iff__less,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.04 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1483_neg__less__0__iff__less,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ zero_z3403309356797280102nteger )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1484_neg__0__less__iff__less,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1485_neg__0__less__iff__less,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1486_neg__0__less__iff__less,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1487_neg__0__less__iff__less,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1488_neg__less__pos,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1489_neg__less__pos,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1490_neg__less__pos,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1491_neg__less__pos,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1492_less__neg__neg,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.04 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1493_less__neg__neg,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1494_less__neg__neg,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1495_less__neg__neg,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1496_diff__numeral__special_I9_J,axiom,
% 9.00/5.04 ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1497_diff__numeral__special_I9_J,axiom,
% 9.00/5.04 ( ( minus_minus_real @ one_one_real @ one_one_real )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1498_diff__numeral__special_I9_J,axiom,
% 9.00/5.04 ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1499_diff__numeral__special_I9_J,axiom,
% 9.00/5.04 ( ( minus_minus_int @ one_one_int @ one_one_int )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1500_div__self,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( A2 != zero_zero_rat )
% 9.00/5.04 => ( ( divide_divide_rat @ A2 @ A2 )
% 9.00/5.04 = one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1501_div__self,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( A2 != zero_zero_int )
% 9.00/5.04 => ( ( divide_divide_int @ A2 @ A2 )
% 9.00/5.04 = one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1502_div__self,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( A2 != zero_zero_nat )
% 9.00/5.04 => ( ( divide_divide_nat @ A2 @ A2 )
% 9.00/5.04 = one_one_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1503_div__self,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( A2 != zero_zero_real )
% 9.00/5.04 => ( ( divide_divide_real @ A2 @ A2 )
% 9.00/5.04 = one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1504_div__self,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( A2 != zero_zero_complex )
% 9.00/5.04 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 9.00/5.04 = one_one_complex ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1505_verit__minus__simplify_I3_J,axiom,
% 9.00/5.04 ! [B3: int] :
% 9.00/5.04 ( ( minus_minus_int @ zero_zero_int @ B3 )
% 9.00/5.04 = ( uminus_uminus_int @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1506_verit__minus__simplify_I3_J,axiom,
% 9.00/5.04 ! [B3: real] :
% 9.00/5.04 ( ( minus_minus_real @ zero_zero_real @ B3 )
% 9.00/5.04 = ( uminus_uminus_real @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1507_verit__minus__simplify_I3_J,axiom,
% 9.00/5.04 ! [B3: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ zero_zero_rat @ B3 )
% 9.00/5.04 = ( uminus_uminus_rat @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1508_verit__minus__simplify_I3_J,axiom,
% 9.00/5.04 ! [B3: code_integer] :
% 9.00/5.04 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.04 = ( uminus1351360451143612070nteger @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1509_verit__minus__simplify_I3_J,axiom,
% 9.00/5.04 ! [B3: complex] :
% 9.00/5.04 ( ( minus_minus_complex @ zero_zero_complex @ B3 )
% 9.00/5.04 = ( uminus1482373934393186551omplex @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1510_diff__0,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ zero_zero_int @ A2 )
% 9.00/5.04 = ( uminus_uminus_int @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1511_diff__0,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ zero_zero_real @ A2 )
% 9.00/5.04 = ( uminus_uminus_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1512_diff__0,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 = ( uminus_uminus_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1513_diff__0,axiom,
% 9.00/5.04 ! [A2: code_integer] :
% 9.00/5.04 ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.04 = ( uminus1351360451143612070nteger @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1514_diff__0,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( minus_minus_complex @ zero_zero_complex @ A2 )
% 9.00/5.04 = ( uminus1482373934393186551omplex @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1515_zero__less__diff,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
% 9.00/5.04 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1516_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_complex @ bot_bot_set_complex )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1517_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_list_nat @ bot_bot_set_list_nat )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1518_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_set_nat @ bot_bot_set_set_nat )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1519_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_real @ bot_bot_set_real )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1520_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_o @ bot_bot_set_o )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1521_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_nat @ bot_bot_set_nat )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1522_card_Oempty,axiom,
% 9.00/5.04 ( ( finite_card_int @ bot_bot_set_int )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1523_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ~ ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ( finite_card_list_nat @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1524_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ~ ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ( finite_card_set_nat @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1525_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ( finite_card_nat @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1526_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ~ ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ( finite_card_int @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1527_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ~ ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ( finite_card_complex @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1528_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ~ ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1529_card_Oinfinite,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1530_diff__is__0__eq,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ( minus_minus_nat @ M @ N )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1531_diff__is__0__eq_H,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ( minus_minus_nat @ M @ N )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1532_diff__Suc__1,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
% 9.00/5.04 = N ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1533_negative__eq__positive,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.04 = ( semiri1314217659103216013at_int @ M ) )
% 9.00/5.04 = ( ( N = zero_zero_nat )
% 9.00/5.04 & ( M = zero_zero_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1534_negative__zle,axiom,
% 9.00/5.04 ! [N: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1535_diff__numeral__special_I12_J,axiom,
% 9.00/5.04 ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1536_diff__numeral__special_I12_J,axiom,
% 9.00/5.04 ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1537_diff__numeral__special_I12_J,axiom,
% 9.00/5.04 ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1538_diff__numeral__special_I12_J,axiom,
% 9.00/5.04 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.00/5.04 = zero_z3403309356797280102nteger ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1539_diff__numeral__special_I12_J,axiom,
% 9.00/5.04 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1540_Suc__pred,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.00/5.04 = N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1541_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_list_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1542_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_set_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1543_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ( ( finite_card_complex @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_complex ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1544_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1545_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bo7653980558646680370d_enat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1546_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( finite_finite_real @ A3 )
% 9.00/5.04 => ( ( ( finite_card_real @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1547_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( finite_finite_o @ A3 )
% 9.00/5.04 => ( ( ( finite_card_o @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_o ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1548_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ( ( finite_card_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1549_card__0__eq,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ( ( finite_card_int @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( A3 = bot_bot_set_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1550_negative__zless,axiom,
% 9.00/5.04 ! [N: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1551_finite__enumerate__mono__iff,axiom,
% 9.00/5.04 ! [S: set_Extended_enat,M: nat,N: nat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ M @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ( ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) )
% 9.00/5.04 = ( ord_less_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1552_finite__enumerate__mono__iff,axiom,
% 9.00/5.04 ! [S: set_nat,M: nat,N: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ M @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) )
% 9.00/5.04 = ( ord_less_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1553_diff__eq__diff__eq,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.04 ( ( ( minus_minus_real @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_real @ C @ D ) )
% 9.00/5.04 => ( ( A2 = B3 )
% 9.00/5.04 = ( C = D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1554_diff__eq__diff__eq,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.04 ( ( ( minus_minus_rat @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_rat @ C @ D ) )
% 9.00/5.04 => ( ( A2 = B3 )
% 9.00/5.04 = ( C = D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1555_diff__eq__diff__eq,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.04 ( ( ( minus_minus_int @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_int @ C @ D ) )
% 9.00/5.04 => ( ( A2 = B3 )
% 9.00/5.04 = ( C = D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1556_equation__minus__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_int @ B3 ) )
% 9.00/5.04 = ( B3
% 9.00/5.04 = ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1557_equation__minus__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_real @ B3 ) )
% 9.00/5.04 = ( B3
% 9.00/5.04 = ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1558_equation__minus__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus_uminus_rat @ B3 ) )
% 9.00/5.04 = ( B3
% 9.00/5.04 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1559_equation__minus__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.04 = ( B3
% 9.00/5.04 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1560_equation__minus__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( A2
% 9.00/5.04 = ( uminus1482373934393186551omplex @ B3 ) )
% 9.00/5.04 = ( B3
% 9.00/5.04 = ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1561_minus__equation__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ( uminus_uminus_int @ A2 )
% 9.00/5.04 = B3 )
% 9.00/5.04 = ( ( uminus_uminus_int @ B3 )
% 9.00/5.04 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1562_minus__equation__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( uminus_uminus_real @ A2 )
% 9.00/5.04 = B3 )
% 9.00/5.04 = ( ( uminus_uminus_real @ B3 )
% 9.00/5.04 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1563_minus__equation__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( uminus_uminus_rat @ A2 )
% 9.00/5.04 = B3 )
% 9.00/5.04 = ( ( uminus_uminus_rat @ B3 )
% 9.00/5.04 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1564_minus__equation__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.04 = B3 )
% 9.00/5.04 = ( ( uminus1351360451143612070nteger @ B3 )
% 9.00/5.04 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1565_minus__equation__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.04 = B3 )
% 9.00/5.04 = ( ( uminus1482373934393186551omplex @ B3 )
% 9.00/5.04 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1566_minus__diff__commute,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( minus_minus_int @ ( uminus_uminus_int @ B3 ) @ A2 )
% 9.00/5.04 = ( minus_minus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1567_minus__diff__commute,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( minus_minus_real @ ( uminus_uminus_real @ B3 ) @ A2 )
% 9.00/5.04 = ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1568_minus__diff__commute,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ ( uminus_uminus_rat @ B3 ) @ A2 )
% 9.00/5.04 = ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1569_minus__diff__commute,axiom,
% 9.00/5.04 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.04 ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 )
% 9.00/5.04 = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1570_minus__diff__commute,axiom,
% 9.00/5.04 ! [B3: complex,A2: complex] :
% 9.00/5.04 ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ A2 )
% 9.00/5.04 = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1571_diff__right__commute,axiom,
% 9.00/5.04 ! [A2: real,C: real,B3: real] :
% 9.00/5.04 ( ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 )
% 9.00/5.04 = ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1572_diff__right__commute,axiom,
% 9.00/5.04 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 )
% 9.00/5.04 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1573_diff__right__commute,axiom,
% 9.00/5.04 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B3 )
% 9.00/5.04 = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1574_diff__right__commute,axiom,
% 9.00/5.04 ! [A2: int,C: int,B3: int] :
% 9.00/5.04 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 )
% 9.00/5.04 = ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1575_diff__commute,axiom,
% 9.00/5.04 ! [I: nat,J: nat,K: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.00/5.04 = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1576_size__neq__size__imp__neq,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.00/5.04 ( ( ( size_size_VEBT_VEBT @ X2 )
% 9.00/5.04 != ( size_size_VEBT_VEBT @ Y3 ) )
% 9.00/5.04 => ( X2 != Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1577_size__neq__size__imp__neq,axiom,
% 9.00/5.04 ! [X2: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 9.00/5.04 ( ( ( size_s6755466524823107622T_VEBT @ X2 )
% 9.00/5.04 != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 9.00/5.04 => ( X2 != Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1578_size__neq__size__imp__neq,axiom,
% 9.00/5.04 ! [X2: num,Y3: num] :
% 9.00/5.04 ( ( ( size_size_num @ X2 )
% 9.00/5.04 != ( size_size_num @ Y3 ) )
% 9.00/5.04 => ( X2 != Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1579_size__neq__size__imp__neq,axiom,
% 9.00/5.04 ! [X2: list_o,Y3: list_o] :
% 9.00/5.04 ( ( ( size_size_list_o @ X2 )
% 9.00/5.04 != ( size_size_list_o @ Y3 ) )
% 9.00/5.04 => ( X2 != Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1580_size__neq__size__imp__neq,axiom,
% 9.00/5.04 ! [X2: list_nat,Y3: list_nat] :
% 9.00/5.04 ( ( ( size_size_list_nat @ X2 )
% 9.00/5.04 != ( size_size_list_nat @ Y3 ) )
% 9.00/5.04 => ( X2 != Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1581_of__nat__diff,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.04 = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1582_of__nat__diff,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.04 = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1583_of__nat__diff,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.04 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1584_of__nat__diff,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.04 = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1585_diff__eq__diff__less__eq,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.04 ( ( ( minus_minus_real @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_real @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1586_diff__eq__diff__less__eq,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.04 ( ( ( minus_minus_rat @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_rat @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1587_diff__eq__diff__less__eq,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.04 ( ( ( minus_minus_int @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_int @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1588_diff__right__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1589_diff__right__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1590_diff__right__mono,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1591_diff__left__mono,axiom,
% 9.00/5.04 ! [B3: real,A2: real,C: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_eq_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1592_diff__left__mono,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1593_diff__left__mono,axiom,
% 9.00/5.04 ! [B3: int,A2: int,C: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1594_diff__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,D: real,C: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_real @ D @ C )
% 9.00/5.04 => ( ord_less_eq_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1595_diff__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,D: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ D @ C )
% 9.00/5.04 => ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1596_diff__mono,axiom,
% 9.00/5.04 ! [A2: int,B3: int,D: int,C: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_int @ D @ C )
% 9.00/5.04 => ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1597_eq__iff__diff__eq__0,axiom,
% 9.00/5.04 ( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
% 9.00/5.04 = ( ^ [A4: real,B4: real] :
% 9.00/5.04 ( ( minus_minus_real @ A4 @ B4 )
% 9.00/5.04 = zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1598_eq__iff__diff__eq__0,axiom,
% 9.00/5.04 ( ( ^ [Y6: rat,Z3: rat] : ( Y6 = Z3 ) )
% 9.00/5.04 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.04 ( ( minus_minus_rat @ A4 @ B4 )
% 9.00/5.04 = zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1599_eq__iff__diff__eq__0,axiom,
% 9.00/5.04 ( ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 ) )
% 9.00/5.04 = ( ^ [A4: int,B4: int] :
% 9.00/5.04 ( ( minus_minus_int @ A4 @ B4 )
% 9.00/5.04 = zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1600_diff__strict__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,D: real,C: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_real @ D @ C )
% 9.00/5.04 => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1601_diff__strict__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,D: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_rat @ D @ C )
% 9.00/5.04 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1602_diff__strict__mono,axiom,
% 9.00/5.04 ! [A2: int,B3: int,D: int,C: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_int @ D @ C )
% 9.00/5.04 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1603_diff__eq__diff__less,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.04 ( ( ( minus_minus_real @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_real @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_real @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1604_diff__eq__diff__less,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.04 ( ( ( minus_minus_rat @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_rat @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_rat @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1605_diff__eq__diff__less,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.04 ( ( ( minus_minus_int @ A2 @ B3 )
% 9.00/5.04 = ( minus_minus_int @ C @ D ) )
% 9.00/5.04 => ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.04 = ( ord_less_int @ C @ D ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1606_diff__strict__left__mono,axiom,
% 9.00/5.04 ! [B3: real,A2: real,C: real] :
% 9.00/5.04 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_real @ ( minus_minus_real @ C @ A2 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1607_diff__strict__left__mono,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_rat @ ( minus_minus_rat @ C @ A2 ) @ ( minus_minus_rat @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1608_diff__strict__left__mono,axiom,
% 9.00/5.04 ! [B3: int,A2: int,C: int] :
% 9.00/5.04 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1609_diff__strict__right__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_real @ ( minus_minus_real @ A2 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1610_diff__strict__right__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_rat @ ( minus_minus_rat @ A2 @ C ) @ ( minus_minus_rat @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1611_diff__strict__right__mono,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B3 @ C ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1612_le__imp__neg__le,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1613_le__imp__neg__le,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 9.00/5.04 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1614_le__imp__neg__le,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1615_le__imp__neg__le,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1616_minus__le__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1617_minus__le__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1618_minus__le__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1619_minus__le__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_eq_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1620_le__minus__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1621_le__minus__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le3102999989581377725nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.04 = ( ord_le3102999989581377725nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1622_le__minus__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1623_le__minus__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1624_verit__negate__coefficient_I2_J,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1625_verit__negate__coefficient_I2_J,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1626_verit__negate__coefficient_I2_J,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.04 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1627_verit__negate__coefficient_I2_J,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
% 9.00/5.04 => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1628_less__minus__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 9.00/5.04 = ( ord_less_int @ B3 @ ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1629_less__minus__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 9.00/5.04 = ( ord_less_real @ B3 @ ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1630_less__minus__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 9.00/5.04 = ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1631_less__minus__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ B3 @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1632_minus__less__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_int @ ( uminus_uminus_int @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1633_minus__less__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1634_minus__less__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1635_minus__less__iff,axiom,
% 9.00/5.04 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.04 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 9.00/5.04 = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B3 ) @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1636_zero__induct__lemma,axiom,
% 9.00/5.04 ! [P: nat > $o,K: nat,I: nat] :
% 9.00/5.04 ( ( P @ K )
% 9.00/5.04 => ( ! [N3: nat] :
% 9.00/5.04 ( ( P @ ( suc @ N3 ) )
% 9.00/5.04 => ( P @ N3 ) )
% 9.00/5.04 => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1637_diffs0__imp__equal,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ( minus_minus_nat @ M @ N )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 => ( ( ( minus_minus_nat @ N @ M )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 => ( M = N ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1638_minus__nat_Odiff__0,axiom,
% 9.00/5.04 ! [M: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ M @ zero_zero_nat )
% 9.00/5.04 = M ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1639_diff__less__mono2,axiom,
% 9.00/5.04 ! [M: nat,N: nat,L: nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ( ord_less_nat @ M @ L )
% 9.00/5.04 => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1640_less__imp__diff__less,axiom,
% 9.00/5.04 ! [J: nat,K: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ J @ K )
% 9.00/5.04 => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1641_diff__le__mono2,axiom,
% 9.00/5.04 ! [M: nat,N: nat,L: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1642_le__diff__iff_H,axiom,
% 9.00/5.04 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ A2 @ C )
% 9.00/5.04 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_nat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1643_diff__le__self,axiom,
% 9.00/5.04 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1644_diff__le__mono,axiom,
% 9.00/5.04 ! [M: nat,N: nat,L: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1645_Nat_Odiff__diff__eq,axiom,
% 9.00/5.04 ! [K: nat,M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.04 => ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.04 => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.00/5.04 = ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1646_le__diff__iff,axiom,
% 9.00/5.04 ! [K: nat,M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.04 => ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.00/5.04 = ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1647_eq__diff__iff,axiom,
% 9.00/5.04 ! [K: nat,M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.04 => ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.04 => ( ( ( minus_minus_nat @ M @ K )
% 9.00/5.04 = ( minus_minus_nat @ N @ K ) )
% 9.00/5.04 = ( M = N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1648_vebt__delete_Osimps_I4_J,axiom,
% 9.00/5.04 ! [Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu: nat] :
% 9.00/5.04 ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Uu )
% 9.00/5.04 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1649_vebt__member_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1650_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 9.00/5.04 ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1651_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 9.00/5.04 ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.00/5.04 ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1652_verit__less__mono__div__int2,axiom,
% 9.00/5.04 ! [A3: int,B2: int,N: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N ) )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N ) @ ( divide_divide_int @ A3 @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1653_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_list_nat,N: nat] :
% 9.00/5.04 ( ~ ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ? [B8: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B8 )
% 9.00/5.04 & ( ( finite_card_list_nat @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_le6045566169113846134st_nat @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1654_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_set_nat,N: nat] :
% 9.00/5.04 ( ~ ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ? [B8: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B8 )
% 9.00/5.04 & ( ( finite_card_set_nat @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_le6893508408891458716et_nat @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1655_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_nat,N: nat] :
% 9.00/5.04 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.04 => ? [B8: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B8 )
% 9.00/5.04 & ( ( finite_card_nat @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_less_eq_set_nat @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1656_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_complex,N: nat] :
% 9.00/5.04 ( ~ ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ? [B8: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B8 )
% 9.00/5.04 & ( ( finite_card_complex @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_le211207098394363844omplex @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1657_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 9.00/5.04 ( ~ ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ? [B8: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B8 )
% 9.00/5.04 & ( ( finite711546835091564841at_nat @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_le3146513528884898305at_nat @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1658_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,N: nat] :
% 9.00/5.04 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ? [B8: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B8 )
% 9.00/5.04 & ( ( finite121521170596916366d_enat @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_le7203529160286727270d_enat @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1659_infinite__arbitrarily__large,axiom,
% 9.00/5.04 ! [A3: set_int,N: nat] :
% 9.00/5.04 ( ~ ( finite_finite_int @ A3 )
% 9.00/5.04 => ? [B8: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B8 )
% 9.00/5.04 & ( ( finite_card_int @ B8 )
% 9.00/5.04 = N )
% 9.00/5.04 & ( ord_less_eq_set_int @ B8 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1660_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.04 = ( finite_card_list_nat @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1661_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.04 = ( finite_card_set_nat @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1662_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite_card_nat @ A3 )
% 9.00/5.04 = ( finite_card_nat @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1663_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite_card_complex @ A3 )
% 9.00/5.04 = ( finite_card_complex @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1664_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.04 = ( finite711546835091564841at_nat @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1665_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.04 = ( finite121521170596916366d_enat @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1666_card__subset__eq,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ( ( finite_card_int @ A3 )
% 9.00/5.04 = ( finite_card_int @ B2 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1667_le__iff__diff__le__0,axiom,
% 9.00/5.04 ( ord_less_eq_real
% 9.00/5.04 = ( ^ [A4: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1668_le__iff__diff__le__0,axiom,
% 9.00/5.04 ( ord_less_eq_rat
% 9.00/5.04 = ( ^ [A4: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1669_le__iff__diff__le__0,axiom,
% 9.00/5.04 ( ord_less_eq_int
% 9.00/5.04 = ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1670_less__iff__diff__less__0,axiom,
% 9.00/5.04 ( ord_less_real
% 9.00/5.04 = ( ^ [A4: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A4 @ B4 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1671_less__iff__diff__less__0,axiom,
% 9.00/5.04 ( ord_less_rat
% 9.00/5.04 = ( ^ [A4: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A4 @ B4 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1672_less__iff__diff__less__0,axiom,
% 9.00/5.04 ( ord_less_int
% 9.00/5.04 = ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1673_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_real,A3: set_real,R2: real > real > $o] :
% 9.00/5.04 ( ( finite_finite_real @ B2 )
% 9.00/5.04 => ( ! [A: real] :
% 9.00/5.04 ( ( member_real @ A @ A3 )
% 9.00/5.04 => ? [B9: real] :
% 9.00/5.04 ( ( member_real @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: real,A22: real,B: real] :
% 9.00/5.04 ( ( member_real @ A1 @ A3 )
% 9.00/5.04 => ( ( member_real @ A22 @ A3 )
% 9.00/5.04 => ( ( member_real @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1674_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_o,A3: set_real,R2: real > $o > $o] :
% 9.00/5.04 ( ( finite_finite_o @ B2 )
% 9.00/5.04 => ( ! [A: real] :
% 9.00/5.04 ( ( member_real @ A @ A3 )
% 9.00/5.04 => ? [B9: $o] :
% 9.00/5.04 ( ( member_o @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: real,A22: real,B: $o] :
% 9.00/5.04 ( ( member_real @ A1 @ A3 )
% 9.00/5.04 => ( ( member_real @ A22 @ A3 )
% 9.00/5.04 => ( ( member_o @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1675_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_real,A3: set_o,R2: $o > real > $o] :
% 9.00/5.04 ( ( finite_finite_real @ B2 )
% 9.00/5.04 => ( ! [A: $o] :
% 9.00/5.04 ( ( member_o @ A @ A3 )
% 9.00/5.04 => ? [B9: real] :
% 9.00/5.04 ( ( member_real @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: $o,A22: $o,B: real] :
% 9.00/5.04 ( ( member_o @ A1 @ A3 )
% 9.00/5.04 => ( ( member_o @ A22 @ A3 )
% 9.00/5.04 => ( ( member_real @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1676_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_o,A3: set_o,R2: $o > $o > $o] :
% 9.00/5.04 ( ( finite_finite_o @ B2 )
% 9.00/5.04 => ( ! [A: $o] :
% 9.00/5.04 ( ( member_o @ A @ A3 )
% 9.00/5.04 => ? [B9: $o] :
% 9.00/5.04 ( ( member_o @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: $o,A22: $o,B: $o] :
% 9.00/5.04 ( ( member_o @ A1 @ A3 )
% 9.00/5.04 => ( ( member_o @ A22 @ A3 )
% 9.00/5.04 => ( ( member_o @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1677_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_real,A3: set_complex,R2: complex > real > $o] :
% 9.00/5.04 ( ( finite_finite_real @ B2 )
% 9.00/5.04 => ( ! [A: complex] :
% 9.00/5.04 ( ( member_complex @ A @ A3 )
% 9.00/5.04 => ? [B9: real] :
% 9.00/5.04 ( ( member_real @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: complex,A22: complex,B: real] :
% 9.00/5.04 ( ( member_complex @ A1 @ A3 )
% 9.00/5.04 => ( ( member_complex @ A22 @ A3 )
% 9.00/5.04 => ( ( member_real @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1678_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_o,A3: set_complex,R2: complex > $o > $o] :
% 9.00/5.04 ( ( finite_finite_o @ B2 )
% 9.00/5.04 => ( ! [A: complex] :
% 9.00/5.04 ( ( member_complex @ A @ A3 )
% 9.00/5.04 => ? [B9: $o] :
% 9.00/5.04 ( ( member_o @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: complex,A22: complex,B: $o] :
% 9.00/5.04 ( ( member_complex @ A1 @ A3 )
% 9.00/5.04 => ( ( member_complex @ A22 @ A3 )
% 9.00/5.04 => ( ( member_o @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1679_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_real,A3: set_nat,R2: nat > real > $o] :
% 9.00/5.04 ( ( finite_finite_real @ B2 )
% 9.00/5.04 => ( ! [A: nat] :
% 9.00/5.04 ( ( member_nat @ A @ A3 )
% 9.00/5.04 => ? [B9: real] :
% 9.00/5.04 ( ( member_real @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: nat,A22: nat,B: real] :
% 9.00/5.04 ( ( member_nat @ A1 @ A3 )
% 9.00/5.04 => ( ( member_nat @ A22 @ A3 )
% 9.00/5.04 => ( ( member_real @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1680_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_o,A3: set_nat,R2: nat > $o > $o] :
% 9.00/5.04 ( ( finite_finite_o @ B2 )
% 9.00/5.04 => ( ! [A: nat] :
% 9.00/5.04 ( ( member_nat @ A @ A3 )
% 9.00/5.04 => ? [B9: $o] :
% 9.00/5.04 ( ( member_o @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: nat,A22: nat,B: $o] :
% 9.00/5.04 ( ( member_nat @ A1 @ A3 )
% 9.00/5.04 => ( ( member_nat @ A22 @ A3 )
% 9.00/5.04 => ( ( member_o @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1681_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_real,A3: set_int,R2: int > real > $o] :
% 9.00/5.04 ( ( finite_finite_real @ B2 )
% 9.00/5.04 => ( ! [A: int] :
% 9.00/5.04 ( ( member_int @ A @ A3 )
% 9.00/5.04 => ? [B9: real] :
% 9.00/5.04 ( ( member_real @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: int,A22: int,B: real] :
% 9.00/5.04 ( ( member_int @ A1 @ A3 )
% 9.00/5.04 => ( ( member_int @ A22 @ A3 )
% 9.00/5.04 => ( ( member_real @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1682_card__le__if__inj__on__rel,axiom,
% 9.00/5.04 ! [B2: set_o,A3: set_int,R2: int > $o > $o] :
% 9.00/5.04 ( ( finite_finite_o @ B2 )
% 9.00/5.04 => ( ! [A: int] :
% 9.00/5.04 ( ( member_int @ A @ A3 )
% 9.00/5.04 => ? [B9: $o] :
% 9.00/5.04 ( ( member_o @ B9 @ B2 )
% 9.00/5.04 & ( R2 @ A @ B9 ) ) )
% 9.00/5.04 => ( ! [A1: int,A22: int,B: $o] :
% 9.00/5.04 ( ( member_int @ A1 @ A3 )
% 9.00/5.04 => ( ( member_int @ A22 @ A3 )
% 9.00/5.04 => ( ( member_o @ B @ B2 )
% 9.00/5.04 => ( ( R2 @ A1 @ B )
% 9.00/5.04 => ( ( R2 @ A22 @ B )
% 9.00/5.04 => ( A1 = A22 ) ) ) ) ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1683_le__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1684_le__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1685_le__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1686_le__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1687_le__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1688_le__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1689_le__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1690_le__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1691_zero__neq__neg__one,axiom,
% 9.00/5.04 ( zero_zero_int
% 9.00/5.04 != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1692_zero__neq__neg__one,axiom,
% 9.00/5.04 ( zero_zero_real
% 9.00/5.04 != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1693_zero__neq__neg__one,axiom,
% 9.00/5.04 ( zero_zero_rat
% 9.00/5.04 != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1694_zero__neq__neg__one,axiom,
% 9.00/5.04 ( zero_z3403309356797280102nteger
% 9.00/5.04 != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1695_zero__neq__neg__one,axiom,
% 9.00/5.04 ( zero_zero_complex
% 9.00/5.04 != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1696_less__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1697_less__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1698_less__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1699_less__minus__one__simps_I2_J,axiom,
% 9.00/5.04 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1700_less__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1701_less__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1702_less__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1703_less__minus__one__simps_I4_J,axiom,
% 9.00/5.04 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1704_diff__less__Suc,axiom,
% 9.00/5.04 ! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1705_Suc__diff__Suc,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ N @ M )
% 9.00/5.04 => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
% 9.00/5.04 = ( minus_minus_nat @ M @ N ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1706_diff__less,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.04 => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1707_Suc__diff__le,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.00/5.04 = ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1708_diff__less__mono,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.04 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.04 => ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1709_less__diff__iff,axiom,
% 9.00/5.04 ! [K: nat,M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.04 => ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.04 => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
% 9.00/5.04 = ( ord_less_nat @ M @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1710_diff__Suc__eq__diff__pred,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.00/5.04 = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1711_not__int__zless__negative,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1712_vebt__member_Osimps_I3_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1713_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 9.00/5.04 ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1714_vebt__mint_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1715_vebt__maxt_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1716_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT] :
% 9.00/5.04 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.04 => ( ! [Uv2: $o] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.00/5.04 => ( ! [Uu2: $o] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.00/5.04 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1717_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.04 => ( ( X2
% 9.00/5.04 != ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.04 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1718_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT,Y3: $o] :
% 9.00/5.04 ( ( ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.04 = Y3 )
% 9.00/5.04 => ( ( ( X2
% 9.00/5.04 = ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.04 => ~ Y3 )
% 9.00/5.04 => ( ( ? [Uv2: $o] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.00/5.04 => Y3 )
% 9.00/5.04 => ( ( ? [Uu2: $o] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.00/5.04 => Y3 )
% 9.00/5.04 => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.00/5.04 => ~ Y3 )
% 9.00/5.04 => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.00/5.04 => Y3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1719_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 9.00/5.04 ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 9.00/5.04 ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1720_vebt__pred_Osimps_I4_J,axiom,
% 9.00/5.04 ! [Uy: nat,Uz: list_VEBT_VEBT,Va: vEBT_VEBT,Vb: nat] :
% 9.00/5.04 ( ( vEBT_vebt_pred @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz @ Va ) @ Vb )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1721_vebt__succ_Osimps_I3_J,axiom,
% 9.00/5.04 ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
% 9.00/5.04 ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1722_frac__ge__0,axiom,
% 9.00/5.04 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1723_frac__ge__0,axiom,
% 9.00/5.04 ! [X2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1724_frac__lt__1,axiom,
% 9.00/5.04 ! [X2: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X2 ) @ one_one_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1725_frac__lt__1,axiom,
% 9.00/5.04 ! [X2: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X2 ) @ one_one_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1726_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_list_nat )
% 9.00/5.04 | ~ ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1727_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_set_nat )
% 9.00/5.04 | ~ ( finite1152437895449049373et_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1728_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ( ( finite_card_complex @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_complex )
% 9.00/5.04 | ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1729_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bo2099793752762293965at_nat )
% 9.00/5.04 | ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1730_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bo7653980558646680370d_enat )
% 9.00/5.04 | ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1731_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( ( finite_card_real @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_real )
% 9.00/5.04 | ~ ( finite_finite_real @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1732_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( ( finite_card_o @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_o )
% 9.00/5.04 | ~ ( finite_finite_o @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1733_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( ( finite_card_nat @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_nat )
% 9.00/5.04 | ~ ( finite_finite_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1734_card__eq__0__iff,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( ( finite_card_int @ A3 )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( A3 = bot_bot_set_int )
% 9.00/5.04 | ~ ( finite_finite_int @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1735_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
% 9.00/5.04 => ( finite8100373058378681591st_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1736_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
% 9.00/5.04 => ( finite1152437895449049373et_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1737_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 9.00/5.04 => ( finite_finite_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1738_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 9.00/5.04 => ( finite_finite_int @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1739_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 9.00/5.04 => ( finite3207457112153483333omplex @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1740_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
% 9.00/5.04 => ( finite6177210948735845034at_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1741_card__ge__0__finite,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
% 9.00/5.04 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1742_card__mono,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1743_card__mono,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1744_card__mono,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1745_card__mono,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1746_card__mono,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1747_card__mono,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1748_card__mono,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1749_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B2 ) @ ( finite_card_list_nat @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1750_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B2 ) @ ( finite_card_set_nat @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1751_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_nat @ B2 ) @ ( finite_card_nat @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1752_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_complex @ B2 ) @ ( finite_card_complex @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1753_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ B2 ) @ ( finite711546835091564841at_nat @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1754_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ B2 ) @ ( finite121521170596916366d_enat @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1755_card__seteq,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_int @ B2 ) @ ( finite_card_int @ A3 ) )
% 9.00/5.04 => ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1756_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_list_nat,C2: nat] :
% 9.00/5.04 ( ! [G: set_list_nat] :
% 9.00/5.04 ( ( ord_le6045566169113846134st_nat @ G @ F2 )
% 9.00/5.04 => ( ( finite8100373058378681591st_nat @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite8100373058378681591st_nat @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite_card_list_nat @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1757_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_set_nat,C2: nat] :
% 9.00/5.04 ( ! [G: set_set_nat] :
% 9.00/5.04 ( ( ord_le6893508408891458716et_nat @ G @ F2 )
% 9.00/5.04 => ( ( finite1152437895449049373et_nat @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite1152437895449049373et_nat @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite_card_set_nat @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1758_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_nat,C2: nat] :
% 9.00/5.04 ( ! [G: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ G @ F2 )
% 9.00/5.04 => ( ( finite_finite_nat @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite_finite_nat @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1759_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_complex,C2: nat] :
% 9.00/5.04 ( ! [G: set_complex] :
% 9.00/5.04 ( ( ord_le211207098394363844omplex @ G @ F2 )
% 9.00/5.04 => ( ( finite3207457112153483333omplex @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite_card_complex @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1760_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_Pr1261947904930325089at_nat,C2: nat] :
% 9.00/5.04 ( ! [G: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ G @ F2 )
% 9.00/5.04 => ( ( finite6177210948735845034at_nat @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1761_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_Extended_enat,C2: nat] :
% 9.00/5.04 ( ! [G: set_Extended_enat] :
% 9.00/5.04 ( ( ord_le7203529160286727270d_enat @ G @ F2 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1762_finite__if__finite__subsets__card__bdd,axiom,
% 9.00/5.04 ! [F2: set_int,C2: nat] :
% 9.00/5.04 ( ! [G: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ G @ F2 )
% 9.00/5.04 => ( ( finite_finite_int @ G )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C2 ) ) )
% 9.00/5.04 => ( ( finite_finite_int @ F2 )
% 9.00/5.04 & ( ord_less_eq_nat @ ( finite_card_int @ F2 ) @ C2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1763_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_list_nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S ) )
% 9.00/5.04 => ~ ! [T4: set_list_nat] :
% 9.00/5.04 ( ( ord_le6045566169113846134st_nat @ T4 @ S )
% 9.00/5.04 => ( ( ( finite_card_list_nat @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite8100373058378681591st_nat @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1764_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_set_nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S ) )
% 9.00/5.04 => ~ ! [T4: set_set_nat] :
% 9.00/5.04 ( ( ord_le6893508408891458716et_nat @ T4 @ S )
% 9.00/5.04 => ( ( ( finite_card_set_nat @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite1152437895449049373et_nat @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1765_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ~ ! [T4: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ T4 @ S )
% 9.00/5.04 => ( ( ( finite_card_nat @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1766_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_complex] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S ) )
% 9.00/5.04 => ~ ! [T4: set_complex] :
% 9.00/5.04 ( ( ord_le211207098394363844omplex @ T4 @ S )
% 9.00/5.04 => ( ( ( finite_card_complex @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite3207457112153483333omplex @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1767_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ S ) )
% 9.00/5.04 => ~ ! [T4: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ T4 @ S )
% 9.00/5.04 => ( ( ( finite711546835091564841at_nat @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite6177210948735845034at_nat @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1768_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_Extended_enat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ~ ! [T4: set_Extended_enat] :
% 9.00/5.04 ( ( ord_le7203529160286727270d_enat @ T4 @ S )
% 9.00/5.04 => ( ( ( finite121521170596916366d_enat @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite4001608067531595151d_enat @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1769_obtain__subset__with__card__n,axiom,
% 9.00/5.04 ! [N: nat,S: set_int] :
% 9.00/5.04 ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S ) )
% 9.00/5.04 => ~ ! [T4: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ T4 @ S )
% 9.00/5.04 => ( ( ( finite_card_int @ T4 )
% 9.00/5.04 = N )
% 9.00/5.04 => ~ ( finite_finite_int @ T4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1770_le__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1771_le__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1772_le__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1773_le__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1774_le__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1775_le__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1776_le__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1777_le__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1778_less__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1779_less__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1780_less__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1781_less__minus__one__simps_I1_J,axiom,
% 9.00/5.04 ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1782_less__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1783_less__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1784_less__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1785_less__minus__one__simps_I3_J,axiom,
% 9.00/5.04 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1786_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le1190675801316882794st_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1787_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_set_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1788_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1789_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1790_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_less_set_complex @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1791_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1792_psubset__card__mono,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1793_diff__Suc__less,axiom,
% 9.00/5.04 ! [N: nat,I: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1794_finite__enumerate__in__set,axiom,
% 9.00/5.04 ! [S: set_Extended_enat,N: nat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ( member_Extended_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ S ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1795_finite__enumerate__in__set,axiom,
% 9.00/5.04 ! [S: set_nat,N: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( member_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ S ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1796_finite__enumerate__Ex,axiom,
% 9.00/5.04 ! [S: set_Extended_enat,S2: extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ( member_Extended_enat @ S2 @ S )
% 9.00/5.04 => ? [N3: nat] :
% 9.00/5.04 ( ( ord_less_nat @ N3 @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 & ( ( infini7641415182203889163d_enat @ S @ N3 )
% 9.00/5.04 = S2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1797_finite__enumerate__Ex,axiom,
% 9.00/5.04 ! [S: set_nat,S2: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( member_nat @ S2 @ S )
% 9.00/5.04 => ? [N3: nat] :
% 9.00/5.04 ( ( ord_less_nat @ N3 @ ( finite_card_nat @ S ) )
% 9.00/5.04 & ( ( infini8530281810654367211te_nat @ S @ N3 )
% 9.00/5.04 = S2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1798_finite__enum__ext,axiom,
% 9.00/5.04 ! [X6: set_Extended_enat,Y7: set_Extended_enat] :
% 9.00/5.04 ( ! [I2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I2 @ ( finite121521170596916366d_enat @ X6 ) )
% 9.00/5.04 => ( ( infini7641415182203889163d_enat @ X6 @ I2 )
% 9.00/5.04 = ( infini7641415182203889163d_enat @ Y7 @ I2 ) ) )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ X6 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ Y7 )
% 9.00/5.04 => ( ( ( finite121521170596916366d_enat @ X6 )
% 9.00/5.04 = ( finite121521170596916366d_enat @ Y7 ) )
% 9.00/5.04 => ( X6 = Y7 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1799_finite__enum__ext,axiom,
% 9.00/5.04 ! [X6: set_nat,Y7: set_nat] :
% 9.00/5.04 ( ! [I2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ I2 @ ( finite_card_nat @ X6 ) )
% 9.00/5.04 => ( ( infini8530281810654367211te_nat @ X6 @ I2 )
% 9.00/5.04 = ( infini8530281810654367211te_nat @ Y7 @ I2 ) ) )
% 9.00/5.04 => ( ( finite_finite_nat @ X6 )
% 9.00/5.04 => ( ( finite_finite_nat @ Y7 )
% 9.00/5.04 => ( ( ( finite_card_nat @ X6 )
% 9.00/5.04 = ( finite_card_nat @ Y7 ) )
% 9.00/5.04 => ( X6 = Y7 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1800_int__cases4,axiom,
% 9.00/5.04 ! [M: int] :
% 9.00/5.04 ( ! [N3: nat] :
% 9.00/5.04 ( M
% 9.00/5.04 != ( semiri1314217659103216013at_int @ N3 ) )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.04 => ( M
% 9.00/5.04 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1801_vebt__member_Osimps_I4_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X2 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1802_int__zle__neg,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.00/5.04 = ( ( N = zero_zero_nat )
% 9.00/5.04 & ( M = zero_zero_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1803_negative__zle__0,axiom,
% 9.00/5.04 ! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1804_nonpos__int__cases,axiom,
% 9.00/5.04 ! [K: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ( K
% 9.00/5.04 != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1805_vebt__pred_Osimps_I5_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
% 9.00/5.04 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1806_vebt__succ_Osimps_I4_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
% 9.00/5.04 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1807_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_list_nat )
% 9.00/5.04 & ( finite8100373058378681591st_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1808_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_set_nat )
% 9.00/5.04 & ( finite1152437895449049373et_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1809_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_complex )
% 9.00/5.04 & ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1810_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite711546835091564841at_nat @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bo2099793752762293965at_nat )
% 9.00/5.04 & ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1811_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite121521170596916366d_enat @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.04 & ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1812_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_real )
% 9.00/5.04 & ( finite_finite_real @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1813_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_o )
% 9.00/5.04 & ( finite_finite_o @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1814_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_nat )
% 9.00/5.04 & ( finite_finite_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1815_card__gt__0__iff,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 9.00/5.04 = ( ( A3 != bot_bot_set_int )
% 9.00/5.04 & ( finite_finite_int @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1816_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: list_nat] :
% 9.00/5.04 ( ( member_list_nat @ X @ A3 )
% 9.00/5.04 => ! [Y: list_nat] :
% 9.00/5.04 ( ( member_list_nat @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1817_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: set_nat] :
% 9.00/5.04 ( ( member_set_nat @ X @ A3 )
% 9.00/5.04 => ! [Y: set_nat] :
% 9.00/5.04 ( ( member_set_nat @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1818_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: nat] :
% 9.00/5.04 ( ( member_nat @ X @ A3 )
% 9.00/5.04 => ! [Y: nat] :
% 9.00/5.04 ( ( member_nat @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1819_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: int] :
% 9.00/5.04 ( ( member_int @ X @ A3 )
% 9.00/5.04 => ! [Y: int] :
% 9.00/5.04 ( ( member_int @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1820_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: complex] :
% 9.00/5.04 ( ( member_complex @ X @ A3 )
% 9.00/5.04 => ! [Y: complex] :
% 9.00/5.04 ( ( member_complex @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1821_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: product_prod_nat_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ X @ A3 )
% 9.00/5.04 => ! [Y: product_prod_nat_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1822_card__le__Suc0__iff__eq,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = ( ! [X: extended_enat] :
% 9.00/5.04 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.04 => ! [Y: extended_enat] :
% 9.00/5.04 ( ( member_Extended_enat @ Y @ A3 )
% 9.00/5.04 => ( X = Y ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1823_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 9.00/5.04 => ( ord_le1190675801316882794st_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1824_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1825_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1826_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 9.00/5.04 => ( ord_less_set_complex @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1827_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 9.00/5.04 => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1828_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 9.00/5.04 => ( ord_le2529575680413868914d_enat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1829_card__psubset,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 9.00/5.04 => ( ord_less_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1830_finite__enumerate__mono,axiom,
% 9.00/5.04 ! [M: nat,N: nat,S: set_Extended_enat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ M ) @ ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1831_finite__enumerate__mono,axiom,
% 9.00/5.04 ! [M: nat,N: nat,S: set_nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ M ) @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1832_Suc__diff__eq__diff__pred,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( minus_minus_nat @ ( suc @ M ) @ N )
% 9.00/5.04 = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1833_Suc__pred_H,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( N
% 9.00/5.04 = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1834_int__cases3,axiom,
% 9.00/5.04 ! [K: int] :
% 9.00/5.04 ( ( K != zero_zero_int )
% 9.00/5.04 => ( ! [N3: nat] :
% 9.00/5.04 ( ( K
% 9.00/5.04 = ( semiri1314217659103216013at_int @ N3 ) )
% 9.00/5.04 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ( ( K
% 9.00/5.04 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.00/5.04 => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1835_not__zle__0__negative,axiom,
% 9.00/5.04 ! [N: nat] :
% 9.00/5.04 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1836_negD,axiom,
% 9.00/5.04 ! [X2: int] :
% 9.00/5.04 ( ( ord_less_int @ X2 @ zero_zero_int )
% 9.00/5.04 => ? [N3: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1837_negative__zless__0,axiom,
% 9.00/5.04 ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1838_vebt__pred_Osimps_I6_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
% 9.00/5.04 ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1839_vebt__succ_Osimps_I5_J,axiom,
% 9.00/5.04 ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
% 9.00/5.04 ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
% 9.00/5.04 = none_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1840_finite__le__enumerate,axiom,
% 9.00/5.04 ! [S: set_nat,N: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ N @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1841_nat__approx__posE,axiom,
% 9.00/5.04 ! [E2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1842_nat__approx__posE,axiom,
% 9.00/5.04 ! [E2: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.04 => ~ ! [N3: nat] :
% 9.00/5.04 ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1843_finite__enumerate__step,axiom,
% 9.00/5.04 ! [S: set_Extended_enat,N: nat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ S ) )
% 9.00/5.04 => ( ord_le72135733267957522d_enat @ ( infini7641415182203889163d_enat @ S @ N ) @ ( infini7641415182203889163d_enat @ S @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1844_finite__enumerate__step,axiom,
% 9.00/5.04 ! [S: set_nat,N: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ S )
% 9.00/5.04 => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S ) )
% 9.00/5.04 => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S @ N ) @ ( infini8530281810654367211te_nat @ S @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1845_le__divide__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1846_le__divide__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1847_le__divide__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1848_le__divide__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1849_divide__le__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1850_divide__le__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1851_divide__le__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1852_divide__le__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1853_zero__less__divide__1__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1854_zero__less__divide__1__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1855_less__divide__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1856_less__divide__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1857_less__divide__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1858_less__divide__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1859_divide__less__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1860_divide__less__eq__1__pos,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1861_divide__less__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1862_divide__less__eq__1__neg,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1863_divide__less__0__1__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.04 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1864_divide__less__0__1__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
% 9.00/5.04 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1865_Diff__empty,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( minus_minus_set_real @ A3 @ bot_bot_set_real )
% 9.00/5.04 = A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1866_Diff__empty,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( minus_minus_set_o @ A3 @ bot_bot_set_o )
% 9.00/5.04 = A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1867_Diff__empty,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( minus_minus_set_int @ A3 @ bot_bot_set_int )
% 9.00/5.04 = A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1868_Diff__empty,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( minus_minus_set_nat @ A3 @ bot_bot_set_nat )
% 9.00/5.04 = A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1869_empty__Diff,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( minus_minus_set_real @ bot_bot_set_real @ A3 )
% 9.00/5.04 = bot_bot_set_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1870_empty__Diff,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( minus_minus_set_o @ bot_bot_set_o @ A3 )
% 9.00/5.04 = bot_bot_set_o ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1871_empty__Diff,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( minus_minus_set_int @ bot_bot_set_int @ A3 )
% 9.00/5.04 = bot_bot_set_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1872_empty__Diff,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( minus_minus_set_nat @ bot_bot_set_nat @ A3 )
% 9.00/5.04 = bot_bot_set_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1873_Diff__cancel,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( minus_minus_set_real @ A3 @ A3 )
% 9.00/5.04 = bot_bot_set_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1874_Diff__cancel,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( minus_minus_set_o @ A3 @ A3 )
% 9.00/5.04 = bot_bot_set_o ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1875_Diff__cancel,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( minus_minus_set_int @ A3 @ A3 )
% 9.00/5.04 = bot_bot_set_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1876_Diff__cancel,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( minus_minus_set_nat @ A3 @ A3 )
% 9.00/5.04 = bot_bot_set_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1877_finite__Diff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1878_finite__Diff,axiom,
% 9.00/5.04 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1879_finite__Diff,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1880_finite__Diff,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1881_finite__Diff,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1882_finite__Diff2,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 = ( finite_finite_int @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1883_finite__Diff2,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.04 = ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1884_finite__Diff2,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1885_finite__Diff2,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.04 = ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1886_finite__Diff2,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( finite_finite_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1887_Compl__anti__mono,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1888_Compl__subset__Compl__iff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 9.00/5.04 = ( ord_less_eq_set_int @ B2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1889_divide__eq__0__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ A2 @ B3 )
% 9.00/5.04 = zero_zero_rat )
% 9.00/5.04 = ( ( A2 = zero_zero_rat )
% 9.00/5.04 | ( B3 = zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1890_divide__eq__0__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ A2 @ B3 )
% 9.00/5.04 = zero_zero_real )
% 9.00/5.04 = ( ( A2 = zero_zero_real )
% 9.00/5.04 | ( B3 = zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1891_divide__eq__0__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 9.00/5.04 = zero_zero_complex )
% 9.00/5.04 = ( ( A2 = zero_zero_complex )
% 9.00/5.04 | ( B3 = zero_zero_complex ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1892_divide__cancel__left,axiom,
% 9.00/5.04 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ C @ A2 )
% 9.00/5.04 = ( divide_divide_rat @ C @ B3 ) )
% 9.00/5.04 = ( ( C = zero_zero_rat )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1893_divide__cancel__left,axiom,
% 9.00/5.04 ! [C: real,A2: real,B3: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ C @ A2 )
% 9.00/5.04 = ( divide_divide_real @ C @ B3 ) )
% 9.00/5.04 = ( ( C = zero_zero_real )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1894_divide__cancel__left,axiom,
% 9.00/5.04 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( divide1717551699836669952omplex @ C @ A2 )
% 9.00/5.04 = ( divide1717551699836669952omplex @ C @ B3 ) )
% 9.00/5.04 = ( ( C = zero_zero_complex )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1895_divide__cancel__right,axiom,
% 9.00/5.04 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ A2 @ C )
% 9.00/5.04 = ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.04 = ( ( C = zero_zero_rat )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1896_divide__cancel__right,axiom,
% 9.00/5.04 ! [A2: real,C: real,B3: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ A2 @ C )
% 9.00/5.04 = ( divide_divide_real @ B3 @ C ) )
% 9.00/5.04 = ( ( C = zero_zero_real )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1897_divide__cancel__right,axiom,
% 9.00/5.04 ! [A2: complex,C: complex,B3: complex] :
% 9.00/5.04 ( ( ( divide1717551699836669952omplex @ A2 @ C )
% 9.00/5.04 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.04 = ( ( C = zero_zero_complex )
% 9.00/5.04 | ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1898_division__ring__divide__zero,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( divide_divide_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 = zero_zero_rat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1899_division__ring__divide__zero,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( divide_divide_real @ A2 @ zero_zero_real )
% 9.00/5.04 = zero_zero_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1900_division__ring__divide__zero,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( divide1717551699836669952omplex @ A2 @ zero_zero_complex )
% 9.00/5.04 = zero_zero_complex ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1901_Diff__eq__empty__iff,axiom,
% 9.00/5.04 ! [A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( ( minus_minus_set_real @ A3 @ B2 )
% 9.00/5.04 = bot_bot_set_real )
% 9.00/5.04 = ( ord_less_eq_set_real @ A3 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1902_Diff__eq__empty__iff,axiom,
% 9.00/5.04 ! [A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( ( minus_minus_set_o @ A3 @ B2 )
% 9.00/5.04 = bot_bot_set_o )
% 9.00/5.04 = ( ord_less_eq_set_o @ A3 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1903_Diff__eq__empty__iff,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( ( minus_minus_set_nat @ A3 @ B2 )
% 9.00/5.04 = bot_bot_set_nat )
% 9.00/5.04 = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1904_Diff__eq__empty__iff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( ( minus_minus_set_int @ A3 @ B2 )
% 9.00/5.04 = bot_bot_set_int )
% 9.00/5.04 = ( ord_less_eq_set_int @ A3 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1905_divide__eq__1__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ A2 @ B3 )
% 9.00/5.04 = one_one_rat )
% 9.00/5.04 = ( ( B3 != zero_zero_rat )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1906_divide__eq__1__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ A2 @ B3 )
% 9.00/5.04 = one_one_real )
% 9.00/5.04 = ( ( B3 != zero_zero_real )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1907_divide__eq__1__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 9.00/5.04 = one_one_complex )
% 9.00/5.04 = ( ( B3 != zero_zero_complex )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1908_one__eq__divide__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( one_one_rat
% 9.00/5.04 = ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( ( B3 != zero_zero_rat )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1909_one__eq__divide__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( one_one_real
% 9.00/5.04 = ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.04 = ( ( B3 != zero_zero_real )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1910_one__eq__divide__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( one_one_complex
% 9.00/5.04 = ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 9.00/5.04 = ( ( B3 != zero_zero_complex )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1911_divide__self,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( A2 != zero_zero_rat )
% 9.00/5.04 => ( ( divide_divide_rat @ A2 @ A2 )
% 9.00/5.04 = one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1912_divide__self,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( A2 != zero_zero_real )
% 9.00/5.04 => ( ( divide_divide_real @ A2 @ A2 )
% 9.00/5.04 = one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1913_divide__self,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( A2 != zero_zero_complex )
% 9.00/5.04 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 9.00/5.04 = one_one_complex ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1914_divide__self__if,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ( A2 = zero_zero_rat )
% 9.00/5.04 => ( ( divide_divide_rat @ A2 @ A2 )
% 9.00/5.04 = zero_zero_rat ) )
% 9.00/5.04 & ( ( A2 != zero_zero_rat )
% 9.00/5.04 => ( ( divide_divide_rat @ A2 @ A2 )
% 9.00/5.04 = one_one_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1915_divide__self__if,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ( A2 = zero_zero_real )
% 9.00/5.04 => ( ( divide_divide_real @ A2 @ A2 )
% 9.00/5.04 = zero_zero_real ) )
% 9.00/5.04 & ( ( A2 != zero_zero_real )
% 9.00/5.04 => ( ( divide_divide_real @ A2 @ A2 )
% 9.00/5.04 = one_one_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1916_divide__self__if,axiom,
% 9.00/5.04 ! [A2: complex] :
% 9.00/5.04 ( ( ( A2 = zero_zero_complex )
% 9.00/5.04 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 9.00/5.04 = zero_zero_complex ) )
% 9.00/5.04 & ( ( A2 != zero_zero_complex )
% 9.00/5.04 => ( ( divide1717551699836669952omplex @ A2 @ A2 )
% 9.00/5.04 = one_one_complex ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1917_divide__eq__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ B3 @ A2 )
% 9.00/5.04 = one_one_rat )
% 9.00/5.04 = ( ( A2 != zero_zero_rat )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1918_divide__eq__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ B3 @ A2 )
% 9.00/5.04 = one_one_real )
% 9.00/5.04 = ( ( A2 != zero_zero_real )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1919_eq__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( one_one_rat
% 9.00/5.04 = ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ( A2 != zero_zero_rat )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1920_eq__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( one_one_real
% 9.00/5.04 = ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ( A2 != zero_zero_real )
% 9.00/5.04 & ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1921_one__divide__eq__0__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ one_one_rat @ A2 )
% 9.00/5.04 = zero_zero_rat )
% 9.00/5.04 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1922_one__divide__eq__0__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ one_one_real @ A2 )
% 9.00/5.04 = zero_zero_real )
% 9.00/5.04 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1923_zero__eq__1__divide__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( zero_zero_rat
% 9.00/5.04 = ( divide_divide_rat @ one_one_rat @ A2 ) )
% 9.00/5.04 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1924_zero__eq__1__divide__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( zero_zero_real
% 9.00/5.04 = ( divide_divide_real @ one_one_real @ A2 ) )
% 9.00/5.04 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1925_zle__diff1__eq,axiom,
% 9.00/5.04 ! [W2: int,Z: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z @ one_one_int ) )
% 9.00/5.04 = ( ord_less_int @ W2 @ Z ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1926_zero__le__divide__1__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1927_zero__le__divide__1__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) )
% 9.00/5.04 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1928_divide__le__0__1__iff,axiom,
% 9.00/5.04 ! [A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A2 ) @ zero_zero_real )
% 9.00/5.04 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1929_divide__le__0__1__iff,axiom,
% 9.00/5.04 ! [A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.04 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1930_Diff__infinite__finite,axiom,
% 9.00/5.04 ! [T2: set_int,S: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ T2 )
% 9.00/5.04 => ( ~ ( finite_finite_int @ S )
% 9.00/5.04 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ T2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1931_Diff__infinite__finite,axiom,
% 9.00/5.04 ! [T2: set_complex,S: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.04 => ( ~ ( finite3207457112153483333omplex @ S )
% 9.00/5.04 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ T2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1932_Diff__infinite__finite,axiom,
% 9.00/5.04 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ T2 )
% 9.00/5.04 => ( ~ ( finite6177210948735845034at_nat @ S )
% 9.00/5.04 => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ T2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1933_Diff__infinite__finite,axiom,
% 9.00/5.04 ! [T2: set_Extended_enat,S: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.04 => ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.04 => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ T2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1934_Diff__infinite__finite,axiom,
% 9.00/5.04 ! [T2: set_nat,S: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ T2 )
% 9.00/5.04 => ( ~ ( finite_finite_nat @ S )
% 9.00/5.04 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ T2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1935_Diff__mono,axiom,
% 9.00/5.04 ! [A3: set_nat,C2: set_nat,D4: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ D4 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1936_Diff__mono,axiom,
% 9.00/5.04 ! [A3: set_int,C2: set_int,D4: set_int,B2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ D4 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ C2 @ D4 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1937_Diff__subset,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1938_Diff__subset,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ A3 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1939_double__diff,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 9.00/5.04 => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C2 @ A3 ) )
% 9.00/5.04 = A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1940_double__diff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.00/5.04 => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C2 @ A3 ) )
% 9.00/5.04 = A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1941_psubset__imp__ex__mem,axiom,
% 9.00/5.04 ! [A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( ord_less_set_real @ A3 @ B2 )
% 9.00/5.04 => ? [B: real] : ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1942_psubset__imp__ex__mem,axiom,
% 9.00/5.04 ! [A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( ord_less_set_o @ A3 @ B2 )
% 9.00/5.04 => ? [B: $o] : ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1943_psubset__imp__ex__mem,axiom,
% 9.00/5.04 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( ord_less_set_set_nat @ A3 @ B2 )
% 9.00/5.04 => ? [B: set_nat] : ( member_set_nat @ B @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1944_psubset__imp__ex__mem,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.04 => ? [B: int] : ( member_int @ B @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1945_psubset__imp__ex__mem,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( ord_less_set_nat @ A3 @ B2 )
% 9.00/5.04 => ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1946_subset__Compl__self__eq,axiom,
% 9.00/5.04 ! [A3: set_real] :
% 9.00/5.04 ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
% 9.00/5.04 = ( A3 = bot_bot_set_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1947_subset__Compl__self__eq,axiom,
% 9.00/5.04 ! [A3: set_o] :
% 9.00/5.04 ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
% 9.00/5.04 = ( A3 = bot_bot_set_o ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1948_subset__Compl__self__eq,axiom,
% 9.00/5.04 ! [A3: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
% 9.00/5.04 = ( A3 = bot_bot_set_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1949_subset__Compl__self__eq,axiom,
% 9.00/5.04 ! [A3: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
% 9.00/5.04 = ( A3 = bot_bot_set_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1950_int__le__induct,axiom,
% 9.00/5.04 ! [I: int,K: int,P: int > $o] :
% 9.00/5.04 ( ( ord_less_eq_int @ I @ K )
% 9.00/5.04 => ( ( P @ K )
% 9.00/5.04 => ( ! [I2: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ I2 @ K )
% 9.00/5.04 => ( ( P @ I2 )
% 9.00/5.04 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.04 => ( P @ I ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1951_int__less__induct,axiom,
% 9.00/5.04 ! [I: int,K: int,P: int > $o] :
% 9.00/5.04 ( ( ord_less_int @ I @ K )
% 9.00/5.04 => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 9.00/5.04 => ( ! [I2: int] :
% 9.00/5.04 ( ( ord_less_int @ I2 @ K )
% 9.00/5.04 => ( ( P @ I2 )
% 9.00/5.04 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.04 => ( P @ I ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1952_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1953_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1954_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1955_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1956_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1957_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1958_card__less__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1959_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1960_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1961_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1962_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1963_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1964_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1965_card__le__sym__Diff,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 9.00/5.04 => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B2 @ A3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1966_linordered__field__no__ub,axiom,
% 9.00/5.04 ! [X4: real] :
% 9.00/5.04 ? [X_1: real] : ( ord_less_real @ X4 @ X_1 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1967_linordered__field__no__ub,axiom,
% 9.00/5.04 ! [X4: rat] :
% 9.00/5.04 ? [X_1: rat] : ( ord_less_rat @ X4 @ X_1 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1968_linordered__field__no__lb,axiom,
% 9.00/5.04 ! [X4: real] :
% 9.00/5.04 ? [Y4: real] : ( ord_less_real @ Y4 @ X4 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1969_linordered__field__no__lb,axiom,
% 9.00/5.04 ! [X4: rat] :
% 9.00/5.04 ? [Y4: rat] : ( ord_less_rat @ Y4 @ X4 ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1970_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6045566169113846134st_nat @ B2 @ A3 )
% 9.00/5.04 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1971_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ B2 @ A3 )
% 9.00/5.04 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1972_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.04 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1973_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 9.00/5.04 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1974_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.04 => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1975_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.04 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1976_card__Diff__subset,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.04 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1977_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_list_nat,A3: set_list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1978_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1979_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1980_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1981_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1982_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_Extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1983_diff__card__le__card__Diff,axiom,
% 9.00/5.04 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ B2 )
% 9.00/5.04 => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1984_int__ops_I6_J,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat] :
% 9.00/5.04 ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
% 9.00/5.04 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.04 = zero_zero_int ) )
% 9.00/5.04 & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) )
% 9.00/5.04 => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.04 = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1985_VEBT_Osize_I4_J,axiom,
% 9.00/5.04 ! [X21: $o,X222: $o] :
% 9.00/5.04 ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1986_zdiff__int__split,axiom,
% 9.00/5.04 ! [P: int > $o,X2: nat,Y3: nat] :
% 9.00/5.04 ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y3 ) ) )
% 9.00/5.04 = ( ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.04 => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 9.00/5.04 & ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.04 => ( P @ zero_zero_int ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1987_divide__right__mono__neg,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1988_divide__right__mono__neg,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( divide_divide_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1989_divide__nonpos__nonpos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1990_divide__nonpos__nonpos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1991_divide__nonpos__nonneg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1992_divide__nonpos__nonneg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1993_divide__nonneg__nonpos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1994_divide__nonneg__nonpos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1995_divide__nonneg__nonneg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1996_divide__nonneg__nonneg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1997_zero__le__divide__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.04 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 9.00/5.04 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1998_zero__le__divide__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 9.00/5.04 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_1999_divide__right__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2000_divide__right__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2001_divide__le__0__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
% 9.00/5.04 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 9.00/5.04 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2002_divide__le__0__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.04 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 9.00/5.04 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2003_divide__neg__neg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2004_divide__neg__neg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2005_divide__neg__pos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2006_divide__neg__pos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2007_divide__pos__neg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2008_divide__pos__neg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2009_divide__pos__pos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2010_divide__pos__pos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2011_divide__less__0__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_rat @ B3 @ zero_zero_rat ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2012_divide__less__0__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ ( divide_divide_real @ A2 @ B3 ) @ zero_zero_real )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_real @ B3 @ zero_zero_real ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2013_divide__less__cancel,axiom,
% 9.00/5.04 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.04 => ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.04 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_rat @ B3 @ A2 ) )
% 9.00/5.04 & ( C != zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2014_divide__less__cancel,axiom,
% 9.00/5.04 ! [A2: real,C: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.04 => ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.04 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.04 => ( ord_less_real @ B3 @ A2 ) )
% 9.00/5.04 & ( C != zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2015_zero__less__divide__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_rat @ zero_zero_rat @ B3 ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2016_zero__less__divide__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_real @ zero_zero_real @ B3 ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2017_divide__strict__right__mono,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2018_divide__strict__right__mono,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real] :
% 9.00/5.04 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2019_divide__strict__right__mono__neg,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.04 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.04 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2020_divide__strict__right__mono__neg,axiom,
% 9.00/5.04 ! [B3: real,A2: real,C: real] :
% 9.00/5.04 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.04 => ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2021_right__inverse__eq,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( B3 != zero_zero_rat )
% 9.00/5.04 => ( ( ( divide_divide_rat @ A2 @ B3 )
% 9.00/5.04 = one_one_rat )
% 9.00/5.04 = ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2022_right__inverse__eq,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( B3 != zero_zero_real )
% 9.00/5.04 => ( ( ( divide_divide_real @ A2 @ B3 )
% 9.00/5.04 = one_one_real )
% 9.00/5.04 = ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2023_right__inverse__eq,axiom,
% 9.00/5.04 ! [B3: complex,A2: complex] :
% 9.00/5.04 ( ( B3 != zero_zero_complex )
% 9.00/5.04 => ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 9.00/5.04 = one_one_complex )
% 9.00/5.04 = ( A2 = B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2024_nonzero__minus__divide__divide,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( B3 != zero_zero_real )
% 9.00/5.04 => ( ( divide_divide_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) )
% 9.00/5.04 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2025_nonzero__minus__divide__divide,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( B3 != zero_zero_rat )
% 9.00/5.04 => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) )
% 9.00/5.04 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2026_nonzero__minus__divide__divide,axiom,
% 9.00/5.04 ! [B3: complex,A2: complex] :
% 9.00/5.04 ( ( B3 != zero_zero_complex )
% 9.00/5.04 => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) )
% 9.00/5.04 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2027_nonzero__minus__divide__right,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( B3 != zero_zero_real )
% 9.00/5.04 => ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.04 = ( divide_divide_real @ A2 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2028_nonzero__minus__divide__right,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( B3 != zero_zero_rat )
% 9.00/5.04 => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.04 = ( divide_divide_rat @ A2 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2029_nonzero__minus__divide__right,axiom,
% 9.00/5.04 ! [B3: complex,A2: complex] :
% 9.00/5.04 ( ( B3 != zero_zero_complex )
% 9.00/5.04 => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 9.00/5.04 = ( divide1717551699836669952omplex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2030_frac__le,axiom,
% 9.00/5.04 ! [Y3: real,X2: real,W2: real,Z: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ W2 @ Z )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2031_frac__le,axiom,
% 9.00/5.04 ! [Y3: rat,X2: rat,W2: rat,Z: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ W2 @ Z )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2032_frac__less,axiom,
% 9.00/5.04 ! [X2: real,Y3: real,W2: real,Z: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ W2 @ Z )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2033_frac__less,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ W2 @ Z )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2034_frac__less2,axiom,
% 9.00/5.04 ! [X2: real,Y3: real,W2: real,Z: real] :
% 9.00/5.04 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.00/5.04 => ( ( ord_less_real @ W2 @ Z )
% 9.00/5.04 => ( ord_less_real @ ( divide_divide_real @ X2 @ Z ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2035_frac__less2,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat,W2: rat,Z: rat] :
% 9.00/5.04 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.00/5.04 => ( ( ord_less_rat @ W2 @ Z )
% 9.00/5.04 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Z ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2036_divide__le__cancel,axiom,
% 9.00/5.04 ! [A2: real,C: real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ C ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.04 => ( ord_less_eq_real @ A2 @ B3 ) )
% 9.00/5.04 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2037_divide__le__cancel,axiom,
% 9.00/5.04 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ C ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.04 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.04 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2038_divide__nonneg__neg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2039_divide__nonneg__neg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2040_divide__nonneg__pos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2041_divide__nonneg__pos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2042_divide__nonpos__neg,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.00/5.04 => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2043_divide__nonpos__neg,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.00/5.04 => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2044_divide__nonpos__pos,axiom,
% 9.00/5.04 ! [X2: real,Y3: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.04 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.04 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2045_divide__nonpos__pos,axiom,
% 9.00/5.04 ! [X2: rat,Y3: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 9.00/5.04 => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.04 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2046_divide__less__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_rat @ B3 @ A2 ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.04 | ( A2 = zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2047_divide__less__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_real @ B3 @ A2 ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.04 | ( A2 = zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2048_less__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2049_less__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2050_divide__eq__minus__1__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( divide_divide_real @ A2 @ B3 )
% 9.00/5.04 = ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.04 = ( ( B3 != zero_zero_real )
% 9.00/5.04 & ( A2
% 9.00/5.04 = ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2051_divide__eq__minus__1__iff,axiom,
% 9.00/5.04 ! [A2: rat,B3: rat] :
% 9.00/5.04 ( ( ( divide_divide_rat @ A2 @ B3 )
% 9.00/5.04 = ( uminus_uminus_rat @ one_one_rat ) )
% 9.00/5.04 = ( ( B3 != zero_zero_rat )
% 9.00/5.04 & ( A2
% 9.00/5.04 = ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2052_divide__eq__minus__1__iff,axiom,
% 9.00/5.04 ! [A2: complex,B3: complex] :
% 9.00/5.04 ( ( ( divide1717551699836669952omplex @ A2 @ B3 )
% 9.00/5.04 = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.00/5.04 = ( ( B3 != zero_zero_complex )
% 9.00/5.04 & ( A2
% 9.00/5.04 = ( uminus1482373934393186551omplex @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2053_divide__le__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ A2 ) @ one_one_real )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_eq_real @ B3 @ A2 ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_eq_real @ A2 @ B3 ) )
% 9.00/5.04 | ( A2 = zero_zero_real ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2054_divide__le__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ A2 ) @ one_one_rat )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_eq_rat @ B3 @ A2 ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.04 | ( A2 = zero_zero_rat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2055_le__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B3 @ A2 ) )
% 9.00/5.04 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.04 & ( ord_less_eq_real @ A2 @ B3 ) )
% 9.00/5.04 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.04 & ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2056_le__divide__eq__1,axiom,
% 9.00/5.04 ! [B3: rat,A2: rat] :
% 9.00/5.04 ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B3 @ A2 ) )
% 9.00/5.04 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.04 & ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.04 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.04 & ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2057_div__pos__pos__trivial,axiom,
% 9.00/5.04 ! [K: int,L: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.04 => ( ( ord_less_int @ K @ L )
% 9.00/5.04 => ( ( divide_divide_int @ K @ L )
% 9.00/5.04 = zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2058_div__neg__neg__trivial,axiom,
% 9.00/5.04 ! [K: int,L: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_int @ L @ K )
% 9.00/5.04 => ( ( divide_divide_int @ K @ L )
% 9.00/5.04 = zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2059_div__eq__minus1,axiom,
% 9.00/5.04 ! [B3: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
% 9.00/5.04 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2060_geqmaxNone,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.04 => ( ( ord_less_eq_nat @ Ma @ X2 )
% 9.00/5.04 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.04 = none_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2061_le__div__geq,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 => ( ( divide_divide_nat @ M @ N )
% 9.00/5.04 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2062_div__less,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ( divide_divide_nat @ M @ N )
% 9.00/5.04 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2063_div__by__Suc__0,axiom,
% 9.00/5.04 ! [M: nat] :
% 9.00/5.04 ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.00/5.04 = M ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2064_compl__le__compl__iff,axiom,
% 9.00/5.04 ! [X2: set_int,Y3: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 9.00/5.04 = ( ord_less_eq_set_int @ Y3 @ X2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2065_bits__div__0,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( divide_divide_int @ zero_zero_int @ A2 )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2066_bits__div__0,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( divide_divide_nat @ zero_zero_nat @ A2 )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2067_DiffI,axiom,
% 9.00/5.04 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ C @ A3 )
% 9.00/5.04 => ( ~ ( member_real @ C @ B2 )
% 9.00/5.04 => ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2068_DiffI,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ C @ A3 )
% 9.00/5.04 => ( ~ ( member_o @ C @ B2 )
% 9.00/5.04 => ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2069_DiffI,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ A3 )
% 9.00/5.04 => ( ~ ( member_set_nat @ C @ B2 )
% 9.00/5.04 => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2070_DiffI,axiom,
% 9.00/5.04 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ C @ A3 )
% 9.00/5.04 => ( ~ ( member_int @ C @ B2 )
% 9.00/5.04 => ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2071_DiffI,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ A3 )
% 9.00/5.04 => ( ~ ( member_nat @ C @ B2 )
% 9.00/5.04 => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2072_Diff__iff,axiom,
% 9.00/5.04 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 9.00/5.04 = ( ( member_real @ C @ A3 )
% 9.00/5.04 & ~ ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2073_Diff__iff,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 9.00/5.04 = ( ( member_o @ C @ A3 )
% 9.00/5.04 & ~ ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2074_Diff__iff,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( ( member_set_nat @ C @ A3 )
% 9.00/5.04 & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2075_Diff__iff,axiom,
% 9.00/5.04 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 = ( ( member_int @ C @ A3 )
% 9.00/5.04 & ~ ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2076_Diff__iff,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 = ( ( member_nat @ C @ A3 )
% 9.00/5.04 & ~ ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2077_Diff__idemp,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ B2 )
% 9.00/5.04 = ( minus_minus_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2078_ComplI,axiom,
% 9.00/5.04 ! [C: real,A3: set_real] :
% 9.00/5.04 ( ~ ( member_real @ C @ A3 )
% 9.00/5.04 => ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2079_ComplI,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o] :
% 9.00/5.04 ( ~ ( member_o @ C @ A3 )
% 9.00/5.04 => ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2080_ComplI,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ~ ( member_set_nat @ C @ A3 )
% 9.00/5.04 => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2081_ComplI,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat] :
% 9.00/5.04 ( ~ ( member_nat @ C @ A3 )
% 9.00/5.04 => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2082_ComplI,axiom,
% 9.00/5.04 ! [C: int,A3: set_int] :
% 9.00/5.04 ( ~ ( member_int @ C @ A3 )
% 9.00/5.04 => ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2083_Compl__iff,axiom,
% 9.00/5.04 ! [C: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
% 9.00/5.04 = ( ~ ( member_real @ C @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2084_Compl__iff,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
% 9.00/5.04 = ( ~ ( member_o @ C @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2085_Compl__iff,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
% 9.00/5.04 = ( ~ ( member_set_nat @ C @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2086_Compl__iff,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
% 9.00/5.04 = ( ~ ( member_nat @ C @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2087_Compl__iff,axiom,
% 9.00/5.04 ! [C: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
% 9.00/5.04 = ( ~ ( member_int @ C @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2088_bits__div__by__0,axiom,
% 9.00/5.04 ! [A2: int] :
% 9.00/5.04 ( ( divide_divide_int @ A2 @ zero_zero_int )
% 9.00/5.04 = zero_zero_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2089_bits__div__by__0,axiom,
% 9.00/5.04 ! [A2: nat] :
% 9.00/5.04 ( ( divide_divide_nat @ A2 @ zero_zero_nat )
% 9.00/5.04 = zero_zero_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2090_real__of__nat__div3,axiom,
% 9.00/5.04 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) @ one_one_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2091_DiffE,axiom,
% 9.00/5.04 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( ( member_real @ C @ A3 )
% 9.00/5.04 => ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2092_DiffE,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( ( member_o @ C @ A3 )
% 9.00/5.04 => ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2093_DiffE,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( ( member_set_nat @ C @ A3 )
% 9.00/5.04 => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2094_DiffE,axiom,
% 9.00/5.04 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( ( member_int @ C @ A3 )
% 9.00/5.04 => ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2095_DiffE,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( ( member_nat @ C @ A3 )
% 9.00/5.04 => ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2096_ComplD,axiom,
% 9.00/5.04 ! [C: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( uminus612125837232591019t_real @ A3 ) )
% 9.00/5.04 => ~ ( member_real @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2097_ComplD,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( uminus_uminus_set_o @ A3 ) )
% 9.00/5.04 => ~ ( member_o @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2098_ComplD,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A3 ) )
% 9.00/5.04 => ~ ( member_set_nat @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2099_ComplD,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A3 ) )
% 9.00/5.04 => ~ ( member_nat @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2100_ComplD,axiom,
% 9.00/5.04 ! [C: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A3 ) )
% 9.00/5.04 => ~ ( member_int @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2101_DiffD1,axiom,
% 9.00/5.04 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 9.00/5.04 => ( member_real @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2102_DiffD1,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 9.00/5.04 => ( member_o @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2103_DiffD1,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.04 => ( member_set_nat @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2104_DiffD1,axiom,
% 9.00/5.04 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 => ( member_int @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2105_DiffD1,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 => ( member_nat @ C @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2106_DiffD2,axiom,
% 9.00/5.04 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ C @ ( minus_minus_set_real @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( member_real @ C @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2107_DiffD2,axiom,
% 9.00/5.04 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ C @ ( minus_minus_set_o @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( member_o @ C @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2108_DiffD2,axiom,
% 9.00/5.04 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( member_set_nat @ C @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2109_DiffD2,axiom,
% 9.00/5.04 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ C @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( member_int @ C @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2110_DiffD2,axiom,
% 9.00/5.04 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.04 => ~ ( member_nat @ C @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2111_real__of__nat__div4,axiom,
% 9.00/5.04 ! [N: nat,X2: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2112_real__of__nat__div2,axiom,
% 9.00/5.04 ! [N: nat,X2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ X2 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ X2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2113_subrelI,axiom,
% 9.00/5.04 ! [R2: set_Pr6588086440996610945on_nat,S2: set_Pr6588086440996610945on_nat] :
% 9.00/5.04 ( ! [X5: option_nat,Y4: option_nat] :
% 9.00/5.04 ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ R2 )
% 9.00/5.04 => ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X5 @ Y4 ) @ S2 ) )
% 9.00/5.04 => ( ord_le6406482658798684961on_nat @ R2 @ S2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2114_subrelI,axiom,
% 9.00/5.04 ! [R2: set_Pr7459493094073627847at_nat,S2: set_Pr7459493094073627847at_nat] :
% 9.00/5.04 ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] :
% 9.00/5.04 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ R2 )
% 9.00/5.04 => ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X5 @ Y4 ) @ S2 ) )
% 9.00/5.04 => ( ord_le5997549366648089703at_nat @ R2 @ S2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2115_subrelI,axiom,
% 9.00/5.04 ! [R2: set_Pr4329608150637261639at_nat,S2: set_Pr4329608150637261639at_nat] :
% 9.00/5.04 ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ R2 )
% 9.00/5.04 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ S2 ) )
% 9.00/5.04 => ( ord_le1268244103169919719at_nat @ R2 @ S2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2116_subrelI,axiom,
% 9.00/5.04 ! [R2: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ! [X5: nat,Y4: nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
% 9.00/5.04 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S2 ) )
% 9.00/5.04 => ( ord_le3146513528884898305at_nat @ R2 @ S2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2117_subrelI,axiom,
% 9.00/5.04 ! [R2: set_Pr958786334691620121nt_int,S2: set_Pr958786334691620121nt_int] :
% 9.00/5.04 ( ! [X5: int,Y4: int] :
% 9.00/5.04 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
% 9.00/5.04 => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S2 ) )
% 9.00/5.04 => ( ord_le2843351958646193337nt_int @ R2 @ S2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2118_div__le__dividend,axiom,
% 9.00/5.04 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ M ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2119_div__le__mono,axiom,
% 9.00/5.04 ! [M: nat,N: nat,K: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N @ K ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2120_vebt__delete_Osimps_I5_J,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X2 )
% 9.00/5.04 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2121_vebt__mint_Ocases,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT] :
% 9.00/5.04 ( ! [A: $o,B: $o] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Leaf @ A @ B ) )
% 9.00/5.04 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.04 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2122_vebt__mint_Osimps_I3_J,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.00/5.04 = ( some_nat @ Mi ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2123_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X2 )
% 9.00/5.04 = ( ( X2 = Mi )
% 9.00/5.04 | ( X2 = Ma ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2124_vebt__maxt_Osimps_I3_J,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.00/5.04 = ( some_nat @ Ma ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2125_vebt__delete_Osimps_I6_J,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X2: nat] :
% 9.00/5.04 ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X2 )
% 9.00/5.04 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2126_compl__le__swap2,axiom,
% 9.00/5.04 ! [Y3: set_int,X2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X2 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2127_compl__le__swap1,axiom,
% 9.00/5.04 ! [Y3: set_int,X2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) )
% 9.00/5.04 => ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2128_compl__mono,axiom,
% 9.00/5.04 ! [X2: set_int,Y3: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2129_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ( divide_divide_nat @ M @ N )
% 9.00/5.04 = zero_zero_nat )
% 9.00/5.04 = ( ( ord_less_nat @ M @ N )
% 9.00/5.04 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2130_Suc__div__le__mono,axiom,
% 9.00/5.04 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N ) @ ( divide_divide_nat @ ( suc @ M ) @ N ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2131_vebt__mint_Oelims,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 9.00/5.04 ( ( ( vEBT_vebt_mint @ X2 )
% 9.00/5.04 = Y3 )
% 9.00/5.04 => ( ! [A: $o,B: $o] :
% 9.00/5.04 ( ( X2
% 9.00/5.04 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.04 => ~ ( ( A
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.04 & ( ~ A
% 9.00/5.04 => ( ( B
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.04 & ( ~ B
% 9.00/5.04 => ( Y3 = none_nat ) ) ) ) ) )
% 9.00/5.04 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.04 => ( Y3 != none_nat ) )
% 9.00/5.04 => ~ ! [Mi2: nat] :
% 9.00/5.04 ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.04 => ( Y3
% 9.00/5.04 != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2132_vebt__maxt_Oelims,axiom,
% 9.00/5.04 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 9.00/5.04 ( ( ( vEBT_vebt_maxt @ X2 )
% 9.00/5.04 = Y3 )
% 9.00/5.04 => ( ! [A: $o,B: $o] :
% 9.00/5.04 ( ( X2
% 9.00/5.04 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.04 => ~ ( ( B
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.04 & ( ~ B
% 9.00/5.04 => ( ( A
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.04 & ( ~ A
% 9.00/5.04 => ( Y3 = none_nat ) ) ) ) ) )
% 9.00/5.04 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.04 => ( Y3 != none_nat ) )
% 9.00/5.04 => ~ ! [Mi2: nat,Ma2: nat] :
% 9.00/5.04 ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.04 ( X2
% 9.00/5.04 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.04 => ( Y3
% 9.00/5.04 != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2133_diff__shunt__var,axiom,
% 9.00/5.04 ! [X2: set_real,Y3: set_real] :
% 9.00/5.04 ( ( ( minus_minus_set_real @ X2 @ Y3 )
% 9.00/5.04 = bot_bot_set_real )
% 9.00/5.04 = ( ord_less_eq_set_real @ X2 @ Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2134_diff__shunt__var,axiom,
% 9.00/5.04 ! [X2: set_o,Y3: set_o] :
% 9.00/5.04 ( ( ( minus_minus_set_o @ X2 @ Y3 )
% 9.00/5.04 = bot_bot_set_o )
% 9.00/5.04 = ( ord_less_eq_set_o @ X2 @ Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2135_diff__shunt__var,axiom,
% 9.00/5.04 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.04 ( ( ( minus_minus_set_nat @ X2 @ Y3 )
% 9.00/5.04 = bot_bot_set_nat )
% 9.00/5.04 = ( ord_less_eq_set_nat @ X2 @ Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2136_diff__shunt__var,axiom,
% 9.00/5.04 ! [X2: set_int,Y3: set_int] :
% 9.00/5.04 ( ( ( minus_minus_set_int @ X2 @ Y3 )
% 9.00/5.04 = bot_bot_set_int )
% 9.00/5.04 = ( ord_less_eq_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2137_div__greater__zero__iff,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N ) )
% 9.00/5.04 = ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.04 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2138_div__le__mono2,axiom,
% 9.00/5.04 ! [M: nat,N: nat,K: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.04 => ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.04 => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2139_div__eq__dividend__iff,axiom,
% 9.00/5.04 ! [M: nat,N: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.04 => ( ( ( divide_divide_nat @ M @ N )
% 9.00/5.04 = M )
% 9.00/5.04 = ( N = one_one_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2140_div__less__dividend,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ one_one_nat @ N )
% 9.00/5.04 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.04 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ M ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2141_pos__imp__zdiv__neg__iff,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
% 9.00/5.04 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2142_neg__imp__zdiv__neg__iff,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int )
% 9.00/5.04 = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2143_div__neg__pos__less0,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ord_less_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2144_div__positive,axiom,
% 9.00/5.04 ! [B3: nat,A2: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2145_div__positive,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.04 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2146_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat] :
% 9.00/5.04 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.04 => ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.04 => ( ( divide_divide_nat @ A2 @ B3 )
% 9.00/5.04 = zero_zero_nat ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2147_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.04 => ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.04 => ( ( divide_divide_int @ A2 @ B3 )
% 9.00/5.04 = zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2148_div__geq,axiom,
% 9.00/5.04 ! [N: nat,M: nat] :
% 9.00/5.04 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.04 => ( ~ ( ord_less_nat @ M @ N )
% 9.00/5.04 => ( ( divide_divide_nat @ M @ N )
% 9.00/5.04 = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2149_div__if,axiom,
% 9.00/5.04 ( divide_divide_nat
% 9.00/5.04 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.04 ( if_nat
% 9.00/5.04 @ ( ( ord_less_nat @ M2 @ N2 )
% 9.00/5.04 | ( N2 = zero_zero_nat ) )
% 9.00/5.04 @ zero_zero_nat
% 9.00/5.04 @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2150_zdiv__mono1,axiom,
% 9.00/5.04 ! [A2: int,A7: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ A7 )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A7 @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2151_zdiv__mono2,axiom,
% 9.00/5.04 ! [A2: int,B7: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.00/5.04 => ( ( ord_less_eq_int @ B7 @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ ( divide_divide_int @ A2 @ B7 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2152_zdiv__eq__0__iff,axiom,
% 9.00/5.04 ! [I: int,K: int] :
% 9.00/5.04 ( ( ( divide_divide_int @ I @ K )
% 9.00/5.04 = zero_zero_int )
% 9.00/5.04 = ( ( K = zero_zero_int )
% 9.00/5.04 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.00/5.04 & ( ord_less_int @ I @ K ) )
% 9.00/5.04 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.00/5.04 & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2153_zdiv__mono1__neg,axiom,
% 9.00/5.04 ! [A2: int,A7: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ A7 )
% 9.00/5.04 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A7 @ B3 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2154_zdiv__mono2__neg,axiom,
% 9.00/5.04 ! [A2: int,B7: int,B3: int] :
% 9.00/5.04 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.00/5.04 => ( ( ord_less_eq_int @ B7 @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B7 ) @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2155_div__int__pos__iff,axiom,
% 9.00/5.04 ! [K: int,L: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 9.00/5.04 = ( ( K = zero_zero_int )
% 9.00/5.04 | ( L = zero_zero_int )
% 9.00/5.04 | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.04 & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 9.00/5.04 | ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.04 & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2156_div__positive__int,axiom,
% 9.00/5.04 ! [L: int,K: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ L @ K )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.04 => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2157_div__nonneg__neg__le0,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.04 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2158_div__nonpos__pos__le0,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2159_pos__imp__zdiv__pos__iff,axiom,
% 9.00/5.04 ! [K: int,I: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 9.00/5.04 = ( ord_less_eq_int @ K @ I ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2160_neg__imp__zdiv__nonneg__iff,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.04 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2161_pos__imp__zdiv__nonneg__iff,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.04 => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 9.00/5.04 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2162_nonneg1__imp__zdiv__pos__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.04 => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A2 @ B3 ) )
% 9.00/5.04 = ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.04 & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2163_int__div__less__self,axiom,
% 9.00/5.04 ! [X2: int,K: int] :
% 9.00/5.04 ( ( ord_less_int @ zero_zero_int @ X2 )
% 9.00/5.04 => ( ( ord_less_int @ one_one_int @ K )
% 9.00/5.04 => ( ord_less_int @ ( divide_divide_int @ X2 @ K ) @ X2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2164_delete__correct,axiom,
% 9.00/5.04 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.04 => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
% 9.00/5.04 = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2165_mi__eq__ma__no__ch,axiom,
% 9.00/5.04 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 9.00/5.04 => ( ( Mi = Ma )
% 9.00/5.04 => ( ! [X4: vEBT_VEBT] :
% 9.00/5.04 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.04 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
% 9.00/5.04 & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2166_divides__aux__eq,axiom,
% 9.00/5.04 ! [Q3: nat,R2: nat] :
% 9.00/5.04 ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.00/5.04 = ( R2 = zero_zero_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2167_divides__aux__eq,axiom,
% 9.00/5.04 ! [Q3: int,R2: int] :
% 9.00/5.04 ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.04 = ( R2 = zero_zero_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2168_delete__correct_H,axiom,
% 9.00/5.04 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.04 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.04 => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X2 ) )
% 9.00/5.04 = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2169_dbl__dec__simps_I2_J,axiom,
% 9.00/5.04 ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 9.00/5.04 = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2170_dbl__dec__simps_I2_J,axiom,
% 9.00/5.04 ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 9.00/5.04 = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2171_dbl__dec__simps_I2_J,axiom,
% 9.00/5.04 ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 9.00/5.04 = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2172_dbl__dec__simps_I2_J,axiom,
% 9.00/5.04 ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 9.00/5.04 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2173_dbl__dec__simps_I2_J,axiom,
% 9.00/5.04 ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 9.00/5.04 = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2174_prod__decode__aux_Osimps,axiom,
% 9.00/5.04 ( nat_prod_decode_aux
% 9.00/5.04 = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2175_prod__decode__aux_Oelims,axiom,
% 9.00/5.04 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 9.00/5.04 = Y3 )
% 9.00/5.04 => ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 9.00/5.04 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 9.00/5.04 => ( Y3
% 9.00/5.04 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2176_insert__absorb2,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2177_insert__absorb2,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real] :
% 9.00/5.04 ( ( insert_real @ X2 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.04 = ( insert_real @ X2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2178_insert__absorb2,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o] :
% 9.00/5.04 ( ( insert_o @ X2 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.04 = ( insert_o @ X2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2179_insert__absorb2,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat] :
% 9.00/5.04 ( ( insert_nat @ X2 @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( insert_nat @ X2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2180_insert__absorb2,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int] :
% 9.00/5.04 ( ( insert_int @ X2 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.04 = ( insert_int @ X2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2181_insert__iff,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2182_insert__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member_real @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2183_insert__iff,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member_o @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2184_insert__iff,axiom,
% 9.00/5.04 ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member_set_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2185_insert__iff,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2186_insert__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 | ( member_int @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2187_insertCI,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2188_insertCI,axiom,
% 9.00/5.04 ! [A2: real,B2: set_real,B3: real] :
% 9.00/5.04 ( ( ~ ( member_real @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2189_insertCI,axiom,
% 9.00/5.04 ! [A2: $o,B2: set_o,B3: $o] :
% 9.00/5.04 ( ( ~ ( member_o @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2190_insertCI,axiom,
% 9.00/5.04 ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
% 9.00/5.04 ( ( ~ ( member_set_nat @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2191_insertCI,axiom,
% 9.00/5.04 ! [A2: nat,B2: set_nat,B3: nat] :
% 9.00/5.04 ( ( ~ ( member_nat @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2192_insertCI,axiom,
% 9.00/5.04 ! [A2: int,B2: set_int,B3: int] :
% 9.00/5.04 ( ( ~ ( member_int @ A2 @ B2 )
% 9.00/5.04 => ( A2 = B3 ) )
% 9.00/5.04 => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2193_singletonI,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2194_singletonI,axiom,
% 9.00/5.04 ! [A2: set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2195_singletonI,axiom,
% 9.00/5.04 ! [A2: real] : ( member_real @ A2 @ ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2196_singletonI,axiom,
% 9.00/5.04 ! [A2: $o] : ( member_o @ A2 @ ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2197_singletonI,axiom,
% 9.00/5.04 ! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2198_singletonI,axiom,
% 9.00/5.04 ! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2199_finite__insert,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real] :
% 9.00/5.04 ( ( finite_finite_real @ ( insert_real @ A2 @ A3 ) )
% 9.00/5.04 = ( finite_finite_real @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2200_finite__insert,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o] :
% 9.00/5.04 ( ( finite_finite_o @ ( insert_o @ A2 @ A3 ) )
% 9.00/5.04 = ( finite_finite_o @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2201_finite__insert,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) )
% 9.00/5.04 = ( finite_finite_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2202_finite__insert,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ ( insert_int @ A2 @ A3 ) )
% 9.00/5.04 = ( finite_finite_int @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2203_finite__insert,axiom,
% 9.00/5.04 ! [A2: complex,A3: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) )
% 9.00/5.04 = ( finite3207457112153483333omplex @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2204_finite__insert,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 9.00/5.04 = ( finite6177210948735845034at_nat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2205_finite__insert,axiom,
% 9.00/5.04 ! [A2: extended_enat,A3: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) )
% 9.00/5.04 = ( finite4001608067531595151d_enat @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2206_insert__subset,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.04 & ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2207_insert__subset,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( ord_less_eq_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member_real @ X2 @ B2 )
% 9.00/5.04 & ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2208_insert__subset,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( ord_less_eq_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member_o @ X2 @ B2 )
% 9.00/5.04 & ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2209_insert__subset,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member_set_nat @ X2 @ B2 )
% 9.00/5.04 & ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2210_insert__subset,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member_nat @ X2 @ B2 )
% 9.00/5.04 & ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2211_insert__subset,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( ( member_int @ X2 @ B2 )
% 9.00/5.04 & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2212_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2213_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: real,B2: set_real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2214_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: $o,B2: set_o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2215_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2216_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: int,B2: set_int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2217_insert__Diff1,axiom,
% 9.00/5.04 ! [X2: nat,B2: set_nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.04 = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2218_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2219_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2220_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2221_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2222_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2223_Diff__insert0,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2224_singleton__insert__inj__eq,axiom,
% 9.00/5.04 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2225_singleton__insert__inj__eq,axiom,
% 9.00/5.04 ! [B3: real,A2: real,A3: set_real] :
% 9.00/5.04 ( ( ( insert_real @ B3 @ bot_bot_set_real )
% 9.00/5.04 = ( insert_real @ A2 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2226_singleton__insert__inj__eq,axiom,
% 9.00/5.04 ! [B3: $o,A2: $o,A3: set_o] :
% 9.00/5.04 ( ( ( insert_o @ B3 @ bot_bot_set_o )
% 9.00/5.04 = ( insert_o @ A2 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2227_singleton__insert__inj__eq,axiom,
% 9.00/5.04 ! [B3: nat,A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( ( insert_nat @ B3 @ bot_bot_set_nat )
% 9.00/5.04 = ( insert_nat @ A2 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2228_singleton__insert__inj__eq,axiom,
% 9.00/5.04 ! [B3: int,A2: int,A3: set_int] :
% 9.00/5.04 ( ( ( insert_int @ B3 @ bot_bot_set_int )
% 9.00/5.04 = ( insert_int @ A2 @ A3 ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2229_singleton__insert__inj__eq_H,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2230_singleton__insert__inj__eq_H,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real,B3: real] :
% 9.00/5.04 ( ( ( insert_real @ A2 @ A3 )
% 9.00/5.04 = ( insert_real @ B3 @ bot_bot_set_real ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2231_singleton__insert__inj__eq_H,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o,B3: $o] :
% 9.00/5.04 ( ( ( insert_o @ A2 @ A3 )
% 9.00/5.04 = ( insert_o @ B3 @ bot_bot_set_o ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2232_singleton__insert__inj__eq_H,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat,B3: nat] :
% 9.00/5.04 ( ( ( insert_nat @ A2 @ A3 )
% 9.00/5.04 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2233_singleton__insert__inj__eq_H,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int,B3: int] :
% 9.00/5.04 ( ( ( insert_int @ A2 @ A3 )
% 9.00/5.04 = ( insert_int @ B3 @ bot_bot_set_int ) )
% 9.00/5.04 = ( ( A2 = B3 )
% 9.00/5.04 & ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2234_insert__Diff__single,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2235_insert__Diff__single,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real] :
% 9.00/5.04 ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.04 = ( insert_real @ A2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2236_insert__Diff__single,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o] :
% 9.00/5.04 ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.04 = ( insert_o @ A2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2237_insert__Diff__single,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int] :
% 9.00/5.04 ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.04 = ( insert_int @ A2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2238_insert__Diff__single,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.04 = ( insert_nat @ A2 @ A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2239_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_real,A2: real,B2: set_real] :
% 9.00/5.04 ( ( finite_finite_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite_finite_real @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2240_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_o,A2: $o,B2: set_o] :
% 9.00/5.04 ( ( finite_finite_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite_finite_o @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2241_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_int,A2: int,B2: set_int] :
% 9.00/5.04 ( ( finite_finite_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite_finite_int @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2242_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_complex,A2: complex,B2: set_complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2243_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2244_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,A2: extended_enat,B2: set_Extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2245_finite__Diff__insert,axiom,
% 9.00/5.04 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 9.00/5.04 ( ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( finite_finite_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2246_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_real,X2: real] :
% 9.00/5.04 ( ( finite_finite_real @ A3 )
% 9.00/5.04 => ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2247_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_o,X2: $o] :
% 9.00/5.04 ( ( finite_finite_o @ A3 )
% 9.00/5.04 => ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2248_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.04 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.04 => ( ~ ( member_list_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2249_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.04 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.04 => ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2250_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_nat,X2: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2251_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_int,X2: int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2252_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_complex,X2: complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.04 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2253_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2254_card__insert__disjoint,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( ~ ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.04 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.04 = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2255_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.04 = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2256_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_set_nat,B3: set_nat] :
% 9.00/5.04 ( ( ord_le6893508408891458716et_nat @ A3 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B3 @ bot_bot_set_set_nat ) ) )
% 9.00/5.04 = ( ~ ( member_set_nat @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2257_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ ( insert_real @ B3 @ bot_bot_set_real ) ) )
% 9.00/5.04 = ( ~ ( member_real @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2258_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_o,B3: $o] :
% 9.00/5.04 ( ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ ( insert_o @ B3 @ bot_bot_set_o ) ) )
% 9.00/5.04 = ( ~ ( member_o @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2259_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_nat,B3: nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B3 @ bot_bot_set_nat ) ) )
% 9.00/5.04 = ( ~ ( member_nat @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2260_subset__Compl__singleton,axiom,
% 9.00/5.04 ! [A3: set_int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( insert_int @ B3 @ bot_bot_set_int ) ) )
% 9.00/5.04 = ( ~ ( member_int @ B3 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2261_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.04 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2262_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ( member_real @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_real @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2263_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ( member_o @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_o @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2264_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: complex,A3: set_complex,B2: set_complex] :
% 9.00/5.04 ( ( member_complex @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_complex @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2265_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: list_nat,A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.04 ( ( member_list_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_list_nat @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2266_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_set_nat @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2267_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ( member_int @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_int @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2268_card__Diff__insert,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_nat @ A2 @ B2 )
% 9.00/5.04 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) ) )
% 9.00/5.04 = ( minus_minus_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ one_one_nat ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2269_less__eq__real__def,axiom,
% 9.00/5.04 ( ord_less_eq_real
% 9.00/5.04 = ( ^ [X: real,Y: real] :
% 9.00/5.04 ( ( ord_less_real @ X @ Y )
% 9.00/5.04 | ( X = Y ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2270_complete__real,axiom,
% 9.00/5.04 ! [S: set_real] :
% 9.00/5.04 ( ? [X4: real] : ( member_real @ X4 @ S )
% 9.00/5.04 => ( ? [Z5: real] :
% 9.00/5.04 ! [X5: real] :
% 9.00/5.04 ( ( member_real @ X5 @ S )
% 9.00/5.04 => ( ord_less_eq_real @ X5 @ Z5 ) )
% 9.00/5.04 => ? [Y4: real] :
% 9.00/5.04 ( ! [X4: real] :
% 9.00/5.04 ( ( member_real @ X4 @ S )
% 9.00/5.04 => ( ord_less_eq_real @ X4 @ Y4 ) )
% 9.00/5.04 & ! [Z5: real] :
% 9.00/5.04 ( ! [X5: real] :
% 9.00/5.04 ( ( member_real @ X5 @ S )
% 9.00/5.04 => ( ord_less_eq_real @ X5 @ Z5 ) )
% 9.00/5.04 => ( ord_less_eq_real @ Y4 @ Z5 ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2271_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert8211810215607154385at_nat @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member8440522571783428010at_nat @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2272_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_real] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_real @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member_real @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2273_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_o] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_o @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member_o @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2274_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_set_nat @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member_set_nat @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2275_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_nat @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member_nat @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2276_mk__disjoint__insert,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ A2 @ A3 )
% 9.00/5.04 => ? [B8: set_int] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_int @ A2 @ B8 ) )
% 9.00/5.04 & ~ ( member_int @ A2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2277_insert__commute,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( insert8211810215607154385at_nat @ X2 @ ( insert8211810215607154385at_nat @ Y3 @ A3 ) )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2278_insert__commute,axiom,
% 9.00/5.04 ! [X2: real,Y3: real,A3: set_real] :
% 9.00/5.04 ( ( insert_real @ X2 @ ( insert_real @ Y3 @ A3 ) )
% 9.00/5.04 = ( insert_real @ Y3 @ ( insert_real @ X2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2279_insert__commute,axiom,
% 9.00/5.04 ! [X2: $o,Y3: $o,A3: set_o] :
% 9.00/5.04 ( ( insert_o @ X2 @ ( insert_o @ Y3 @ A3 ) )
% 9.00/5.04 = ( insert_o @ Y3 @ ( insert_o @ X2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2280_insert__commute,axiom,
% 9.00/5.04 ! [X2: nat,Y3: nat,A3: set_nat] :
% 9.00/5.04 ( ( insert_nat @ X2 @ ( insert_nat @ Y3 @ A3 ) )
% 9.00/5.04 = ( insert_nat @ Y3 @ ( insert_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2281_insert__commute,axiom,
% 9.00/5.04 ! [X2: int,Y3: int,A3: set_int] :
% 9.00/5.04 ( ( insert_int @ X2 @ ( insert_int @ Y3 @ A3 ) )
% 9.00/5.04 = ( insert_int @ Y3 @ ( insert_int @ X2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2282_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member8440522571783428010at_nat @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 != B3 )
% 9.00/5.04 => ? [C4: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert8211810215607154385at_nat @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member8440522571783428010at_nat @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert8211810215607154385at_nat @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member8440522571783428010at_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2283_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real,B3: real,B2: set_real] :
% 9.00/5.04 ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_real @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert_real @ A2 @ A3 )
% 9.00/5.04 = ( insert_real @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 != B3 )
% 9.00/5.04 => ? [C4: set_real] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_real @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member_real @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert_real @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member_real @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2284_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o,B3: $o,B2: set_o] :
% 9.00/5.04 ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_o @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert_o @ A2 @ A3 )
% 9.00/5.04 = ( insert_o @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 = ~ B3 )
% 9.00/5.04 => ? [C4: set_o] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_o @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member_o @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert_o @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member_o @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2285_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: set_nat,A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ~ ( member_set_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_set_nat @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert_set_nat @ A2 @ A3 )
% 9.00/5.04 = ( insert_set_nat @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 != B3 )
% 9.00/5.04 => ? [C4: set_set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_set_nat @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member_set_nat @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert_set_nat @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member_set_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2286_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat,B3: nat,B2: set_nat] :
% 9.00/5.04 ( ~ ( member_nat @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_nat @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert_nat @ A2 @ A3 )
% 9.00/5.04 = ( insert_nat @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 != B3 )
% 9.00/5.04 => ? [C4: set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_nat @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member_nat @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert_nat @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member_nat @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2287_insert__eq__iff,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int,B3: int,B2: set_int] :
% 9.00/5.04 ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.04 => ( ~ ( member_int @ B3 @ B2 )
% 9.00/5.04 => ( ( ( insert_int @ A2 @ A3 )
% 9.00/5.04 = ( insert_int @ B3 @ B2 ) )
% 9.00/5.04 = ( ( ( A2 = B3 )
% 9.00/5.04 => ( A3 = B2 ) )
% 9.00/5.04 & ( ( A2 != B3 )
% 9.00/5.04 => ? [C4: set_int] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_int @ B3 @ C4 ) )
% 9.00/5.04 & ~ ( member_int @ B3 @ C4 )
% 9.00/5.04 & ( B2
% 9.00/5.04 = ( insert_int @ A2 @ C4 ) )
% 9.00/5.04 & ~ ( member_int @ A2 @ C4 ) ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2288_insert__absorb,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.04 => ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2289_insert__absorb,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ A2 @ A3 )
% 9.00/5.04 => ( ( insert_real @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2290_insert__absorb,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ A2 @ A3 )
% 9.00/5.04 => ( ( insert_o @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2291_insert__absorb,axiom,
% 9.00/5.04 ! [A2: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.04 => ( ( insert_set_nat @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2292_insert__absorb,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ A3 )
% 9.00/5.04 => ( ( insert_nat @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2293_insert__absorb,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ A2 @ A3 )
% 9.00/5.04 => ( ( insert_int @ A2 @ A3 )
% 9.00/5.04 = A3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2294_insert__ident,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert8211810215607154385at_nat @ X2 @ A3 )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2295_insert__ident,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member_real @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert_real @ X2 @ A3 )
% 9.00/5.04 = ( insert_real @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2296_insert__ident,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member_o @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert_o @ X2 @ A3 )
% 9.00/5.04 = ( insert_o @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2297_insert__ident,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member_set_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert_set_nat @ X2 @ A3 )
% 9.00/5.04 = ( insert_set_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2298_insert__ident,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member_nat @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert_nat @ X2 @ A3 )
% 9.00/5.04 = ( insert_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2299_insert__ident,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.04 => ( ~ ( member_int @ X2 @ B2 )
% 9.00/5.04 => ( ( ( insert_int @ X2 @ A3 )
% 9.00/5.04 = ( insert_int @ X2 @ B2 ) )
% 9.00/5.04 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2300_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert8211810215607154385at_nat @ X2 @ B8 ) )
% 9.00/5.04 => ( member8440522571783428010at_nat @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2301_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_real] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_real @ X2 @ B8 ) )
% 9.00/5.04 => ( member_real @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2302_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_o] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_o @ X2 @ B8 ) )
% 9.00/5.04 => ( member_o @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2303_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_set_nat @ X2 @ B8 ) )
% 9.00/5.04 => ( member_set_nat @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2304_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_nat] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_nat @ X2 @ B8 ) )
% 9.00/5.04 => ( member_nat @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2305_Set_Oset__insert,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ X2 @ A3 )
% 9.00/5.04 => ~ ! [B8: set_int] :
% 9.00/5.04 ( ( A3
% 9.00/5.04 = ( insert_int @ X2 @ B8 ) )
% 9.00/5.04 => ( member_int @ X2 @ B8 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2306_insertI2,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.04 => ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2307_insertI2,axiom,
% 9.00/5.04 ! [A2: real,B2: set_real,B3: real] :
% 9.00/5.04 ( ( member_real @ A2 @ B2 )
% 9.00/5.04 => ( member_real @ A2 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2308_insertI2,axiom,
% 9.00/5.04 ! [A2: $o,B2: set_o,B3: $o] :
% 9.00/5.04 ( ( member_o @ A2 @ B2 )
% 9.00/5.04 => ( member_o @ A2 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2309_insertI2,axiom,
% 9.00/5.04 ! [A2: set_nat,B2: set_set_nat,B3: set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ B2 )
% 9.00/5.04 => ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2310_insertI2,axiom,
% 9.00/5.04 ! [A2: nat,B2: set_nat,B3: nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ B2 )
% 9.00/5.04 => ( member_nat @ A2 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2311_insertI2,axiom,
% 9.00/5.04 ! [A2: int,B2: set_int,B3: int] :
% 9.00/5.04 ( ( member_int @ A2 @ B2 )
% 9.00/5.04 => ( member_int @ A2 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2312_insertI1,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2313_insertI1,axiom,
% 9.00/5.04 ! [A2: real,B2: set_real] : ( member_real @ A2 @ ( insert_real @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2314_insertI1,axiom,
% 9.00/5.04 ! [A2: $o,B2: set_o] : ( member_o @ A2 @ ( insert_o @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2315_insertI1,axiom,
% 9.00/5.04 ! [A2: set_nat,B2: set_set_nat] : ( member_set_nat @ A2 @ ( insert_set_nat @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2316_insertI1,axiom,
% 9.00/5.04 ! [A2: nat,B2: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2317_insertI1,axiom,
% 9.00/5.04 ! [A2: int,B2: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2318_insertE,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 != B3 )
% 9.00/5.04 => ( member8440522571783428010at_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2319_insertE,axiom,
% 9.00/5.04 ! [A2: real,B3: real,A3: set_real] :
% 9.00/5.04 ( ( member_real @ A2 @ ( insert_real @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 != B3 )
% 9.00/5.04 => ( member_real @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2320_insertE,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o,A3: set_o] :
% 9.00/5.04 ( ( member_o @ A2 @ ( insert_o @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 = ~ B3 )
% 9.00/5.04 => ( member_o @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2321_insertE,axiom,
% 9.00/5.04 ! [A2: set_nat,B3: set_nat,A3: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ A2 @ ( insert_set_nat @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 != B3 )
% 9.00/5.04 => ( member_set_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2322_insertE,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat,A3: set_nat] :
% 9.00/5.04 ( ( member_nat @ A2 @ ( insert_nat @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 != B3 )
% 9.00/5.04 => ( member_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2323_insertE,axiom,
% 9.00/5.04 ! [A2: int,B3: int,A3: set_int] :
% 9.00/5.04 ( ( member_int @ A2 @ ( insert_int @ B3 @ A3 ) )
% 9.00/5.04 => ( ( A2 != B3 )
% 9.00/5.04 => ( member_int @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2324_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc8306885398267862888on_nat] :
% 9.00/5.04 ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.00/5.04 => ~ ! [F3: nat > nat > nat,A: nat,B: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc8929957630744042906on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2325_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc5542196010084753463at_nat] :
% 9.00/5.04 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.00/5.04 => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc2899441246263362727at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2326_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc1193250871479095198on_num] :
% 9.00/5.04 ( ! [Uu2: num > num > num,Uv2: option_num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: num > num > num,V2: num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.00/5.04 => ~ ! [F3: num > num > num,A: num,B: num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc5778274026573060048on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2327_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc2233624965454879586on_nat] :
% 9.00/5.04 ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.00/5.04 => ~ ! [F3: nat > nat > $o,X5: nat,Y4: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc4035269172776083154on_nat @ F3 @ ( produc5098337634421038937on_nat @ ( some_nat @ X5 ) @ ( some_nat @ Y4 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2328_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc5491161045314408544at_nat] :
% 9.00/5.04 ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.00/5.04 => ~ ! [F3: product_prod_nat_nat > product_prod_nat_nat > $o,X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3994169339658061776at_nat @ F3 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X5 ) @ ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2329_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.00/5.04 ! [X2: produc7036089656553540234on_num] :
% 9.00/5.04 ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.00/5.04 => ( ! [Uw2: num > num > $o,V2: num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.00/5.04 => ~ ! [F3: num > num > $o,X5: num,Y4: num] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3576312749637752826on_num @ F3 @ ( produc8585076106096196333on_num @ ( some_num @ X5 ) @ ( some_num @ Y4 ) ) ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2330_singletonD,axiom,
% 9.00/5.04 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2331_singletonD,axiom,
% 9.00/5.04 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.04 ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2332_singletonD,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2333_singletonD,axiom,
% 9.00/5.04 ! [B3: $o,A2: $o] :
% 9.00/5.04 ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2334_singletonD,axiom,
% 9.00/5.04 ! [B3: nat,A2: nat] :
% 9.00/5.04 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2335_singletonD,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 9.00/5.04 => ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2336_singleton__iff,axiom,
% 9.00/5.04 ! [B3: product_prod_nat_nat,A2: product_prod_nat_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ B3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2337_singleton__iff,axiom,
% 9.00/5.04 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.04 ( ( member_set_nat @ B3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2338_singleton__iff,axiom,
% 9.00/5.04 ! [B3: real,A2: real] :
% 9.00/5.04 ( ( member_real @ B3 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2339_singleton__iff,axiom,
% 9.00/5.04 ! [B3: $o,A2: $o] :
% 9.00/5.04 ( ( member_o @ B3 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2340_singleton__iff,axiom,
% 9.00/5.04 ! [B3: nat,A2: nat] :
% 9.00/5.04 ( ( member_nat @ B3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2341_singleton__iff,axiom,
% 9.00/5.04 ! [B3: int,A2: int] :
% 9.00/5.04 ( ( member_int @ B3 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 9.00/5.04 = ( B3 = A2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2342_doubleton__eq__iff,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 9.00/5.04 ( ( ( insert8211810215607154385at_nat @ A2 @ ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.04 = ( ( ( A2 = C )
% 9.00/5.04 & ( B3 = D ) )
% 9.00/5.04 | ( ( A2 = D )
% 9.00/5.04 & ( B3 = C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2343_doubleton__eq__iff,axiom,
% 9.00/5.04 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.04 ( ( ( insert_real @ A2 @ ( insert_real @ B3 @ bot_bot_set_real ) )
% 9.00/5.04 = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 9.00/5.04 = ( ( ( A2 = C )
% 9.00/5.04 & ( B3 = D ) )
% 9.00/5.04 | ( ( A2 = D )
% 9.00/5.04 & ( B3 = C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2344_doubleton__eq__iff,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o,C: $o,D: $o] :
% 9.00/5.04 ( ( ( insert_o @ A2 @ ( insert_o @ B3 @ bot_bot_set_o ) )
% 9.00/5.04 = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
% 9.00/5.04 = ( ( ( A2 = C )
% 9.00/5.04 & ( B3 = D ) )
% 9.00/5.04 | ( ( A2 = D )
% 9.00/5.04 & ( B3 = C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2345_doubleton__eq__iff,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.04 ( ( ( insert_nat @ A2 @ ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 9.00/5.04 = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 9.00/5.04 = ( ( ( A2 = C )
% 9.00/5.04 & ( B3 = D ) )
% 9.00/5.04 | ( ( A2 = D )
% 9.00/5.04 & ( B3 = C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2346_doubleton__eq__iff,axiom,
% 9.00/5.04 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.04 ( ( ( insert_int @ A2 @ ( insert_int @ B3 @ bot_bot_set_int ) )
% 9.00/5.04 = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 9.00/5.04 = ( ( ( A2 = C )
% 9.00/5.04 & ( B3 = D ) )
% 9.00/5.04 | ( ( A2 = D )
% 9.00/5.04 & ( B3 = C ) ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2347_insert__not__empty,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( insert8211810215607154385at_nat @ A2 @ A3 )
% 9.00/5.04 != bot_bo2099793752762293965at_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2348_insert__not__empty,axiom,
% 9.00/5.04 ! [A2: real,A3: set_real] :
% 9.00/5.04 ( ( insert_real @ A2 @ A3 )
% 9.00/5.04 != bot_bot_set_real ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2349_insert__not__empty,axiom,
% 9.00/5.04 ! [A2: $o,A3: set_o] :
% 9.00/5.04 ( ( insert_o @ A2 @ A3 )
% 9.00/5.04 != bot_bot_set_o ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2350_insert__not__empty,axiom,
% 9.00/5.04 ! [A2: nat,A3: set_nat] :
% 9.00/5.04 ( ( insert_nat @ A2 @ A3 )
% 9.00/5.04 != bot_bot_set_nat ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2351_insert__not__empty,axiom,
% 9.00/5.04 ! [A2: int,A3: set_int] :
% 9.00/5.04 ( ( insert_int @ A2 @ A3 )
% 9.00/5.04 != bot_bot_set_int ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2352_singleton__inject,axiom,
% 9.00/5.04 ! [A2: product_prod_nat_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 9.00/5.04 = ( insert8211810215607154385at_nat @ B3 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.04 => ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2353_singleton__inject,axiom,
% 9.00/5.04 ! [A2: real,B3: real] :
% 9.00/5.04 ( ( ( insert_real @ A2 @ bot_bot_set_real )
% 9.00/5.04 = ( insert_real @ B3 @ bot_bot_set_real ) )
% 9.00/5.04 => ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2354_singleton__inject,axiom,
% 9.00/5.04 ! [A2: $o,B3: $o] :
% 9.00/5.04 ( ( ( insert_o @ A2 @ bot_bot_set_o )
% 9.00/5.04 = ( insert_o @ B3 @ bot_bot_set_o ) )
% 9.00/5.04 => ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2355_singleton__inject,axiom,
% 9.00/5.04 ! [A2: nat,B3: nat] :
% 9.00/5.04 ( ( ( insert_nat @ A2 @ bot_bot_set_nat )
% 9.00/5.04 = ( insert_nat @ B3 @ bot_bot_set_nat ) )
% 9.00/5.04 => ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2356_singleton__inject,axiom,
% 9.00/5.04 ! [A2: int,B3: int] :
% 9.00/5.04 ( ( ( insert_int @ A2 @ bot_bot_set_int )
% 9.00/5.04 = ( insert_int @ B3 @ bot_bot_set_int ) )
% 9.00/5.04 => ( A2 = B3 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2357_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_real,A2: real] :
% 9.00/5.04 ( ( finite_finite_real @ A3 )
% 9.00/5.04 => ( finite_finite_real @ ( insert_real @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2358_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_o,A2: $o] :
% 9.00/5.04 ( ( finite_finite_o @ A3 )
% 9.00/5.04 => ( finite_finite_o @ ( insert_o @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2359_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_nat,A2: nat] :
% 9.00/5.04 ( ( finite_finite_nat @ A3 )
% 9.00/5.04 => ( finite_finite_nat @ ( insert_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2360_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_int,A2: int] :
% 9.00/5.04 ( ( finite_finite_int @ A3 )
% 9.00/5.04 => ( finite_finite_int @ ( insert_int @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2361_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_complex,A2: complex] :
% 9.00/5.04 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.04 => ( finite3207457112153483333omplex @ ( insert_complex @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2362_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 9.00/5.04 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.04 => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2363_finite_OinsertI,axiom,
% 9.00/5.04 ! [A3: set_Extended_enat,A2: extended_enat] :
% 9.00/5.04 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.04 => ( finite4001608067531595151d_enat @ ( insert_Extended_enat @ A2 @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2364_insert__mono,axiom,
% 9.00/5.04 ! [C2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ C2 @ D4 )
% 9.00/5.04 => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A2 @ C2 ) @ ( insert8211810215607154385at_nat @ A2 @ D4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2365_insert__mono,axiom,
% 9.00/5.04 ! [C2: set_real,D4: set_real,A2: real] :
% 9.00/5.04 ( ( ord_less_eq_set_real @ C2 @ D4 )
% 9.00/5.04 => ( ord_less_eq_set_real @ ( insert_real @ A2 @ C2 ) @ ( insert_real @ A2 @ D4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2366_insert__mono,axiom,
% 9.00/5.04 ! [C2: set_o,D4: set_o,A2: $o] :
% 9.00/5.04 ( ( ord_less_eq_set_o @ C2 @ D4 )
% 9.00/5.04 => ( ord_less_eq_set_o @ ( insert_o @ A2 @ C2 ) @ ( insert_o @ A2 @ D4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2367_insert__mono,axiom,
% 9.00/5.04 ! [C2: set_nat,D4: set_nat,A2: nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ C2 @ D4 )
% 9.00/5.04 => ( ord_less_eq_set_nat @ ( insert_nat @ A2 @ C2 ) @ ( insert_nat @ A2 @ D4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2368_insert__mono,axiom,
% 9.00/5.04 ! [C2: set_int,D4: set_int,A2: int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ C2 @ D4 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( insert_int @ A2 @ C2 ) @ ( insert_int @ A2 @ D4 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2369_subset__insert,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2370_subset__insert,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real,B2: set_real] :
% 9.00/5.04 ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2371_subset__insert,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o,B2: set_o] :
% 9.00/5.04 ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2372_subset__insert,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.04 ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2373_subset__insert,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.04 ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2374_subset__insert,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int,B2: set_int] :
% 9.00/5.04 ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 9.00/5.04 = ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2375_subset__insertI,axiom,
% 9.00/5.04 ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2376_subset__insertI,axiom,
% 9.00/5.04 ! [B2: set_real,A2: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2377_subset__insertI,axiom,
% 9.00/5.04 ! [B2: set_o,A2: $o] : ( ord_less_eq_set_o @ B2 @ ( insert_o @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2378_subset__insertI,axiom,
% 9.00/5.04 ! [B2: set_nat,A2: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2379_subset__insertI,axiom,
% 9.00/5.04 ! [B2: set_int,A2: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A2 @ B2 ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2380_subset__insertI2,axiom,
% 9.00/5.04 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.00/5.04 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2381_subset__insertI2,axiom,
% 9.00/5.04 ! [A3: set_real,B2: set_real,B3: real] :
% 9.00/5.04 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2382_subset__insertI2,axiom,
% 9.00/5.04 ! [A3: set_o,B2: set_o,B3: $o] :
% 9.00/5.04 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2383_subset__insertI2,axiom,
% 9.00/5.04 ! [A3: set_nat,B2: set_nat,B3: nat] :
% 9.00/5.04 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2384_subset__insertI2,axiom,
% 9.00/5.04 ! [A3: set_int,B2: set_int,B3: int] :
% 9.00/5.04 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.04 => ( ord_less_eq_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2385_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,X6: set_Pr1261947904930325089at_nat] :
% 9.00/5.04 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_le3146513528884898305at_nat @ X6 @ A3 )
% 9.00/5.04 => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2386_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: real,A3: set_real,X6: set_real] :
% 9.00/5.04 ( ( member_real @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_real @ X6 @ A3 )
% 9.00/5.04 => ( ord_less_eq_set_real @ ( insert_real @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2387_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: $o,A3: set_o,X6: set_o] :
% 9.00/5.04 ( ( member_o @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_o @ X6 @ A3 )
% 9.00/5.04 => ( ord_less_eq_set_o @ ( insert_o @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2388_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: set_nat,A3: set_set_nat,X6: set_set_nat] :
% 9.00/5.04 ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_le6893508408891458716et_nat @ X6 @ A3 )
% 9.00/5.04 => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2389_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: nat,A3: set_nat,X6: set_nat] :
% 9.00/5.04 ( ( member_nat @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_nat @ X6 @ A3 )
% 9.00/5.04 => ( ord_less_eq_set_nat @ ( insert_nat @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2390_insert__subsetI,axiom,
% 9.00/5.04 ! [X2: int,A3: set_int,X6: set_int] :
% 9.00/5.04 ( ( member_int @ X2 @ A3 )
% 9.00/5.04 => ( ( ord_less_eq_set_int @ X6 @ A3 )
% 9.00/5.04 => ( ord_less_eq_set_int @ ( insert_int @ X2 @ X6 ) @ A3 ) ) ) ).
% 9.00/5.04
% 9.00/5.04 thf(fact_2391_fold__atLeastAtMost__nat_Ocases,axiom,
% 9.00/5.04 ! [X2: produc4471711990508489141at_nat] :
% 9.00/5.04 ~ ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
% 9.00/5.04 ( X2
% 9.00/5.04 != ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2392_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert8211810215607154385at_nat @ X2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2393_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: real,B2: set_real,A3: set_real] :
% 9.00/5.05 ( ( ( member_real @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_minus_set_real @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert_real @ X2 @ ( minus_minus_set_real @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2394_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: $o,B2: set_o,A3: set_o] :
% 9.00/5.05 ( ( ( member_o @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_minus_set_o @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert_o @ X2 @ ( minus_minus_set_o @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2395_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.05 ( ( ( member_set_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert_set_nat @ X2 @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2396_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: int,B2: set_int,A3: set_int] :
% 9.00/5.05 ( ( ( member_int @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_minus_set_int @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert_int @ X2 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2397_insert__Diff__if,axiom,
% 9.00/5.05 ! [X2: nat,B2: set_nat,A3: set_nat] :
% 9.00/5.05 ( ( ( member_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( minus_minus_set_nat @ A3 @ B2 ) ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ B2 )
% 9.00/5.05 = ( insert_nat @ X2 @ ( minus_minus_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2398_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bot_set_complex )
% 9.00/5.05 => ~ ! [A5: set_complex] :
% 9.00/5.05 ( ? [A: complex] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_complex @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite3207457112153483333omplex @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2399_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ~ ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ? [A: product_prod_nat_nat] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert8211810215607154385at_nat @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite6177210948735845034at_nat @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2400_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ~ ! [A5: set_Extended_enat] :
% 9.00/5.05 ( ? [A: extended_enat] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_Extended_enat @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite4001608067531595151d_enat @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2401_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bot_set_real )
% 9.00/5.05 => ~ ! [A5: set_real] :
% 9.00/5.05 ( ? [A: real] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_real @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite_finite_real @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2402_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bot_set_o )
% 9.00/5.05 => ~ ! [A5: set_o] :
% 9.00/5.05 ( ? [A: $o] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_o @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite_finite_o @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2403_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bot_set_nat )
% 9.00/5.05 => ~ ! [A5: set_nat] :
% 9.00/5.05 ( ? [A: nat] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_nat @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite_finite_nat @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2404_finite_Ocases,axiom,
% 9.00/5.05 ! [A2: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A2 )
% 9.00/5.05 => ( ( A2 != bot_bot_set_int )
% 9.00/5.05 => ~ ! [A5: set_int] :
% 9.00/5.05 ( ? [A: int] :
% 9.00/5.05 ( A2
% 9.00/5.05 = ( insert_int @ A @ A5 ) )
% 9.00/5.05 => ~ ( finite_finite_int @ A5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2405_finite_Osimps,axiom,
% 9.00/5.05 ( finite3207457112153483333omplex
% 9.00/5.05 = ( ^ [A4: set_complex] :
% 9.00/5.05 ( ( A4 = bot_bot_set_complex )
% 9.00/5.05 | ? [A6: set_complex,B4: complex] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_complex @ B4 @ A6 ) )
% 9.00/5.05 & ( finite3207457112153483333omplex @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2406_finite_Osimps,axiom,
% 9.00/5.05 ( finite6177210948735845034at_nat
% 9.00/5.05 = ( ^ [A4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( A4 = bot_bo2099793752762293965at_nat )
% 9.00/5.05 | ? [A6: set_Pr1261947904930325089at_nat,B4: product_prod_nat_nat] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert8211810215607154385at_nat @ B4 @ A6 ) )
% 9.00/5.05 & ( finite6177210948735845034at_nat @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2407_finite_Osimps,axiom,
% 9.00/5.05 ( finite4001608067531595151d_enat
% 9.00/5.05 = ( ^ [A4: set_Extended_enat] :
% 9.00/5.05 ( ( A4 = bot_bo7653980558646680370d_enat )
% 9.00/5.05 | ? [A6: set_Extended_enat,B4: extended_enat] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_Extended_enat @ B4 @ A6 ) )
% 9.00/5.05 & ( finite4001608067531595151d_enat @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2408_finite_Osimps,axiom,
% 9.00/5.05 ( finite_finite_real
% 9.00/5.05 = ( ^ [A4: set_real] :
% 9.00/5.05 ( ( A4 = bot_bot_set_real )
% 9.00/5.05 | ? [A6: set_real,B4: real] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_real @ B4 @ A6 ) )
% 9.00/5.05 & ( finite_finite_real @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2409_finite_Osimps,axiom,
% 9.00/5.05 ( finite_finite_o
% 9.00/5.05 = ( ^ [A4: set_o] :
% 9.00/5.05 ( ( A4 = bot_bot_set_o )
% 9.00/5.05 | ? [A6: set_o,B4: $o] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_o @ B4 @ A6 ) )
% 9.00/5.05 & ( finite_finite_o @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2410_finite_Osimps,axiom,
% 9.00/5.05 ( finite_finite_nat
% 9.00/5.05 = ( ^ [A4: set_nat] :
% 9.00/5.05 ( ( A4 = bot_bot_set_nat )
% 9.00/5.05 | ? [A6: set_nat,B4: nat] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_nat @ B4 @ A6 ) )
% 9.00/5.05 & ( finite_finite_nat @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2411_finite_Osimps,axiom,
% 9.00/5.05 ( finite_finite_int
% 9.00/5.05 = ( ^ [A4: set_int] :
% 9.00/5.05 ( ( A4 = bot_bot_set_int )
% 9.00/5.05 | ? [A6: set_int,B4: int] :
% 9.00/5.05 ( ( A4
% 9.00/5.05 = ( insert_int @ B4 @ A6 ) )
% 9.00/5.05 & ( finite_finite_int @ A6 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2412_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [X5: set_nat,F4: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F4 )
% 9.00/5.05 => ( ~ ( member_set_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2413_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [X5: complex,F4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F4 )
% 9.00/5.05 => ( ~ ( member_complex @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2414_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F4 )
% 9.00/5.05 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2415_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F4 )
% 9.00/5.05 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2416_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real,F4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ F4 )
% 9.00/5.05 => ( ~ ( member_real @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2417_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o,F4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ F4 )
% 9.00/5.05 => ( ~ ( member_o @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2418_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [X5: nat,F4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ F4 )
% 9.00/5.05 => ( ~ ( member_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2419_finite__induct,axiom,
% 9.00/5.05 ! [F2: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ F2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [X5: int,F4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ F4 )
% 9.00/5.05 => ( ~ ( member_int @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2420_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [X5: set_nat] : ( P @ ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) )
% 9.00/5.05 => ( ! [X5: set_nat,F4: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_set_nat )
% 9.00/5.05 => ( ~ ( member_set_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2421_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_complex )
% 9.00/5.05 => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
% 9.00/5.05 => ( ! [X5: complex,F4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_complex )
% 9.00/5.05 => ( ~ ( member_complex @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2422_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [X5: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.05 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2423_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [X5: extended_enat] : ( P @ ( insert_Extended_enat @ X5 @ bot_bo7653980558646680370d_enat ) )
% 9.00/5.05 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2424_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
% 9.00/5.05 => ( ! [X5: real,F4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_real )
% 9.00/5.05 => ( ~ ( member_real @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2425_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o] : ( P @ ( insert_o @ X5 @ bot_bot_set_o ) )
% 9.00/5.05 => ( ! [X5: $o,F4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_o )
% 9.00/5.05 => ( ~ ( member_o @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2426_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_nat )
% 9.00/5.05 => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
% 9.00/5.05 => ( ! [X5: nat,F4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_nat )
% 9.00/5.05 => ( ~ ( member_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2427_finite__ne__induct,axiom,
% 9.00/5.05 ! [F2: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ F2 )
% 9.00/5.05 => ( ( F2 != bot_bot_set_int )
% 9.00/5.05 => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
% 9.00/5.05 => ( ! [X5: int,F4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ F4 )
% 9.00/5.05 => ( ( F4 != bot_bot_set_int )
% 9.00/5.05 => ( ~ ( member_int @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2428_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_set_nat > $o,A3: set_set_nat] :
% 9.00/5.05 ( ! [A5: set_set_nat] :
% 9.00/5.05 ( ~ ( finite1152437895449049373et_nat @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [X5: set_nat,F4: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F4 )
% 9.00/5.05 => ( ~ ( member_set_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_set_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2429_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_complex > $o,A3: set_complex] :
% 9.00/5.05 ( ! [A5: set_complex] :
% 9.00/5.05 ( ~ ( finite3207457112153483333omplex @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [X5: complex,F4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F4 )
% 9.00/5.05 => ( ~ ( member_complex @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_complex @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2430_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ~ ( finite6177210948735845034at_nat @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [X5: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F4 )
% 9.00/5.05 => ( ~ ( member8440522571783428010at_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2431_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_Extended_enat > $o,A3: set_Extended_enat] :
% 9.00/5.05 ( ! [A5: set_Extended_enat] :
% 9.00/5.05 ( ~ ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [X5: extended_enat,F4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F4 )
% 9.00/5.05 => ( ~ ( member_Extended_enat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2432_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_real > $o,A3: set_real] :
% 9.00/5.05 ( ! [A5: set_real] :
% 9.00/5.05 ( ~ ( finite_finite_real @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real,F4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ F4 )
% 9.00/5.05 => ( ~ ( member_real @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_real @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2433_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_o > $o,A3: set_o] :
% 9.00/5.05 ( ! [A5: set_o] :
% 9.00/5.05 ( ~ ( finite_finite_o @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o,F4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ F4 )
% 9.00/5.05 => ( ~ ( member_o @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_o @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2434_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_nat > $o,A3: set_nat] :
% 9.00/5.05 ( ! [A5: set_nat] :
% 9.00/5.05 ( ~ ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [X5: nat,F4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ F4 )
% 9.00/5.05 => ( ~ ( member_nat @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_nat @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2435_infinite__finite__induct,axiom,
% 9.00/5.05 ! [P: set_int > $o,A3: set_int] :
% 9.00/5.05 ( ! [A5: set_int] :
% 9.00/5.05 ( ~ ( finite_finite_int @ A5 )
% 9.00/5.05 => ( P @ A5 ) )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [X5: int,F4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ F4 )
% 9.00/5.05 => ( ~ ( member_int @ X5 @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_int @ X5 @ F4 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2436_subset__singletonD,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.05 => ( ( A3 = bot_bo2099793752762293965at_nat )
% 9.00/5.05 | ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2437_subset__singletonD,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.05 => ( ( A3 = bot_bot_set_real )
% 9.00/5.05 | ( A3
% 9.00/5.05 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2438_subset__singletonD,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.05 => ( ( A3 = bot_bot_set_o )
% 9.00/5.05 | ( A3
% 9.00/5.05 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2439_subset__singletonD,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.05 => ( ( A3 = bot_bot_set_nat )
% 9.00/5.05 | ( A3
% 9.00/5.05 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2440_subset__singletonD,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.05 => ( ( A3 = bot_bot_set_int )
% 9.00/5.05 | ( A3
% 9.00/5.05 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2441_subset__singleton__iff,axiom,
% 9.00/5.05 ! [X6: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 9.00/5.05 ( ( ord_le3146513528884898305at_nat @ X6 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.05 = ( ( X6 = bot_bo2099793752762293965at_nat )
% 9.00/5.05 | ( X6
% 9.00/5.05 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2442_subset__singleton__iff,axiom,
% 9.00/5.05 ! [X6: set_real,A2: real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ X6 @ ( insert_real @ A2 @ bot_bot_set_real ) )
% 9.00/5.05 = ( ( X6 = bot_bot_set_real )
% 9.00/5.05 | ( X6
% 9.00/5.05 = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2443_subset__singleton__iff,axiom,
% 9.00/5.05 ! [X6: set_o,A2: $o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A2 @ bot_bot_set_o ) )
% 9.00/5.05 = ( ( X6 = bot_bot_set_o )
% 9.00/5.05 | ( X6
% 9.00/5.05 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2444_subset__singleton__iff,axiom,
% 9.00/5.05 ! [X6: set_nat,A2: nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
% 9.00/5.05 = ( ( X6 = bot_bot_set_nat )
% 9.00/5.05 | ( X6
% 9.00/5.05 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2445_subset__singleton__iff,axiom,
% 9.00/5.05 ! [X6: set_int,A2: int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A2 @ bot_bot_set_int ) )
% 9.00/5.05 = ( ( X6 = bot_bot_set_int )
% 9.00/5.05 | ( X6
% 9.00/5.05 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2446_Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2447_Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_real,A2: real,B2: set_real] :
% 9.00/5.05 ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ B2 ) @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2448_Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_o,A2: $o,B2: set_o] :
% 9.00/5.05 ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ B2 ) @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2449_Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_int,A2: int,B2: set_int] :
% 9.00/5.05 ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2450_Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 9.00/5.05 ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2451_insert__Diff,axiom,
% 9.00/5.05 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.05 => ( ( insert8211810215607154385at_nat @ A2 @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2452_insert__Diff,axiom,
% 9.00/5.05 ! [A2: set_nat,A3: set_set_nat] :
% 9.00/5.05 ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.05 => ( ( insert_set_nat @ A2 @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2453_insert__Diff,axiom,
% 9.00/5.05 ! [A2: real,A3: set_real] :
% 9.00/5.05 ( ( member_real @ A2 @ A3 )
% 9.00/5.05 => ( ( insert_real @ A2 @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2454_insert__Diff,axiom,
% 9.00/5.05 ! [A2: $o,A3: set_o] :
% 9.00/5.05 ( ( member_o @ A2 @ A3 )
% 9.00/5.05 => ( ( insert_o @ A2 @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2455_insert__Diff,axiom,
% 9.00/5.05 ! [A2: int,A3: set_int] :
% 9.00/5.05 ( ( member_int @ A2 @ A3 )
% 9.00/5.05 => ( ( insert_int @ A2 @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2456_insert__Diff,axiom,
% 9.00/5.05 ! [A2: nat,A3: set_nat] :
% 9.00/5.05 ( ( member_nat @ A2 @ A3 )
% 9.00/5.05 => ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2457_Diff__insert2,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2458_Diff__insert2,axiom,
% 9.00/5.05 ! [A3: set_real,A2: real,B2: set_real] :
% 9.00/5.05 ( ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) @ B2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2459_Diff__insert2,axiom,
% 9.00/5.05 ! [A3: set_o,A2: $o,B2: set_o] :
% 9.00/5.05 ( ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) @ B2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2460_Diff__insert2,axiom,
% 9.00/5.05 ! [A3: set_int,A2: int,B2: set_int] :
% 9.00/5.05 ( ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) @ B2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2461_Diff__insert2,axiom,
% 9.00/5.05 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 9.00/5.05 ( ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.05 = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2462_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2463_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: set_nat,A3: set_set_nat] :
% 9.00/5.05 ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X2 @ A3 ) @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2464_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: real,A3: set_real] :
% 9.00/5.05 ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_minus_set_real @ ( insert_real @ X2 @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2465_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: $o,A3: set_o] :
% 9.00/5.05 ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_minus_set_o @ ( insert_o @ X2 @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2466_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: int,A3: set_int] :
% 9.00/5.05 ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_minus_set_int @ ( insert_int @ X2 @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2467_Diff__insert__absorb,axiom,
% 9.00/5.05 ! [X2: nat,A3: set_nat] :
% 9.00/5.05 ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( minus_minus_set_nat @ ( insert_nat @ X2 @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2468_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_le3146513528884898305at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2469_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_real,B2: set_real,X2: real,C2: set_real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_set_real @ A3 @ ( minus_minus_set_real @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member_real @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2470_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_o,B2: set_o,X2: $o,C2: set_o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ ( insert_o @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_set_o @ A3 @ ( minus_minus_set_o @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member_o @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2471_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_set_nat,B2: set_set_nat,X2: set_nat,C2: set_set_nat] :
% 9.00/5.05 ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_le6893508408891458716et_nat @ A3 @ ( minus_2163939370556025621et_nat @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member_set_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2472_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_nat,B2: set_nat,X2: nat,C2: set_nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2473_subset__Diff__insert,axiom,
% 9.00/5.05 ! [A3: set_int,B2: set_int,X2: int,C2: set_int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X2 @ C2 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_set_int @ A3 @ ( minus_minus_set_int @ B2 @ C2 ) )
% 9.00/5.05 & ~ ( member_int @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2474_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2475_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ ( insert_real @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2476_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ ( insert_o @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2477_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2478_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2479_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2480_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2481_card__insert__le,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( insert_int @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2482_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_complex,P: set_complex > $o,F: complex > rat] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [X5: complex,S4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ S4 )
% 9.00/5.05 => ( ! [Y5: complex] :
% 9.00/5.05 ( ( member_complex @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2483_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > rat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [X5: extended_enat,S4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ S4 )
% 9.00/5.05 => ( ! [Y5: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2484_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_real,P: set_real > $o,F: real > rat] :
% 9.00/5.05 ( ( finite_finite_real @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real,S4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ S4 )
% 9.00/5.05 => ( ! [Y5: real] :
% 9.00/5.05 ( ( member_real @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2485_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_o,P: set_o > $o,F: $o > rat] :
% 9.00/5.05 ( ( finite_finite_o @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o,S4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ S4 )
% 9.00/5.05 => ( ! [Y5: $o] :
% 9.00/5.05 ( ( member_o @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2486_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_nat,P: set_nat > $o,F: nat > rat] :
% 9.00/5.05 ( ( finite_finite_nat @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [X5: nat,S4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ S4 )
% 9.00/5.05 => ( ! [Y5: nat] :
% 9.00/5.05 ( ( member_nat @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2487_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_int,P: set_int > $o,F: int > rat] :
% 9.00/5.05 ( ( finite_finite_int @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [X5: int,S4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ S4 )
% 9.00/5.05 => ( ! [Y5: int] :
% 9.00/5.05 ( ( member_int @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2488_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_complex,P: set_complex > $o,F: complex > num] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [X5: complex,S4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ S4 )
% 9.00/5.05 => ( ! [Y5: complex] :
% 9.00/5.05 ( ( member_complex @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2489_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_Extended_enat,P: set_Extended_enat > $o,F: extended_enat > num] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [X5: extended_enat,S4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ S4 )
% 9.00/5.05 => ( ! [Y5: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2490_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_real,P: set_real > $o,F: real > num] :
% 9.00/5.05 ( ( finite_finite_real @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real,S4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ S4 )
% 9.00/5.05 => ( ! [Y5: real] :
% 9.00/5.05 ( ( member_real @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2491_finite__ranking__induct,axiom,
% 9.00/5.05 ! [S: set_o,P: set_o > $o,F: $o > num] :
% 9.00/5.05 ( ( finite_finite_o @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o,S4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ S4 )
% 9.00/5.05 => ( ! [Y5: $o] :
% 9.00/5.05 ( ( member_o @ Y5 @ S4 )
% 9.00/5.05 => ( ord_less_eq_num @ ( F @ Y5 ) @ ( F @ X5 ) ) )
% 9.00/5.05 => ( ( P @ S4 )
% 9.00/5.05 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2492_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [B: extended_enat,A5: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( ! [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ A5 )
% 9.00/5.05 => ( ord_le72135733267957522d_enat @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2493_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [B: $o,A5: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A5 )
% 9.00/5.05 => ( ! [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_o @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2494_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [B: real,A5: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A5 )
% 9.00/5.05 => ( ! [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_real @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2495_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_rat,P: set_rat > $o] :
% 9.00/5.05 ( ( finite_finite_rat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_rat )
% 9.00/5.05 => ( ! [B: rat,A5: set_rat] :
% 9.00/5.05 ( ( finite_finite_rat @ A5 )
% 9.00/5.05 => ( ! [X4: rat] :
% 9.00/5.05 ( ( member_rat @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_rat @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2496_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_num,P: set_num > $o] :
% 9.00/5.05 ( ( finite_finite_num @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_num )
% 9.00/5.05 => ( ! [B: num,A5: set_num] :
% 9.00/5.05 ( ( finite_finite_num @ A5 )
% 9.00/5.05 => ( ! [X4: num] :
% 9.00/5.05 ( ( member_num @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_num @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2497_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [B: nat,A5: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( ! [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_nat @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2498_finite__linorder__max__induct,axiom,
% 9.00/5.05 ! [A3: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [B: int,A5: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A5 )
% 9.00/5.05 => ( ! [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_int @ X4 @ B ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2499_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [B: extended_enat,A5: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( ! [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ A5 )
% 9.00/5.05 => ( ord_le72135733267957522d_enat @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2500_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [B: $o,A5: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A5 )
% 9.00/5.05 => ( ! [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_o @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_o @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2501_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [B: real,A5: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A5 )
% 9.00/5.05 => ( ! [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_real @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_real @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2502_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_rat,P: set_rat > $o] :
% 9.00/5.05 ( ( finite_finite_rat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_rat )
% 9.00/5.05 => ( ! [B: rat,A5: set_rat] :
% 9.00/5.05 ( ( finite_finite_rat @ A5 )
% 9.00/5.05 => ( ! [X4: rat] :
% 9.00/5.05 ( ( member_rat @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_rat @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_rat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2503_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_num,P: set_num > $o] :
% 9.00/5.05 ( ( finite_finite_num @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_num )
% 9.00/5.05 => ( ! [B: num,A5: set_num] :
% 9.00/5.05 ( ( finite_finite_num @ A5 )
% 9.00/5.05 => ( ! [X4: num] :
% 9.00/5.05 ( ( member_num @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_num @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_num @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2504_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [B: nat,A5: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( ! [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_nat @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_nat @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2505_finite__linorder__min__induct,axiom,
% 9.00/5.05 ! [A3: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [B: int,A5: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A5 )
% 9.00/5.05 => ( ! [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ A5 )
% 9.00/5.05 => ( ord_less_int @ B @ X4 ) )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( insert_int @ B @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2506_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F2 )
% 9.00/5.05 => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [A: set_nat,F4: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F4 )
% 9.00/5.05 => ( ( member_set_nat @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_set_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2507_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.05 => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [A: complex,F4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F4 )
% 9.00/5.05 => ( ( member_complex @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_complex @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2508_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.05 => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F4 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ A @ A3 )
% 9.00/5.05 => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2509_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.05 => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [A: extended_enat,F4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F4 )
% 9.00/5.05 => ( ( member_Extended_enat @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_Extended_enat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2510_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_real,A3: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_real @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [A: real,F4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ F4 )
% 9.00/5.05 => ( ( member_real @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_real @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2511_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_o,A3: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_o @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [A: $o,F4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ F4 )
% 9.00/5.05 => ( ( member_o @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_o @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2512_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [A: nat,F4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ F4 )
% 9.00/5.05 => ( ( member_nat @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2513_finite__subset__induct,axiom,
% 9.00/5.05 ! [F2: set_int,A3: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_int @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [A: int,F4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ F4 )
% 9.00/5.05 => ( ( member_int @ A @ A3 )
% 9.00/5.05 => ( ~ ( member_int @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2514_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_set_nat,A3: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F2 )
% 9.00/5.05 => ( ( ord_le6893508408891458716et_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [A: set_nat,F4: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ F4 )
% 9.00/5.05 => ( ( member_set_nat @ A @ A3 )
% 9.00/5.05 => ( ( ord_le6893508408891458716et_nat @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_set_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_set_nat @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2515_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_complex,A3: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.05 => ( ( ord_le211207098394363844omplex @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [A: complex,F4: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ F4 )
% 9.00/5.05 => ( ( member_complex @ A @ A3 )
% 9.00/5.05 => ( ( ord_le211207098394363844omplex @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_complex @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_complex @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2516_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.05 => ( ( ord_le3146513528884898305at_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [A: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ F4 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ A @ A3 )
% 9.00/5.05 => ( ( ord_le3146513528884898305at_nat @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member8440522571783428010at_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert8211810215607154385at_nat @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2517_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_Extended_enat,A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.05 => ( ( ord_le7203529160286727270d_enat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [A: extended_enat,F4: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ F4 )
% 9.00/5.05 => ( ( member_Extended_enat @ A @ A3 )
% 9.00/5.05 => ( ( ord_le7203529160286727270d_enat @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_Extended_enat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_Extended_enat @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2518_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_real,A3: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_real @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [A: real,F4: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ F4 )
% 9.00/5.05 => ( ( member_real @ A @ A3 )
% 9.00/5.05 => ( ( ord_less_eq_set_real @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_real @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_real @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2519_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_o,A3: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_o @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [A: $o,F4: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ F4 )
% 9.00/5.05 => ( ( member_o @ A @ A3 )
% 9.00/5.05 => ( ( ord_less_eq_set_o @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_o @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_o @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2520_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_nat,A3: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_nat @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [A: nat,F4: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ F4 )
% 9.00/5.05 => ( ( member_nat @ A @ A3 )
% 9.00/5.05 => ( ( ord_less_eq_set_nat @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_nat @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_nat @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2521_finite__subset__induct_H,axiom,
% 9.00/5.05 ! [F2: set_int,A3: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ F2 )
% 9.00/5.05 => ( ( ord_less_eq_set_int @ F2 @ A3 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [A: int,F4: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ F4 )
% 9.00/5.05 => ( ( member_int @ A @ A3 )
% 9.00/5.05 => ( ( ord_less_eq_set_int @ F4 @ A3 )
% 9.00/5.05 => ( ~ ( member_int @ A @ F4 )
% 9.00/5.05 => ( ( P @ F4 )
% 9.00/5.05 => ( P @ ( insert_int @ A @ F4 ) ) ) ) ) ) )
% 9.00/5.05 => ( P @ F2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2522_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_real,K: nat] :
% 9.00/5.05 ( ( ( finite_card_real @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: real,B6: set_real] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_real @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_real @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_real @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite_finite_real @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2523_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_o,K: nat] :
% 9.00/5.05 ( ( ( finite_card_o @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: $o,B6: set_o] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_o @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_o @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_o @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite_finite_o @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2524_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_list_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: list_nat,B6: set_list_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_list_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_list_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_list_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2525_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: set_nat,B6: set_set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_set_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_set_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_set_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2526_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: nat,B6: set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite_finite_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2527_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_int,K: nat] :
% 9.00/5.05 ( ( ( finite_card_int @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: int,B6: set_int] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_int @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_int @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_int @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite_finite_int @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2528_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_complex,K: nat] :
% 9.00/5.05 ( ( ( finite_card_complex @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: complex,B6: set_complex] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_complex @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_complex @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_complex @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2529_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 9.00/5.05 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite711546835091564841at_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2530_card__Suc__eq__finite,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,K: nat] :
% 9.00/5.05 ( ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: extended_enat,B6: set_Extended_enat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_Extended_enat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_Extended_enat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite121521170596916366d_enat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2531_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_real @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_real @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2532_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_o @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_o @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2533_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_list_nat @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_list_nat @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2534_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_set_nat @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_set_nat @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2535_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_nat @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_nat @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2536_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_int @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_int @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2537_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.05 = ( finite_card_complex @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_complex @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2538_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( finite711546835091564841at_nat @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2539_card__insert__if,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.05 = ( finite121521170596916366d_enat @ A3 ) ) )
% 9.00/5.05 & ( ~ ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2540_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_complex,A2: complex] :
% 9.00/5.05 ( ~ ( finite3207457112153483333omplex @ S )
% 9.00/5.05 => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2541_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat] :
% 9.00/5.05 ( ~ ( finite6177210948735845034at_nat @ S )
% 9.00/5.05 => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2542_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_Extended_enat,A2: extended_enat] :
% 9.00/5.05 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.05 => ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2543_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_real,A2: real] :
% 9.00/5.05 ( ~ ( finite_finite_real @ S )
% 9.00/5.05 => ~ ( finite_finite_real @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2544_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_o,A2: $o] :
% 9.00/5.05 ( ~ ( finite_finite_o @ S )
% 9.00/5.05 => ~ ( finite_finite_o @ ( minus_minus_set_o @ S @ ( insert_o @ A2 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2545_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_int,A2: int] :
% 9.00/5.05 ( ~ ( finite_finite_int @ S )
% 9.00/5.05 => ~ ( finite_finite_int @ ( minus_minus_set_int @ S @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2546_infinite__remove,axiom,
% 9.00/5.05 ! [S: set_nat,A2: nat] :
% 9.00/5.05 ( ~ ( finite_finite_nat @ S )
% 9.00/5.05 => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2547_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_complex > $o,A3: set_complex] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_complex] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: complex] :
% 9.00/5.05 ( ( member_complex @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) )
% 9.00/5.05 | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite3207457112153483333omplex @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2548_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_Pr1261947904930325089at_nat > $o,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: product_prod_nat_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.05 | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite6177210948735845034at_nat @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2549_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_Extended_enat > $o,A3: set_Extended_enat] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_Extended_enat] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.05 | ~ ( finite4001608067531595151d_enat @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite4001608067531595151d_enat @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2550_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_real > $o,A3: set_real] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_real] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) )
% 9.00/5.05 | ~ ( finite_finite_real @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite_finite_real @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2551_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_o > $o,A3: set_o] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_o] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) )
% 9.00/5.05 | ~ ( finite_finite_o @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite_finite_o @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2552_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_int > $o,A3: set_int] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_int] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) )
% 9.00/5.05 | ~ ( finite_finite_int @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite_finite_int @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2553_infinite__coinduct,axiom,
% 9.00/5.05 ! [X6: set_nat > $o,A3: set_nat] :
% 9.00/5.05 ( ( X6 @ A3 )
% 9.00/5.05 => ( ! [A5: set_nat] :
% 9.00/5.05 ( ( X6 @ A5 )
% 9.00/5.05 => ? [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ A5 )
% 9.00/5.05 & ( ( X6 @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) )
% 9.00/5.05 | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ) )
% 9.00/5.05 => ~ ( finite_finite_nat @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2554_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: set_nat,A5: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A5 )
% 9.00/5.05 => ( ( member_set_nat @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2555_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: complex,A5: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A5 )
% 9.00/5.05 => ( ( member_complex @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_complex ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2556_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A5 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2557_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: extended_enat,A5: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( ( member_Extended_enat @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ A @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bo7653980558646680370d_enat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2558_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: real,A5: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A5 )
% 9.00/5.05 => ( ( member_real @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2559_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: $o,A5: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A5 )
% 9.00/5.05 => ( ( member_o @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2560_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: int,A5: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A5 )
% 9.00/5.05 => ( ( member_int @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2561_finite__empty__induct,axiom,
% 9.00/5.05 ! [A3: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( P @ A3 )
% 9.00/5.05 => ( ! [A: nat,A5: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( ( member_nat @ A @ A5 )
% 9.00/5.05 => ( ( P @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
% 9.00/5.05 => ( P @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2562_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2563_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
% 9.00/5.05 ( ( ord_le6893508408891458716et_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2564_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real,B2: set_real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2565_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o,B2: set_o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2566_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2567_subset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int,B2: set_int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2568_Diff__single__insert,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 9.00/5.05 => ( ord_le3146513528884898305at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2569_Diff__single__insert,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real,B2: set_real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 )
% 9.00/5.05 => ( ord_less_eq_set_real @ A3 @ ( insert_real @ X2 @ B2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2570_Diff__single__insert,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o,B2: set_o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 )
% 9.00/5.05 => ( ord_less_eq_set_o @ A3 @ ( insert_o @ X2 @ B2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2571_Diff__single__insert,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 )
% 9.00/5.05 => ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2572_Diff__single__insert,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int,B2: set_int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 )
% 9.00/5.05 => ( ord_less_eq_set_int @ A3 @ ( insert_int @ X2 @ B2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2573_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: product_prod_nat_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2574_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_complex] :
% 9.00/5.05 ( ( ( finite_card_complex @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: complex] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_complex @ X5 @ bot_bot_set_complex ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2575_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_list_nat] :
% 9.00/5.05 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: list_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_list_nat @ X5 @ bot_bot_set_list_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2576_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_set_nat] :
% 9.00/5.05 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: set_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_set_nat @ X5 @ bot_bot_set_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2577_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_real] :
% 9.00/5.05 ( ( ( finite_card_real @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: real] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2578_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_o] :
% 9.00/5.05 ( ( ( finite_card_o @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: $o] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2579_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_nat] :
% 9.00/5.05 ( ( ( finite_card_nat @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2580_card__1__singletonE,axiom,
% 9.00/5.05 ! [A3: set_int] :
% 9.00/5.05 ( ( ( finite_card_int @ A3 )
% 9.00/5.05 = one_one_nat )
% 9.00/5.05 => ~ ! [X5: int] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2581_Compl__insert,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2582_Compl__insert,axiom,
% 9.00/5.05 ! [X2: real,A3: set_real] :
% 9.00/5.05 ( ( uminus612125837232591019t_real @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.05 = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A3 ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2583_Compl__insert,axiom,
% 9.00/5.05 ! [X2: $o,A3: set_o] :
% 9.00/5.05 ( ( uminus_uminus_set_o @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.05 = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A3 ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2584_Compl__insert,axiom,
% 9.00/5.05 ! [X2: int,A3: set_int] :
% 9.00/5.05 ( ( uminus1532241313380277803et_int @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.05 = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2585_Compl__insert,axiom,
% 9.00/5.05 ! [X2: nat,A3: set_nat] :
% 9.00/5.05 ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2586_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 9.00/5.05 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member8440522571783428010at_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite711546835091564841at_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2587_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_complex,K: nat] :
% 9.00/5.05 ( ( ( finite_card_complex @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: complex,B6: set_complex] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_complex @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_complex @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_complex @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_complex ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2588_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_list_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: list_nat,B6: set_list_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_list_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_list_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_list_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_list_nat ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2589_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: set_nat,B6: set_set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_set_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_set_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_set_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_set_nat ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2590_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_real,K: nat] :
% 9.00/5.05 ( ( ( finite_card_real @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: real,B6: set_real] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_real @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_real @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_real @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_real ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2591_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_o,K: nat] :
% 9.00/5.05 ( ( ( finite_card_o @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: $o,B6: set_o] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_o @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_o @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_o @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_o ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2592_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: nat,B6: set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_nat @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_nat @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_nat @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2593_card__Suc__eq,axiom,
% 9.00/5.05 ! [A3: set_int,K: nat] :
% 9.00/5.05 ( ( ( finite_card_int @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 = ( ? [B4: int,B6: set_int] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_int @ B4 @ B6 ) )
% 9.00/5.05 & ~ ( member_int @ B4 @ B6 )
% 9.00/5.05 & ( ( finite_card_int @ B6 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B6 = bot_bot_set_int ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2594_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,K: nat] :
% 9.00/5.05 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: product_prod_nat_nat,B8: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ B @ B8 ) )
% 9.00/5.05 & ~ ( member8440522571783428010at_nat @ B @ B8 )
% 9.00/5.05 & ( ( finite711546835091564841at_nat @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2595_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_complex,K: nat] :
% 9.00/5.05 ( ( ( finite_card_complex @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: complex,B8: set_complex] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_complex @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_complex @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_complex @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_complex ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2596_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_list_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: list_nat,B8: set_list_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_list_nat @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_list_nat @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_list_nat @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_list_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2597_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: set_nat,B8: set_set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_set_nat @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_set_nat @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_set_nat @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2598_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_real,K: nat] :
% 9.00/5.05 ( ( ( finite_card_real @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: real,B8: set_real] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_real @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_real @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_real @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2599_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_o,K: nat] :
% 9.00/5.05 ( ( ( finite_card_o @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: $o,B8: set_o] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_o @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_o @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_o @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2600_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_nat,K: nat] :
% 9.00/5.05 ( ( ( finite_card_nat @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: nat,B8: set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_nat @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_nat @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_nat @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2601_card__eq__SucD,axiom,
% 9.00/5.05 ! [A3: set_int,K: nat] :
% 9.00/5.05 ( ( ( finite_card_int @ A3 )
% 9.00/5.05 = ( suc @ K ) )
% 9.00/5.05 => ? [B: int,B8: set_int] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_int @ B @ B8 ) )
% 9.00/5.05 & ~ ( member_int @ B @ B8 )
% 9.00/5.05 & ( ( finite_card_int @ B8 )
% 9.00/5.05 = K )
% 9.00/5.05 & ( ( K = zero_zero_nat )
% 9.00/5.05 => ( B8 = bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2602_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: product_prod_nat_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2603_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_complex] :
% 9.00/5.05 ( ( ( finite_card_complex @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: complex] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2604_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_list_nat] :
% 9.00/5.05 ( ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: list_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2605_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_set_nat] :
% 9.00/5.05 ( ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: set_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2606_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_real] :
% 9.00/5.05 ( ( ( finite_card_real @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: real] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2607_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_o] :
% 9.00/5.05 ( ( ( finite_card_o @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: $o] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2608_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_nat] :
% 9.00/5.05 ( ( ( finite_card_nat @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2609_card__1__singleton__iff,axiom,
% 9.00/5.05 ! [A3: set_int] :
% 9.00/5.05 ( ( ( finite_card_int @ A3 )
% 9.00/5.05 = ( suc @ zero_zero_nat ) )
% 9.00/5.05 = ( ? [X: int] :
% 9.00/5.05 ( A3
% 9.00/5.05 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2610_remove__induct,axiom,
% 9.00/5.05 ! [P: set_set_nat > $o,B2: set_set_nat] :
% 9.00/5.05 ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_set_nat )
% 9.00/5.05 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2611_remove__induct,axiom,
% 9.00/5.05 ! [P: set_complex > $o,B2: set_complex] :
% 9.00/5.05 ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_complex )
% 9.00/5.05 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: complex] :
% 9.00/5.05 ( ( member_complex @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2612_remove__induct,axiom,
% 9.00/5.05 ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: product_prod_nat_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2613_remove__induct,axiom,
% 9.00/5.05 ! [P: set_Extended_enat > $o,B2: set_Extended_enat] :
% 9.00/5.05 ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ( ~ ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2614_remove__induct,axiom,
% 9.00/5.05 ! [P: set_real > $o,B2: set_real] :
% 9.00/5.05 ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ( ~ ( finite_finite_real @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_real )
% 9.00/5.05 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2615_remove__induct,axiom,
% 9.00/5.05 ! [P: set_o > $o,B2: set_o] :
% 9.00/5.05 ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ( ~ ( finite_finite_o @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_o )
% 9.00/5.05 => ( ( ord_less_eq_set_o @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2616_remove__induct,axiom,
% 9.00/5.05 ! [P: set_nat > $o,B2: set_nat] :
% 9.00/5.05 ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ( ~ ( finite_finite_nat @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_nat )
% 9.00/5.05 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2617_remove__induct,axiom,
% 9.00/5.05 ! [P: set_int > $o,B2: set_int] :
% 9.00/5.05 ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ( ~ ( finite_finite_int @ B2 )
% 9.00/5.05 => ( P @ B2 ) )
% 9.00/5.05 => ( ! [A5: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_int )
% 9.00/5.05 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2618_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_set_nat,P: set_set_nat > $o] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [A5: set_set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_set_nat )
% 9.00/5.05 => ( ( ord_le6893508408891458716et_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_2163939370556025621et_nat @ A5 @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2619_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [A5: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_complex )
% 9.00/5.05 => ( ( ord_le211207098394363844omplex @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: complex] :
% 9.00/5.05 ( ( member_complex @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_811609699411566653omplex @ A5 @ ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2620_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [A5: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ( ord_le3146513528884898305at_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: product_prod_nat_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_1356011639430497352at_nat @ A5 @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2621_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [A5: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ( ord_le7203529160286727270d_enat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_925952699566721837d_enat @ A5 @ ( insert_Extended_enat @ X4 @ bot_bo7653980558646680370d_enat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2622_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [A5: set_real] :
% 9.00/5.05 ( ( finite_finite_real @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_real )
% 9.00/5.05 => ( ( ord_less_eq_set_real @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_real @ A5 @ ( insert_real @ X4 @ bot_bot_set_real ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2623_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [A5: set_o] :
% 9.00/5.05 ( ( finite_finite_o @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_o )
% 9.00/5.05 => ( ( ord_less_eq_set_o @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_o @ A5 @ ( insert_o @ X4 @ bot_bot_set_o ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2624_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [A5: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_nat )
% 9.00/5.05 => ( ( ord_less_eq_set_nat @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_nat @ A5 @ ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2625_finite__remove__induct,axiom,
% 9.00/5.05 ! [B2: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ B2 )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [A5: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A5 )
% 9.00/5.05 => ( ( A5 != bot_bot_set_int )
% 9.00/5.05 => ( ( ord_less_eq_set_int @ A5 @ B2 )
% 9.00/5.05 => ( ! [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ A5 )
% 9.00/5.05 => ( P @ ( minus_minus_set_int @ A5 @ ( insert_int @ X4 @ bot_bot_set_int ) ) ) )
% 9.00/5.05 => ( P @ A5 ) ) ) ) )
% 9.00/5.05 => ( P @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2626_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_real] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A3 ) )
% 9.00/5.05 = ( ? [A4: real,B6: set_real] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_real @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_real @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_real @ B6 ) )
% 9.00/5.05 & ( finite_finite_real @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2627_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_o] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_o @ A3 ) )
% 9.00/5.05 = ( ? [A4: $o,B6: set_o] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_o @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_o @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_o @ B6 ) )
% 9.00/5.05 & ( finite_finite_o @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2628_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_list_nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A3 ) )
% 9.00/5.05 = ( ? [A4: list_nat,B6: set_list_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_list_nat @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_list_nat @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B6 ) )
% 9.00/5.05 & ( finite8100373058378681591st_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2629_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_set_nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A3 ) )
% 9.00/5.05 = ( ? [A4: set_nat,B6: set_set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_set_nat @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_set_nat @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B6 ) )
% 9.00/5.05 & ( finite1152437895449049373et_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2630_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A3 ) )
% 9.00/5.05 = ( ? [A4: nat,B6: set_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_nat @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_nat @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B6 ) )
% 9.00/5.05 & ( finite_finite_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2631_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_int] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A3 ) )
% 9.00/5.05 = ( ? [A4: int,B6: set_int] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_int @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_int @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_int @ B6 ) )
% 9.00/5.05 & ( finite_finite_int @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2632_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_complex] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A3 ) )
% 9.00/5.05 = ( ? [A4: complex,B6: set_complex] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_complex @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_complex @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B6 ) )
% 9.00/5.05 & ( finite3207457112153483333omplex @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2633_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite711546835091564841at_nat @ A3 ) )
% 9.00/5.05 = ( ? [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert8211810215607154385at_nat @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member8440522571783428010at_nat @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite711546835091564841at_nat @ B6 ) )
% 9.00/5.05 & ( finite6177210948735845034at_nat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2634_card__le__Suc__iff,axiom,
% 9.00/5.05 ! [N: nat,A3: set_Extended_enat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite121521170596916366d_enat @ A3 ) )
% 9.00/5.05 = ( ? [A4: extended_enat,B6: set_Extended_enat] :
% 9.00/5.05 ( ( A3
% 9.00/5.05 = ( insert_Extended_enat @ A4 @ B6 ) )
% 9.00/5.05 & ~ ( member_Extended_enat @ A4 @ B6 )
% 9.00/5.05 & ( ord_less_eq_nat @ N @ ( finite121521170596916366d_enat @ B6 ) )
% 9.00/5.05 & ( finite4001608067531595151d_enat @ B6 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2635_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2636_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2637_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2638_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2639_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2640_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] : ( ord_less_eq_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2641_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2642_card__Diff1__le,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2643_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_complex,P: set_complex > $o] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_complex )
% 9.00/5.05 => ( ! [T4: set_complex] :
% 9.00/5.05 ( ( ord_less_set_complex @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: complex] :
% 9.00/5.05 ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_complex @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2644_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ S )
% 9.00/5.05 => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.00/5.05 => ( ! [T4: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_le7866589430770878221at_nat @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: product_prod_nat_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ X4 @ ( minus_1356011639430497352at_nat @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert8211810215607154385at_nat @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2645_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_Extended_enat,P: set_Extended_enat > $o] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.05 => ( ( P @ bot_bo7653980558646680370d_enat )
% 9.00/5.05 => ( ! [T4: set_Extended_enat] :
% 9.00/5.05 ( ( ord_le2529575680413868914d_enat @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: extended_enat] :
% 9.00/5.05 ( ( member_Extended_enat @ X4 @ ( minus_925952699566721837d_enat @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_Extended_enat @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2646_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_real,P: set_real > $o] :
% 9.00/5.05 ( ( finite_finite_real @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_real )
% 9.00/5.05 => ( ! [T4: set_real] :
% 9.00/5.05 ( ( ord_less_set_real @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: real] :
% 9.00/5.05 ( ( member_real @ X4 @ ( minus_minus_set_real @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_real @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2647_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_o,P: set_o > $o] :
% 9.00/5.05 ( ( finite_finite_o @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_o )
% 9.00/5.05 => ( ! [T4: set_o] :
% 9.00/5.05 ( ( ord_less_set_o @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: $o] :
% 9.00/5.05 ( ( member_o @ X4 @ ( minus_minus_set_o @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_o @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2648_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_int,P: set_int > $o] :
% 9.00/5.05 ( ( finite_finite_int @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_int )
% 9.00/5.05 => ( ! [T4: set_int] :
% 9.00/5.05 ( ( ord_less_set_int @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: int] :
% 9.00/5.05 ( ( member_int @ X4 @ ( minus_minus_set_int @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_int @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2649_finite__induct__select,axiom,
% 9.00/5.05 ! [S: set_nat,P: set_nat > $o] :
% 9.00/5.05 ( ( finite_finite_nat @ S )
% 9.00/5.05 => ( ( P @ bot_bot_set_nat )
% 9.00/5.05 => ( ! [T4: set_nat] :
% 9.00/5.05 ( ( ord_less_set_nat @ T4 @ S )
% 9.00/5.05 => ( ( P @ T4 )
% 9.00/5.05 => ? [X4: nat] :
% 9.00/5.05 ( ( member_nat @ X4 @ ( minus_minus_set_nat @ S @ T4 ) )
% 9.00/5.05 & ( P @ ( insert_nat @ X4 @ T4 ) ) ) ) )
% 9.00/5.05 => ( P @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2650_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ord_le7866589430770878221at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.05 => ( ord_le7866589430770878221at_nat @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le3146513528884898305at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2651_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat,B2: set_set_nat] :
% 9.00/5.05 ( ( ord_less_set_set_nat @ A3 @ ( insert_set_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_set_nat @ X2 @ B2 )
% 9.00/5.05 => ( ord_less_set_set_nat @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_le6893508408891458716et_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2652_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real,B2: set_real] :
% 9.00/5.05 ( ( ord_less_set_real @ A3 @ ( insert_real @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_real @ X2 @ B2 )
% 9.00/5.05 => ( ord_less_set_real @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_real @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2653_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o,B2: set_o] :
% 9.00/5.05 ( ( ord_less_set_o @ A3 @ ( insert_o @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_o @ X2 @ B2 )
% 9.00/5.05 => ( ord_less_set_o @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_o @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2654_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat,B2: set_nat] :
% 9.00/5.05 ( ( ord_less_set_nat @ A3 @ ( insert_nat @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_nat @ X2 @ B2 )
% 9.00/5.05 => ( ord_less_set_nat @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2655_psubset__insert__iff,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int,B2: set_int] :
% 9.00/5.05 ( ( ord_less_set_int @ A3 @ ( insert_int @ X2 @ B2 ) )
% 9.00/5.05 = ( ( ( member_int @ X2 @ B2 )
% 9.00/5.05 => ( ord_less_set_int @ A3 @ B2 ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ B2 )
% 9.00/5.05 => ( ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ B2 ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2656_VEBT__internal_Onaive__member_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [A: $o,B: $o,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 9.00/5.05 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 9.00/5.05 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ X5 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2657_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2658_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2659_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2660_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ A3 )
% 9.00/5.05 = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2661_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite121521170596916366d_enat @ A3 )
% 9.00/5.05 = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2662_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2663_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2664_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2665_card_Oremove,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ A3 )
% 9.00/5.05 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2666_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( insert_list_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2667_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( insert_set_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2668_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2669_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2670_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( finite121521170596916366d_enat @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2671_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2672_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2673_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2674_card_Oinsert__remove,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.05 = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2675_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) )
% 9.00/5.05 = ( finite_card_list_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2676_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 9.00/5.05 = ( finite_card_set_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2677_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 9.00/5.05 = ( finite_card_complex @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2678_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.00/5.05 = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2679_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) )
% 9.00/5.05 = ( finite121521170596916366d_enat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2680_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 9.00/5.05 = ( finite_card_real @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2681_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) )
% 9.00/5.05 = ( finite_card_o @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2682_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 9.00/5.05 = ( finite_card_int @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2683_card__Suc__Diff1,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 9.00/5.05 = ( finite_card_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2684_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2685_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2686_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2687_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2688_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2689_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2690_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2691_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2692_card__Diff1__less,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2693_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat,Y3: list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 => ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( member_list_nat @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) @ ( insert_list_nat @ Y3 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2694_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat,Y3: set_nat] :
% 9.00/5.05 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) @ ( insert_set_nat @ Y3 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2695_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex,Y3: complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( member_complex @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) @ ( insert_complex @ Y3 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2696_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( member8440522571783428010at_nat @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) @ ( insert8211810215607154385at_nat @ Y3 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2697_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat,Y3: extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.05 => ( ( member_Extended_enat @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) @ ( insert_Extended_enat @ Y3 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2698_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real,Y3: real] :
% 9.00/5.05 ( ( finite_finite_real @ A3 )
% 9.00/5.05 => ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( member_real @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) @ ( insert_real @ Y3 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2699_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o,Y3: $o] :
% 9.00/5.05 ( ( finite_finite_o @ A3 )
% 9.00/5.05 => ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( member_o @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) @ ( insert_o @ Y3 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2700_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int,Y3: int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( member_int @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) @ ( insert_int @ Y3 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2701_card__Diff2__less,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat,Y3: nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( member_nat @ Y3 @ A3 )
% 9.00/5.05 => ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( insert_nat @ Y3 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2702_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A3 ) )
% 9.00/5.05 = ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.05 & ( member_list_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2703_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A3 ) )
% 9.00/5.05 = ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.05 & ( member_set_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2704_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_complex,X2: complex] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A3 ) )
% 9.00/5.05 = ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 & ( member_complex @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2705_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A3 ) )
% 9.00/5.05 = ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 & ( member8440522571783428010at_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2706_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat,X2: extended_enat] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) @ ( finite121521170596916366d_enat @ A3 ) )
% 9.00/5.05 = ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 & ( member_Extended_enat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2707_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_real,X2: real] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A3 ) )
% 9.00/5.05 = ( ( finite_finite_real @ A3 )
% 9.00/5.05 & ( member_real @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2708_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_o,X2: $o] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) @ ( finite_card_o @ A3 ) )
% 9.00/5.05 = ( ( finite_finite_o @ A3 )
% 9.00/5.05 & ( member_o @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2709_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_int,X2: int] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A3 ) )
% 9.00/5.05 = ( ( finite_finite_int @ A3 )
% 9.00/5.05 & ( member_int @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2710_card__Diff1__less__iff,axiom,
% 9.00/5.05 ! [A3: set_nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A3 ) )
% 9.00/5.05 = ( ( finite_finite_nat @ A3 )
% 9.00/5.05 & ( member_nat @ X2 @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2711_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2712_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: complex,A3: set_complex] :
% 9.00/5.05 ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2713_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: list_nat,A3: set_list_nat] :
% 9.00/5.05 ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2714_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: set_nat,A3: set_set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2715_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: real,A3: set_real] :
% 9.00/5.05 ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2716_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: $o,A3: set_o] :
% 9.00/5.05 ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2717_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: int,A3: set_int] :
% 9.00/5.05 ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2718_card__Diff__singleton,axiom,
% 9.00/5.05 ! [X2: nat,A3: set_nat] :
% 9.00/5.05 ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2719_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.05 = ( finite711546835091564841at_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2720_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: complex,A3: set_complex] :
% 9.00/5.05 ( ( ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 9.00/5.05 = ( finite_card_complex @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2721_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: list_nat,A3: set_list_nat] :
% 9.00/5.05 ( ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_list_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ ( insert_list_nat @ X2 @ bot_bot_set_list_nat ) ) )
% 9.00/5.05 = ( finite_card_list_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2722_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: set_nat,A3: set_set_nat] :
% 9.00/5.05 ( ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_set_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.05 = ( finite_card_set_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2723_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: real,A3: set_real] :
% 9.00/5.05 ( ( ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_real @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_real @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.05 = ( finite_card_real @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2724_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: $o,A3: set_o] :
% 9.00/5.05 ( ( ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_o @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_o @ ( minus_minus_set_o @ A3 @ ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.05 = ( finite_card_o @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2725_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: int,A3: set_int] :
% 9.00/5.05 ( ( ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.05 = ( finite_card_int @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2726_card__Diff__singleton__if,axiom,
% 9.00/5.05 ! [X2: nat,A3: set_nat] :
% 9.00/5.05 ( ( ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.05 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.05 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.05 = ( finite_card_nat @ A3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2727_enumerate__Suc_H,axiom,
% 9.00/5.05 ! [S: set_nat,N: nat] :
% 9.00/5.05 ( ( infini8530281810654367211te_nat @ S @ ( suc @ N ) )
% 9.00/5.05 = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S @ ( insert_nat @ ( infini8530281810654367211te_nat @ S @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2728_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2729_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_real,X2: real] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_real @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_real @ ( insert_real @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2730_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_o,X2: $o] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_o @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_o @ ( insert_o @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2731_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_complex,X2: complex] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2732_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_list_nat,X2: list_nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2733_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_set_nat,X2: set_nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2734_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2735_card__insert__le__m1,axiom,
% 9.00/5.05 ! [N: nat,Y3: set_int,X2: int] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( finite_card_int @ Y3 ) @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.05 => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X2 @ Y3 ) ) @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2736_vebt__pred_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
% 9.00/5.05 => ( ! [A: $o,Uw2: $o] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) )
% 9.00/5.05 => ( ! [A: $o,B: $o,Va2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) )
% 9.00/5.05 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT,Vb2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Vb2 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2737_vebt__succ_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [Uu2: $o,B: $o] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) )
% 9.00/5.05 => ( ! [Uv2: $o,Uw2: $o,N3: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) )
% 9.00/5.05 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va3: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va3 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2738_VEBT__internal_Omembermima_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 9.00/5.05 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 9.00/5.05 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
% 9.00/5.05 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X5 ) )
% 9.00/5.05 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X5 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2739_vebt__member_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [A: $o,B: $o,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 9.00/5.05 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X5 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
% 9.00/5.05 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2740_vebt__delete_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [A: $o,B: $o] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) )
% 9.00/5.05 => ( ! [A: $o,B: $o] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) )
% 9.00/5.05 => ( ! [A: $o,B: $o,N3: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) )
% 9.00/5.05 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
% 9.00/5.05 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X5 ) )
% 9.00/5.05 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X5 ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2741_vebt__insert_Ocases,axiom,
% 9.00/5.05 ! [X2: produc9072475918466114483BT_nat] :
% 9.00/5.05 ( ! [A: $o,B: $o,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ X5 ) )
% 9.00/5.05 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ X5 ) )
% 9.00/5.05 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ X5 ) )
% 9.00/5.05 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ X5 ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X5: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ X5 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2742_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2743_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2744_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2745_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2746_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2747_List_Ofinite__set,axiom,
% 9.00/5.05 ! [Xs: list_Extended_enat] : ( finite4001608067531595151d_enat @ ( set_Extended_enat2 @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2748_Bolzano,axiom,
% 9.00/5.05 ! [A2: real,B3: real,P: real > real > $o] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.05 => ( ! [A: real,B: real,C3: real] :
% 9.00/5.05 ( ( P @ A @ B )
% 9.00/5.05 => ( ( P @ B @ C3 )
% 9.00/5.05 => ( ( ord_less_eq_real @ A @ B )
% 9.00/5.05 => ( ( ord_less_eq_real @ B @ C3 )
% 9.00/5.05 => ( P @ A @ C3 ) ) ) ) )
% 9.00/5.05 => ( ! [X5: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.05 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.05 => ? [D3: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.00/5.05 & ! [A: real,B: real] :
% 9.00/5.05 ( ( ( ord_less_eq_real @ A @ X5 )
% 9.00/5.05 & ( ord_less_eq_real @ X5 @ B )
% 9.00/5.05 & ( ord_less_real @ ( minus_minus_real @ B @ A ) @ D3 ) )
% 9.00/5.05 => ( P @ A @ B ) ) ) ) )
% 9.00/5.05 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2749_card__length,axiom,
% 9.00/5.05 ! [Xs: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs ) ) @ ( size_s3451745648224563538omplex @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2750_card__length,axiom,
% 9.00/5.05 ! [Xs: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs ) ) @ ( size_s3023201423986296836st_nat @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2751_card__length,axiom,
% 9.00/5.05 ! [Xs: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2752_card__length,axiom,
% 9.00/5.05 ! [Xs: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2753_card__length,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2754_card__length,axiom,
% 9.00/5.05 ! [Xs: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2755_card__length,axiom,
% 9.00/5.05 ! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2756_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: real,Xs: list_real] :
% 9.00/5.05 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2757_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: set_nat,Xs: list_set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2758_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: int,Xs: list_int] :
% 9.00/5.05 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2759_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2760_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: $o,Xs: list_o] :
% 9.00/5.05 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2761_length__pos__if__in__set,axiom,
% 9.00/5.05 ! [X2: nat,Xs: list_nat] :
% 9.00/5.05 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2762_the__elem__eq,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat] :
% 9.00/5.05 ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.05 = X2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2763_the__elem__eq,axiom,
% 9.00/5.05 ! [X2: real] :
% 9.00/5.05 ( ( the_elem_real @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.05 = X2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2764_the__elem__eq,axiom,
% 9.00/5.05 ! [X2: $o] :
% 9.00/5.05 ( ( the_elem_o @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.05 = X2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2765_the__elem__eq,axiom,
% 9.00/5.05 ! [X2: nat] :
% 9.00/5.05 ( ( the_elem_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.05 = X2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2766_the__elem__eq,axiom,
% 9.00/5.05 ! [X2: int] :
% 9.00/5.05 ( ( the_elem_int @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.05 = X2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2767_case4_I1_J,axiom,
% 9.00/5.05 ! [X4: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
% 9.00/5.05 => ( ( vEBT_invar_vebt @ X4 @ na )
% 9.00/5.05 & ! [Xa: vEBT_VEBT] :
% 9.00/5.05 ( ( vEBT_invar_vebt @ Xa @ na )
% 9.00/5.05 => ( ( ( vEBT_VEBT_set_vebt @ X4 )
% 9.00/5.05 = ( vEBT_VEBT_set_vebt @ Xa ) )
% 9.00/5.05 => ( Xa = X4 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2768_vebt__maxt_Opelims,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 9.00/5.05 ( ( ( vEBT_vebt_maxt @ X2 )
% 9.00/5.05 = Y3 )
% 9.00/5.05 => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X2 )
% 9.00/5.05 => ( ! [A: $o,B: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.05 => ( ( ( B
% 9.00/5.05 => ( Y3
% 9.00/5.05 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ B
% 9.00/5.05 => ( ( A
% 9.00/5.05 => ( Y3
% 9.00/5.05 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.05 & ( ~ A
% 9.00/5.05 => ( Y3 = none_nat ) ) ) ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
% 9.00/5.05 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.05 => ( ( Y3 = none_nat )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.05 => ( ( Y3
% 9.00/5.05 = ( some_nat @ Ma2 ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2769_vebt__mint_Opelims,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Y3: option_nat] :
% 9.00/5.05 ( ( ( vEBT_vebt_mint @ X2 )
% 9.00/5.05 = Y3 )
% 9.00/5.05 => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X2 )
% 9.00/5.05 => ( ! [A: $o,B: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.05 => ( ( ( A
% 9.00/5.05 => ( Y3
% 9.00/5.05 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.05 & ( ~ A
% 9.00/5.05 => ( ( B
% 9.00/5.05 => ( Y3
% 9.00/5.05 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.05 & ( ~ B
% 9.00/5.05 => ( Y3 = none_nat ) ) ) ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A @ B ) ) ) )
% 9.00/5.05 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.05 => ( ( Y3 = none_nat )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.00/5.05 => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.05 => ( ( Y3
% 9.00/5.05 = ( some_nat @ Mi2 ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2770_is__singletonI,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2771_is__singletonI,axiom,
% 9.00/5.05 ! [X2: real] : ( is_singleton_real @ ( insert_real @ X2 @ bot_bot_set_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2772_is__singletonI,axiom,
% 9.00/5.05 ! [X2: $o] : ( is_singleton_o @ ( insert_o @ X2 @ bot_bot_set_o ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2773_is__singletonI,axiom,
% 9.00/5.05 ! [X2: nat] : ( is_singleton_nat @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2774_is__singletonI,axiom,
% 9.00/5.05 ! [X2: int] : ( is_singleton_int @ ( insert_int @ X2 @ bot_bot_set_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2775_case4_I5_J,axiom,
% 9.00/5.05 ( m
% 9.00/5.05 = ( suc @ na ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2776_is__singleton__the__elem,axiom,
% 9.00/5.05 ( is_sin2850979758926227957at_nat
% 9.00/5.05 = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A6 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2777_is__singleton__the__elem,axiom,
% 9.00/5.05 ( is_singleton_real
% 9.00/5.05 = ( ^ [A6: set_real] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_real @ ( the_elem_real @ A6 ) @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2778_is__singleton__the__elem,axiom,
% 9.00/5.05 ( is_singleton_o
% 9.00/5.05 = ( ^ [A6: set_o] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_o @ ( the_elem_o @ A6 ) @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2779_is__singleton__the__elem,axiom,
% 9.00/5.05 ( is_singleton_nat
% 9.00/5.05 = ( ^ [A6: set_nat] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_nat @ ( the_elem_nat @ A6 ) @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2780_is__singleton__the__elem,axiom,
% 9.00/5.05 ( is_singleton_int
% 9.00/5.05 = ( ^ [A6: set_int] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_int @ ( the_elem_int @ A6 ) @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2781_is__singletonI_H,axiom,
% 9.00/5.05 ! [A3: set_set_nat] :
% 9.00/5.05 ( ( A3 != bot_bot_set_set_nat )
% 9.00/5.05 => ( ! [X5: set_nat,Y4: set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.05 => ( ( member_set_nat @ Y4 @ A3 )
% 9.00/5.05 => ( X5 = Y4 ) ) )
% 9.00/5.05 => ( is_singleton_set_nat @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2782_is__singletonI_H,axiom,
% 9.00/5.05 ! [A3: set_real] :
% 9.00/5.05 ( ( A3 != bot_bot_set_real )
% 9.00/5.05 => ( ! [X5: real,Y4: real] :
% 9.00/5.05 ( ( member_real @ X5 @ A3 )
% 9.00/5.05 => ( ( member_real @ Y4 @ A3 )
% 9.00/5.05 => ( X5 = Y4 ) ) )
% 9.00/5.05 => ( is_singleton_real @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2783_is__singletonI_H,axiom,
% 9.00/5.05 ! [A3: set_o] :
% 9.00/5.05 ( ( A3 != bot_bot_set_o )
% 9.00/5.05 => ( ! [X5: $o,Y4: $o] :
% 9.00/5.05 ( ( member_o @ X5 @ A3 )
% 9.00/5.05 => ( ( member_o @ Y4 @ A3 )
% 9.00/5.05 => ( X5 = Y4 ) ) )
% 9.00/5.05 => ( is_singleton_o @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2784_is__singletonI_H,axiom,
% 9.00/5.05 ! [A3: set_nat] :
% 9.00/5.05 ( ( A3 != bot_bot_set_nat )
% 9.00/5.05 => ( ! [X5: nat,Y4: nat] :
% 9.00/5.05 ( ( member_nat @ X5 @ A3 )
% 9.00/5.05 => ( ( member_nat @ Y4 @ A3 )
% 9.00/5.05 => ( X5 = Y4 ) ) )
% 9.00/5.05 => ( is_singleton_nat @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2785_is__singletonI_H,axiom,
% 9.00/5.05 ! [A3: set_int] :
% 9.00/5.05 ( ( A3 != bot_bot_set_int )
% 9.00/5.05 => ( ! [X5: int,Y4: int] :
% 9.00/5.05 ( ( member_int @ X5 @ A3 )
% 9.00/5.05 => ( ( member_int @ Y4 @ A3 )
% 9.00/5.05 => ( X5 = Y4 ) ) )
% 9.00/5.05 => ( is_singleton_int @ A3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2786_length__induct,axiom,
% 9.00/5.05 ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ! [Xs3: list_VEBT_VEBT] :
% 9.00/5.05 ( ! [Ys: list_VEBT_VEBT] :
% 9.00/5.05 ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 9.00/5.05 => ( P @ Ys ) )
% 9.00/5.05 => ( P @ Xs3 ) )
% 9.00/5.05 => ( P @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2787_length__induct,axiom,
% 9.00/5.05 ! [P: list_o > $o,Xs: list_o] :
% 9.00/5.05 ( ! [Xs3: list_o] :
% 9.00/5.05 ( ! [Ys: list_o] :
% 9.00/5.05 ( ( ord_less_nat @ ( size_size_list_o @ Ys ) @ ( size_size_list_o @ Xs3 ) )
% 9.00/5.05 => ( P @ Ys ) )
% 9.00/5.05 => ( P @ Xs3 ) )
% 9.00/5.05 => ( P @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2788_length__induct,axiom,
% 9.00/5.05 ! [P: list_nat > $o,Xs: list_nat] :
% 9.00/5.05 ( ! [Xs3: list_nat] :
% 9.00/5.05 ( ! [Ys: list_nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( size_size_list_nat @ Ys ) @ ( size_size_list_nat @ Xs3 ) )
% 9.00/5.05 => ( P @ Ys ) )
% 9.00/5.05 => ( P @ Xs3 ) )
% 9.00/5.05 => ( P @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2789_finite__maxlen,axiom,
% 9.00/5.05 ! [M5: set_list_VEBT_VEBT] :
% 9.00/5.05 ( ( finite3004134309566078307T_VEBT @ M5 )
% 9.00/5.05 => ? [N3: nat] :
% 9.00/5.05 ! [X4: list_VEBT_VEBT] :
% 9.00/5.05 ( ( member2936631157270082147T_VEBT @ X4 @ M5 )
% 9.00/5.05 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X4 ) @ N3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2790_finite__maxlen,axiom,
% 9.00/5.05 ! [M5: set_list_o] :
% 9.00/5.05 ( ( finite_finite_list_o @ M5 )
% 9.00/5.05 => ? [N3: nat] :
% 9.00/5.05 ! [X4: list_o] :
% 9.00/5.05 ( ( member_list_o @ X4 @ M5 )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_o @ X4 ) @ N3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2791_finite__maxlen,axiom,
% 9.00/5.05 ! [M5: set_list_nat] :
% 9.00/5.05 ( ( finite8100373058378681591st_nat @ M5 )
% 9.00/5.05 => ? [N3: nat] :
% 9.00/5.05 ! [X4: list_nat] :
% 9.00/5.05 ( ( member_list_nat @ X4 @ M5 )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_nat @ X4 ) @ N3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2792_is__singleton__def,axiom,
% 9.00/5.05 ( is_sin2850979758926227957at_nat
% 9.00/5.05 = ( ^ [A6: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ? [X: product_prod_nat_nat] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2793_is__singleton__def,axiom,
% 9.00/5.05 ( is_singleton_real
% 9.00/5.05 = ( ^ [A6: set_real] :
% 9.00/5.05 ? [X: real] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2794_is__singleton__def,axiom,
% 9.00/5.05 ( is_singleton_o
% 9.00/5.05 = ( ^ [A6: set_o] :
% 9.00/5.05 ? [X: $o] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2795_is__singleton__def,axiom,
% 9.00/5.05 ( is_singleton_nat
% 9.00/5.05 = ( ^ [A6: set_nat] :
% 9.00/5.05 ? [X: nat] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2796_is__singleton__def,axiom,
% 9.00/5.05 ( is_singleton_int
% 9.00/5.05 = ( ^ [A6: set_int] :
% 9.00/5.05 ? [X: int] :
% 9.00/5.05 ( A6
% 9.00/5.05 = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2797_is__singletonE,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( is_sin2850979758926227957at_nat @ A3 )
% 9.00/5.05 => ~ ! [X5: product_prod_nat_nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2798_is__singletonE,axiom,
% 9.00/5.05 ! [A3: set_real] :
% 9.00/5.05 ( ( is_singleton_real @ A3 )
% 9.00/5.05 => ~ ! [X5: real] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_real @ X5 @ bot_bot_set_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2799_is__singletonE,axiom,
% 9.00/5.05 ! [A3: set_o] :
% 9.00/5.05 ( ( is_singleton_o @ A3 )
% 9.00/5.05 => ~ ! [X5: $o] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2800_is__singletonE,axiom,
% 9.00/5.05 ! [A3: set_nat] :
% 9.00/5.05 ( ( is_singleton_nat @ A3 )
% 9.00/5.05 => ~ ! [X5: nat] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2801_is__singletonE,axiom,
% 9.00/5.05 ! [A3: set_int] :
% 9.00/5.05 ( ( is_singleton_int @ A3 )
% 9.00/5.05 => ~ ! [X5: int] :
% 9.00/5.05 ( A3
% 9.00/5.05 != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2802_is__singleton__altdef,axiom,
% 9.00/5.05 ( is_singleton_complex
% 9.00/5.05 = ( ^ [A6: set_complex] :
% 9.00/5.05 ( ( finite_card_complex @ A6 )
% 9.00/5.05 = one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2803_is__singleton__altdef,axiom,
% 9.00/5.05 ( is_sin2641923865335537900st_nat
% 9.00/5.05 = ( ^ [A6: set_list_nat] :
% 9.00/5.05 ( ( finite_card_list_nat @ A6 )
% 9.00/5.05 = one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2804_is__singleton__altdef,axiom,
% 9.00/5.05 ( is_singleton_set_nat
% 9.00/5.05 = ( ^ [A6: set_set_nat] :
% 9.00/5.05 ( ( finite_card_set_nat @ A6 )
% 9.00/5.05 = one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2805_is__singleton__altdef,axiom,
% 9.00/5.05 ( is_singleton_nat
% 9.00/5.05 = ( ^ [A6: set_nat] :
% 9.00/5.05 ( ( finite_card_nat @ A6 )
% 9.00/5.05 = one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2806_is__singleton__altdef,axiom,
% 9.00/5.05 ( is_singleton_int
% 9.00/5.05 = ( ^ [A6: set_int] :
% 9.00/5.05 ( ( finite_card_int @ A6 )
% 9.00/5.05 = one_one_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2807_finite__list,axiom,
% 9.00/5.05 ! [A3: set_VEBT_VEBT] :
% 9.00/5.05 ( ( finite5795047828879050333T_VEBT @ A3 )
% 9.00/5.05 => ? [Xs3: list_VEBT_VEBT] :
% 9.00/5.05 ( ( set_VEBT_VEBT2 @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2808_finite__list,axiom,
% 9.00/5.05 ! [A3: set_nat] :
% 9.00/5.05 ( ( finite_finite_nat @ A3 )
% 9.00/5.05 => ? [Xs3: list_nat] :
% 9.00/5.05 ( ( set_nat2 @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2809_finite__list,axiom,
% 9.00/5.05 ! [A3: set_int] :
% 9.00/5.05 ( ( finite_finite_int @ A3 )
% 9.00/5.05 => ? [Xs3: list_int] :
% 9.00/5.05 ( ( set_int2 @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2810_finite__list,axiom,
% 9.00/5.05 ! [A3: set_complex] :
% 9.00/5.05 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.05 => ? [Xs3: list_complex] :
% 9.00/5.05 ( ( set_complex2 @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2811_finite__list,axiom,
% 9.00/5.05 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.05 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.05 => ? [Xs3: list_P6011104703257516679at_nat] :
% 9.00/5.05 ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2812_finite__list,axiom,
% 9.00/5.05 ! [A3: set_Extended_enat] :
% 9.00/5.05 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.05 => ? [Xs3: list_Extended_enat] :
% 9.00/5.05 ( ( set_Extended_enat2 @ Xs3 )
% 9.00/5.05 = A3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2813_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_real,B2: set_real] :
% 9.00/5.05 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: real] :
% 9.00/5.05 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.00/5.05 => ( member_real @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2814_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_o,B2: set_o] :
% 9.00/5.05 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: $o] :
% 9.00/5.05 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( member_o @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2815_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_set_nat,B2: set_set_nat] :
% 9.00/5.05 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 => ( member_set_nat @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2816_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.00/5.05 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2817_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_nat,B2: set_nat] :
% 9.00/5.05 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: nat] :
% 9.00/5.05 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( member_nat @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2818_subset__code_I1_J,axiom,
% 9.00/5.05 ! [Xs: list_int,B2: set_int] :
% 9.00/5.05 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 9.00/5.05 = ( ! [X: int] :
% 9.00/5.05 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( member_int @ X @ B2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2819_case4_I8_J,axiom,
% 9.00/5.05 ( ( mi = ma )
% 9.00/5.05 => ! [X4: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList2 ) )
% 9.00/5.05 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2820_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Y3: $o] :
% 9.00/5.05 ( ( ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.05 = Y3 )
% 9.00/5.05 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 9.00/5.05 => ( ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.05 => ( Y3
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 9.00/5.05 => ( ! [Uv2: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.00/5.05 => ( ~ Y3
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 9.00/5.05 => ( ! [Uu2: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.00/5.05 => ( ~ Y3
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 9.00/5.05 => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.00/5.05 => ( Y3
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 9.00/5.05 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.00/5.05 => ( ~ Y3
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2821_case4_I13_J,axiom,
% 9.00/5.05 ( ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) )
% 9.00/5.05 = ( vEBT_VEBT_set_vebt @ sa ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2822_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT] :
% 9.00/5.05 ( ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.05 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 9.00/5.05 => ( ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 9.00/5.05 => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2823_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT] :
% 9.00/5.05 ( ~ ( vEBT_VEBT_minNull @ X2 )
% 9.00/5.05 => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X2 )
% 9.00/5.05 => ( ! [Uv2: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 9.00/5.05 => ( ! [Uu2: $o] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 9.00/5.05 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.05 ( ( X2
% 9.00/5.05 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.00/5.05 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2824_int__power__div__base,axiom,
% 9.00/5.05 ! [M: nat,K: int] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.05 => ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.05 => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 9.00/5.05 = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2825_set__removeAll,axiom,
% 9.00/5.05 ! [X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.00/5.05 ( ( set_Pr5648618587558075414at_nat @ ( remove3673390508374433037at_nat @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2826_set__removeAll,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ( set_VEBT_VEBT2 @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( insert_VEBT_VEBT @ X2 @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2827_set__removeAll,axiom,
% 9.00/5.05 ! [X2: real,Xs: list_real] :
% 9.00/5.05 ( ( set_real2 @ ( removeAll_real @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_minus_set_real @ ( set_real2 @ Xs ) @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2828_set__removeAll,axiom,
% 9.00/5.05 ! [X2: $o,Xs: list_o] :
% 9.00/5.05 ( ( set_o2 @ ( removeAll_o @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_minus_set_o @ ( set_o2 @ Xs ) @ ( insert_o @ X2 @ bot_bot_set_o ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2829_set__removeAll,axiom,
% 9.00/5.05 ! [X2: int,Xs: list_int] :
% 9.00/5.05 ( ( set_int2 @ ( removeAll_int @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_minus_set_int @ ( set_int2 @ Xs ) @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2830_set__removeAll,axiom,
% 9.00/5.05 ! [X2: nat,Xs: list_nat] :
% 9.00/5.05 ( ( set_nat2 @ ( removeAll_nat @ X2 @ Xs ) )
% 9.00/5.05 = ( minus_minus_set_nat @ ( set_nat2 @ Xs ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2831_inthall,axiom,
% 9.00/5.05 ! [Xs: list_real,P: real > $o,N: nat] :
% 9.00/5.05 ( ! [X5: real] :
% 9.00/5.05 ( ( member_real @ X5 @ ( set_real2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_real @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2832_inthall,axiom,
% 9.00/5.05 ! [Xs: list_set_nat,P: set_nat > $o,N: nat] :
% 9.00/5.05 ( ! [X5: set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X5 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_set_nat @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2833_inthall,axiom,
% 9.00/5.05 ! [Xs: list_int,P: int > $o,N: nat] :
% 9.00/5.05 ( ! [X5: int] :
% 9.00/5.05 ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2834_inthall,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N: nat] :
% 9.00/5.05 ( ! [X5: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2835_inthall,axiom,
% 9.00/5.05 ! [Xs: list_o,P: $o > $o,N: nat] :
% 9.00/5.05 ( ! [X5: $o] :
% 9.00/5.05 ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2836_inthall,axiom,
% 9.00/5.05 ! [Xs: list_nat,P: nat > $o,N: nat] :
% 9.00/5.05 ( ! [X5: nat] :
% 9.00/5.05 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2837_nat__ivt__aux,axiom,
% 9.00/5.05 ! [N: nat,F: nat > int,K: int] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ N )
% 9.00/5.05 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.00/5.05 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.00/5.05 => ? [I2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ I2 @ N )
% 9.00/5.05 & ( ( F @ I2 )
% 9.00/5.05 = K ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2838_case4_I9_J,axiom,
% 9.00/5.05 ord_less_eq_nat @ mi @ ma ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2839_case4_I3_J,axiom,
% 9.00/5.05 vEBT_invar_vebt @ summary2 @ m ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2840_abs__idempotent,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
% 9.00/5.05 = ( abs_abs_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2841_abs__idempotent,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( abs_abs_real @ ( abs_abs_real @ A2 ) )
% 9.00/5.05 = ( abs_abs_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2842_abs__idempotent,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( abs_abs_rat @ ( abs_abs_rat @ A2 ) )
% 9.00/5.05 = ( abs_abs_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2843_abs__idempotent,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A2 ) )
% 9.00/5.05 = ( abs_abs_Code_integer @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2844_case4_I2_J,axiom,
% 9.00/5.05 ! [S2: vEBT_VEBT] :
% 9.00/5.05 ( ( vEBT_invar_vebt @ S2 @ m )
% 9.00/5.05 => ( ( ( vEBT_VEBT_set_vebt @ summary2 )
% 9.00/5.05 = ( vEBT_VEBT_set_vebt @ S2 ) )
% 9.00/5.05 => ( S2 = summary2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2845_abs__0,axiom,
% 9.00/5.05 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.00/5.05 = zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2846_abs__0,axiom,
% 9.00/5.05 ( ( abs_abs_real @ zero_zero_real )
% 9.00/5.05 = zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2847_abs__0,axiom,
% 9.00/5.05 ( ( abs_abs_rat @ zero_zero_rat )
% 9.00/5.05 = zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2848_abs__0,axiom,
% 9.00/5.05 ( ( abs_abs_int @ zero_zero_int )
% 9.00/5.05 = zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2849_abs__0__eq,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( zero_z3403309356797280102nteger
% 9.00/5.05 = ( abs_abs_Code_integer @ A2 ) )
% 9.00/5.05 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2850_abs__0__eq,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( zero_zero_real
% 9.00/5.05 = ( abs_abs_real @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2851_abs__0__eq,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( zero_zero_rat
% 9.00/5.05 = ( abs_abs_rat @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2852_abs__0__eq,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( zero_zero_int
% 9.00/5.05 = ( abs_abs_int @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2853_abs__eq__0,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = zero_z3403309356797280102nteger )
% 9.00/5.05 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2854_abs__eq__0,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ( abs_abs_real @ A2 )
% 9.00/5.05 = zero_zero_real )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2855_abs__eq__0,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = zero_zero_rat )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2856_abs__eq__0,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ( abs_abs_int @ A2 )
% 9.00/5.05 = zero_zero_int )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2857_abs__zero,axiom,
% 9.00/5.05 ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.00/5.05 = zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2858_abs__zero,axiom,
% 9.00/5.05 ( ( abs_abs_real @ zero_zero_real )
% 9.00/5.05 = zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2859_abs__zero,axiom,
% 9.00/5.05 ( ( abs_abs_rat @ zero_zero_rat )
% 9.00/5.05 = zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2860_abs__zero,axiom,
% 9.00/5.05 ( ( abs_abs_int @ zero_zero_int )
% 9.00/5.05 = zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2861_abs__minus__cancel,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.05 = ( abs_abs_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2862_abs__minus__cancel,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( abs_abs_real @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.05 = ( abs_abs_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2863_abs__minus__cancel,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( abs_abs_rat @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.05 = ( abs_abs_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2864_abs__minus__cancel,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.05 = ( abs_abs_Code_integer @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2865_abs__of__nat,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
% 9.00/5.05 = ( semiri681578069525770553at_rat @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2866_abs__of__nat,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
% 9.00/5.05 = ( semiri4939895301339042750nteger @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2867_abs__of__nat,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.05 = ( semiri1314217659103216013at_int @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2868_abs__of__nat,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.05 = ( semiri5074537144036343181t_real @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2869_abs__le__zero__iff,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
% 9.00/5.05 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2870_abs__le__zero__iff,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ zero_zero_real )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2871_abs__le__zero__iff,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2872_abs__le__zero__iff,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2873_abs__le__self__iff,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ A2 )
% 9.00/5.05 = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2874_abs__le__self__iff,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ A2 )
% 9.00/5.05 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2875_abs__le__self__iff,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ A2 )
% 9.00/5.05 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2876_abs__le__self__iff,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
% 9.00/5.05 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2877_abs__of__nonneg,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.05 => ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2878_abs__of__nonneg,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.05 => ( ( abs_abs_real @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2879_abs__of__nonneg,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 => ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2880_abs__of__nonneg,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.05 => ( ( abs_abs_int @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2881_zero__less__abs__iff,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) )
% 9.00/5.05 = ( A2 != zero_z3403309356797280102nteger ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2882_zero__less__abs__iff,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A2 ) )
% 9.00/5.05 = ( A2 != zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2883_zero__less__abs__iff,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) )
% 9.00/5.05 = ( A2 != zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2884_zero__less__abs__iff,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
% 9.00/5.05 = ( A2 != zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2885_zero__le__divide__abs__iff,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.05 | ( B3 = zero_zero_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2886_zero__le__divide__abs__iff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) )
% 9.00/5.05 = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 | ( B3 = zero_zero_rat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2887_divide__le__0__abs__iff,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( divide_divide_real @ A2 @ ( abs_abs_real @ B3 ) ) @ zero_zero_real )
% 9.00/5.05 = ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.05 | ( B3 = zero_zero_real ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2888_divide__le__0__abs__iff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( divide_divide_rat @ A2 @ ( abs_abs_rat @ B3 ) ) @ zero_zero_rat )
% 9.00/5.05 = ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.05 | ( B3 = zero_zero_rat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2889_abs__of__nonpos,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.05 => ( ( abs_abs_real @ A2 )
% 9.00/5.05 = ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2890_abs__of__nonpos,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger )
% 9.00/5.05 => ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2891_abs__of__nonpos,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.05 => ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2892_abs__of__nonpos,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.05 => ( ( abs_abs_int @ A2 )
% 9.00/5.05 = ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2893_zabs__less__one__iff,axiom,
% 9.00/5.05 ! [Z: int] :
% 9.00/5.05 ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
% 9.00/5.05 = ( Z = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2894_aa,axiom,
% 9.00/5.05 ord_less_eq_set_nat @ ( insert_nat @ mi @ ( insert_nat @ ma @ bot_bot_set_nat ) ) @ ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2895_case4_I6_J,axiom,
% 9.00/5.05 ( deg
% 9.00/5.05 = ( plus_plus_nat @ na @ m ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2896_abs__le__D1,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2897_abs__le__D1,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_le3102999989581377725nteger @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2898_abs__le__D1,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2899_abs__le__D1,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2900_abs__ge__self,axiom,
% 9.00/5.05 ! [A2: real] : ( ord_less_eq_real @ A2 @ ( abs_abs_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2901_abs__ge__self,axiom,
% 9.00/5.05 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ A2 @ ( abs_abs_Code_integer @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2902_abs__ge__self,axiom,
% 9.00/5.05 ! [A2: rat] : ( ord_less_eq_rat @ A2 @ ( abs_abs_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2903_abs__ge__self,axiom,
% 9.00/5.05 ! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2904_abs__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = zero_z3403309356797280102nteger )
% 9.00/5.05 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2905_abs__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ( abs_abs_real @ A2 )
% 9.00/5.05 = zero_zero_real )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2906_abs__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = zero_zero_rat )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2907_abs__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ( abs_abs_int @ A2 )
% 9.00/5.05 = zero_zero_int )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2908_abs__minus__commute,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) )
% 9.00/5.05 = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2909_abs__minus__commute,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.05 = ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2910_abs__minus__commute,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2911_abs__minus__commute,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.05 = ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2912_infinite__int__iff__unbounded__le,axiom,
% 9.00/5.05 ! [S: set_int] :
% 9.00/5.05 ( ( ~ ( finite_finite_int @ S ) )
% 9.00/5.05 = ( ! [M2: int] :
% 9.00/5.05 ? [N2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N2 ) )
% 9.00/5.05 & ( member_int @ N2 @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2913_infinite__int__iff__unbounded,axiom,
% 9.00/5.05 ! [S: set_int] :
% 9.00/5.05 ( ( ~ ( finite_finite_int @ S ) )
% 9.00/5.05 = ( ! [M2: int] :
% 9.00/5.05 ? [N2: int] :
% 9.00/5.05 ( ( ord_less_int @ M2 @ ( abs_abs_int @ N2 ) )
% 9.00/5.05 & ( member_int @ N2 @ S ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2914_abs__ge__zero,axiom,
% 9.00/5.05 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2915_abs__ge__zero,axiom,
% 9.00/5.05 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2916_abs__ge__zero,axiom,
% 9.00/5.05 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2917_abs__ge__zero,axiom,
% 9.00/5.05 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2918_abs__of__pos,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.05 => ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2919_abs__of__pos,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.05 => ( ( abs_abs_real @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2920_abs__of__pos,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 => ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2921_abs__of__pos,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.05 => ( ( abs_abs_int @ A2 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2922_abs__not__less__zero,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2923_abs__not__less__zero,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ~ ( ord_less_real @ ( abs_abs_real @ A2 ) @ zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2924_abs__not__less__zero,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ~ ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2925_abs__not__less__zero,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2926_abs__triangle__ineq2__sym,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2927_abs__triangle__ineq2__sym,axiom,
% 9.00/5.05 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2928_abs__triangle__ineq2__sym,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2929_abs__triangle__ineq2__sym,axiom,
% 9.00/5.05 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2930_abs__triangle__ineq3,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2931_abs__triangle__ineq3,axiom,
% 9.00/5.05 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2932_abs__triangle__ineq3,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2933_abs__triangle__ineq3,axiom,
% 9.00/5.05 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2934_abs__triangle__ineq2,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2935_abs__triangle__ineq2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2936_abs__triangle__ineq2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2937_abs__triangle__ineq2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2938_nonzero__abs__divide,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( B3 != zero_zero_rat )
% 9.00/5.05 => ( ( abs_abs_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( divide_divide_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2939_nonzero__abs__divide,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( B3 != zero_zero_real )
% 9.00/5.05 => ( ( abs_abs_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.05 = ( divide_divide_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2940_abs__ge__minus__self,axiom,
% 9.00/5.05 ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ ( abs_abs_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2941_abs__ge__minus__self,axiom,
% 9.00/5.05 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( abs_abs_Code_integer @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2942_abs__ge__minus__self,axiom,
% 9.00/5.05 ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ ( abs_abs_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2943_abs__ge__minus__self,axiom,
% 9.00/5.05 ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ ( abs_abs_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2944_abs__le__iff,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2945_abs__le__iff,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 9.00/5.05 & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2946_abs__le__iff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2947_abs__le__iff,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2948_abs__le__D2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2949_abs__le__D2,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2950_abs__le__D2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2951_abs__le__D2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2952_abs__leI,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_real @ ( abs_abs_real @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2953_abs__leI,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ord_le3102999989581377725nteger @ A2 @ B3 )
% 9.00/5.05 => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2954_abs__leI,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_rat @ ( abs_abs_rat @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2955_abs__leI,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 9.00/5.05 => ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2956_abs__less__iff,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2957_abs__less__iff,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_real @ ( uminus_uminus_real @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2958_abs__less__iff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.05 & ( ord_less_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2959_abs__less__iff,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ B3 )
% 9.00/5.05 = ( ( ord_le6747313008572928689nteger @ A2 @ B3 )
% 9.00/5.05 & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2960_nth__equalityI,axiom,
% 9.00/5.05 ! [Xs: list_int,Ys2: list_int] :
% 9.00/5.05 ( ( ( size_size_list_int @ Xs )
% 9.00/5.05 = ( size_size_list_int @ Ys2 ) )
% 9.00/5.05 => ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( ( nth_int @ Xs @ I2 )
% 9.00/5.05 = ( nth_int @ Ys2 @ I2 ) ) )
% 9.00/5.05 => ( Xs = Ys2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2961_nth__equalityI,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.00/5.05 ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.00/5.05 = ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.00/5.05 => ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 9.00/5.05 = ( nth_VEBT_VEBT @ Ys2 @ I2 ) ) )
% 9.00/5.05 => ( Xs = Ys2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2962_nth__equalityI,axiom,
% 9.00/5.05 ! [Xs: list_o,Ys2: list_o] :
% 9.00/5.05 ( ( ( size_size_list_o @ Xs )
% 9.00/5.05 = ( size_size_list_o @ Ys2 ) )
% 9.00/5.05 => ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( ( nth_o @ Xs @ I2 )
% 9.00/5.05 = ( nth_o @ Ys2 @ I2 ) ) )
% 9.00/5.05 => ( Xs = Ys2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2963_nth__equalityI,axiom,
% 9.00/5.05 ! [Xs: list_nat,Ys2: list_nat] :
% 9.00/5.05 ( ( ( size_size_list_nat @ Xs )
% 9.00/5.05 = ( size_size_list_nat @ Ys2 ) )
% 9.00/5.05 => ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( ( nth_nat @ Xs @ I2 )
% 9.00/5.05 = ( nth_nat @ Ys2 @ I2 ) ) )
% 9.00/5.05 => ( Xs = Ys2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2964_Skolem__list__nth,axiom,
% 9.00/5.05 ! [K: nat,P: nat > int > $o] :
% 9.00/5.05 ( ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ? [X8: int] : ( P @ I4 @ X8 ) ) )
% 9.00/5.05 = ( ? [Xs2: list_int] :
% 9.00/5.05 ( ( ( size_size_list_int @ Xs2 )
% 9.00/5.05 = K )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ( P @ I4 @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2965_Skolem__list__nth,axiom,
% 9.00/5.05 ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 9.00/5.05 ( ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ? [X8: vEBT_VEBT] : ( P @ I4 @ X8 ) ) )
% 9.00/5.05 = ( ? [Xs2: list_VEBT_VEBT] :
% 9.00/5.05 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.00/5.05 = K )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2966_Skolem__list__nth,axiom,
% 9.00/5.05 ! [K: nat,P: nat > $o > $o] :
% 9.00/5.05 ( ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ? [X8: $o] : ( P @ I4 @ X8 ) ) )
% 9.00/5.05 = ( ? [Xs2: list_o] :
% 9.00/5.05 ( ( ( size_size_list_o @ Xs2 )
% 9.00/5.05 = K )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ( P @ I4 @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2967_Skolem__list__nth,axiom,
% 9.00/5.05 ! [K: nat,P: nat > nat > $o] :
% 9.00/5.05 ( ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
% 9.00/5.05 = ( ? [Xs2: list_nat] :
% 9.00/5.05 ( ( ( size_size_list_nat @ Xs2 )
% 9.00/5.05 = K )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ K )
% 9.00/5.05 => ( P @ I4 @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2968_list__eq__iff__nth__eq,axiom,
% 9.00/5.05 ( ( ^ [Y6: list_int,Z3: list_int] : ( Y6 = Z3 ) )
% 9.00/5.05 = ( ^ [Xs2: list_int,Ys3: list_int] :
% 9.00/5.05 ( ( ( size_size_list_int @ Xs2 )
% 9.00/5.05 = ( size_size_list_int @ Ys3 ) )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
% 9.00/5.05 => ( ( nth_int @ Xs2 @ I4 )
% 9.00/5.05 = ( nth_int @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2969_list__eq__iff__nth__eq,axiom,
% 9.00/5.05 ( ( ^ [Y6: list_VEBT_VEBT,Z3: list_VEBT_VEBT] : ( Y6 = Z3 ) )
% 9.00/5.05 = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.00/5.05 ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.00/5.05 = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.00/5.05 => ( ( nth_VEBT_VEBT @ Xs2 @ I4 )
% 9.00/5.05 = ( nth_VEBT_VEBT @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2970_list__eq__iff__nth__eq,axiom,
% 9.00/5.05 ( ( ^ [Y6: list_o,Z3: list_o] : ( Y6 = Z3 ) )
% 9.00/5.05 = ( ^ [Xs2: list_o,Ys3: list_o] :
% 9.00/5.05 ( ( ( size_size_list_o @ Xs2 )
% 9.00/5.05 = ( size_size_list_o @ Ys3 ) )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
% 9.00/5.05 => ( ( nth_o @ Xs2 @ I4 )
% 9.00/5.05 = ( nth_o @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2971_list__eq__iff__nth__eq,axiom,
% 9.00/5.05 ( ( ^ [Y6: list_nat,Z3: list_nat] : ( Y6 = Z3 ) )
% 9.00/5.05 = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 9.00/5.05 ( ( ( size_size_list_nat @ Xs2 )
% 9.00/5.05 = ( size_size_list_nat @ Ys3 ) )
% 9.00/5.05 & ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
% 9.00/5.05 => ( ( nth_nat @ Xs2 @ I4 )
% 9.00/5.05 = ( nth_nat @ Ys3 @ I4 ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2972_dense__eq0__I,axiom,
% 9.00/5.05 ! [X2: real] :
% 9.00/5.05 ( ! [E: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ E )
% 9.00/5.05 => ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ E ) )
% 9.00/5.05 => ( X2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2973_dense__eq0__I,axiom,
% 9.00/5.05 ! [X2: rat] :
% 9.00/5.05 ( ! [E: rat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.00/5.05 => ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ E ) )
% 9.00/5.05 => ( X2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2974_eq__abs__iff_H,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( abs_abs_real @ B3 ) )
% 9.00/5.05 = ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.05 & ( ( B3 = A2 )
% 9.00/5.05 | ( B3
% 9.00/5.05 = ( uminus_uminus_real @ A2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2975_eq__abs__iff_H,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( abs_abs_Code_integer @ B3 ) )
% 9.00/5.05 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.05 & ( ( B3 = A2 )
% 9.00/5.05 | ( B3
% 9.00/5.05 = ( uminus1351360451143612070nteger @ A2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2976_eq__abs__iff_H,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( abs_abs_rat @ B3 ) )
% 9.00/5.05 = ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 & ( ( B3 = A2 )
% 9.00/5.05 | ( B3
% 9.00/5.05 = ( uminus_uminus_rat @ A2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2977_eq__abs__iff_H,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( abs_abs_int @ B3 ) )
% 9.00/5.05 = ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.05 & ( ( B3 = A2 )
% 9.00/5.05 | ( B3
% 9.00/5.05 = ( uminus_uminus_int @ A2 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2978_abs__eq__iff_H,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ( abs_abs_real @ A2 )
% 9.00/5.05 = B3 )
% 9.00/5.05 = ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.05 & ( ( A2 = B3 )
% 9.00/5.05 | ( A2
% 9.00/5.05 = ( uminus_uminus_real @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2979_abs__eq__iff_H,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = B3 )
% 9.00/5.05 = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.05 & ( ( A2 = B3 )
% 9.00/5.05 | ( A2
% 9.00/5.05 = ( uminus1351360451143612070nteger @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2980_abs__eq__iff_H,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = B3 )
% 9.00/5.05 = ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.05 & ( ( A2 = B3 )
% 9.00/5.05 | ( A2
% 9.00/5.05 = ( uminus_uminus_rat @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2981_abs__eq__iff_H,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ( abs_abs_int @ A2 )
% 9.00/5.05 = B3 )
% 9.00/5.05 = ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.05 & ( ( A2 = B3 )
% 9.00/5.05 | ( A2
% 9.00/5.05 = ( uminus_uminus_int @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2982_abs__minus__le__zero,axiom,
% 9.00/5.05 ! [A2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A2 ) ) @ zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2983_abs__minus__le__zero,axiom,
% 9.00/5.05 ! [A2: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A2 ) ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2984_abs__minus__le__zero,axiom,
% 9.00/5.05 ! [A2: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A2 ) ) @ zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2985_abs__minus__le__zero,axiom,
% 9.00/5.05 ! [A2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A2 ) ) @ zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2986_abs__div__pos,axiom,
% 9.00/5.05 ! [Y3: rat,X2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.05 => ( ( divide_divide_rat @ ( abs_abs_rat @ X2 ) @ Y3 )
% 9.00/5.05 = ( abs_abs_rat @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2987_abs__div__pos,axiom,
% 9.00/5.05 ! [Y3: real,X2: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.05 => ( ( divide_divide_real @ ( abs_abs_real @ X2 ) @ Y3 )
% 9.00/5.05 = ( abs_abs_real @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2988_abs__if,axiom,
% 9.00/5.05 ( abs_abs_int
% 9.00/5.05 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2989_abs__if,axiom,
% 9.00/5.05 ( abs_abs_real
% 9.00/5.05 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2990_abs__if,axiom,
% 9.00/5.05 ( abs_abs_rat
% 9.00/5.05 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2991_abs__if,axiom,
% 9.00/5.05 ( abs_abs_Code_integer
% 9.00/5.05 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2992_abs__of__neg,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.05 => ( ( abs_abs_int @ A2 )
% 9.00/5.05 = ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2993_abs__of__neg,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.05 => ( ( abs_abs_real @ A2 )
% 9.00/5.05 = ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2994_abs__of__neg,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.05 => ( ( abs_abs_rat @ A2 )
% 9.00/5.05 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2995_abs__of__neg,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 9.00/5.05 => ( ( abs_abs_Code_integer @ A2 )
% 9.00/5.05 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2996_abs__if__raw,axiom,
% 9.00/5.05 ( abs_abs_int
% 9.00/5.05 = ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2997_abs__if__raw,axiom,
% 9.00/5.05 ( abs_abs_real
% 9.00/5.05 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2998_abs__if__raw,axiom,
% 9.00/5.05 ( abs_abs_rat
% 9.00/5.05 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_2999_abs__if__raw,axiom,
% 9.00/5.05 ( abs_abs_Code_integer
% 9.00/5.05 = ( ^ [A4: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A4 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A4 ) @ A4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3000_length__removeAll__less__eq,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3001_length__removeAll__less__eq,axiom,
% 9.00/5.05 ! [X2: $o,Xs: list_o] : ( ord_less_eq_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3002_length__removeAll__less__eq,axiom,
% 9.00/5.05 ! [X2: nat,Xs: list_nat] : ( ord_less_eq_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3003_all__set__conv__all__nth,axiom,
% 9.00/5.05 ! [Xs: list_int,P: int > $o] :
% 9.00/5.05 ( ( ! [X: int] :
% 9.00/5.05 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( P @ X ) ) )
% 9.00/5.05 = ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_int @ Xs @ I4 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3004_all__set__conv__all__nth,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.00/5.05 ( ( ! [X: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( P @ X ) ) )
% 9.00/5.05 = ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3005_all__set__conv__all__nth,axiom,
% 9.00/5.05 ! [Xs: list_o,P: $o > $o] :
% 9.00/5.05 ( ( ! [X: $o] :
% 9.00/5.05 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( P @ X ) ) )
% 9.00/5.05 = ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_o @ Xs @ I4 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3006_all__set__conv__all__nth,axiom,
% 9.00/5.05 ! [Xs: list_nat,P: nat > $o] :
% 9.00/5.05 ( ( ! [X: nat] :
% 9.00/5.05 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X ) ) )
% 9.00/5.05 = ( ! [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3007_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_real,P: real > $o,X2: real] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_real @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3008_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_set_nat,P: set_nat > $o,X2: set_nat] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_set_nat @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3009_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_int,P: int > $o,X2: int] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_int @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3010_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X2: vEBT_VEBT] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3011_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_o,P: $o > $o,X2: $o] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_o @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3012_all__nth__imp__all__set,axiom,
% 9.00/5.05 ! [Xs: list_nat,P: nat > $o,X2: nat] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( P @ ( nth_nat @ Xs @ I2 ) ) )
% 9.00/5.05 => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3013_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: real,Xs: list_real] :
% 9.00/5.05 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
% 9.00/5.05 & ( ( nth_real @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3014_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: set_nat,Xs: list_set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.05 & ( ( nth_set_nat @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3015_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: int,Xs: list_int] :
% 9.00/5.05 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 & ( ( nth_int @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3016_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 & ( ( nth_VEBT_VEBT @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3017_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: $o,Xs: list_o] :
% 9.00/5.05 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 & ( ( nth_o @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3018_in__set__conv__nth,axiom,
% 9.00/5.05 ! [X2: nat,Xs: list_nat] :
% 9.00/5.05 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 = ( ? [I4: nat] :
% 9.00/5.05 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 & ( ( nth_nat @ Xs @ I4 )
% 9.00/5.05 = X2 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3019_list__ball__nth,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_int,P: int > $o] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( ! [X5: int] :
% 9.00/5.05 ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( P @ ( nth_int @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3020_list__ball__nth,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( ! [X5: vEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( P @ ( nth_VEBT_VEBT @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3021_list__ball__nth,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_o,P: $o > $o] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( ! [X5: $o] :
% 9.00/5.05 ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( P @ ( nth_o @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3022_list__ball__nth,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_nat,P: nat > $o] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( ! [X5: nat] :
% 9.00/5.05 ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( P @ X5 ) )
% 9.00/5.05 => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3023_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_real] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.00/5.05 => ( member_real @ ( nth_real @ Xs @ N ) @ ( set_real2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3024_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_set_nat] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.05 => ( member_set_nat @ ( nth_set_nat @ Xs @ N ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3025_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_int] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.05 => ( member_int @ ( nth_int @ Xs @ N ) @ ( set_int2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3026_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.05 => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3027_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_o] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.05 => ( member_o @ ( nth_o @ Xs @ N ) @ ( set_o2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3028_nth__mem,axiom,
% 9.00/5.05 ! [N: nat,Xs: list_nat] :
% 9.00/5.05 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.05 => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3029_zabs__def,axiom,
% 9.00/5.05 ( abs_abs_int
% 9.00/5.05 = ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3030_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: real,Xs: list_real] :
% 9.00/5.05 ( ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_real @ ( removeAll_real @ X2 @ Xs ) ) @ ( size_size_list_real @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3031_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: set_nat,Xs: list_set_nat] :
% 9.00/5.05 ( ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_s3254054031482475050et_nat @ ( removeAll_set_nat @ X2 @ Xs ) ) @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3032_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: int,Xs: list_int] :
% 9.00/5.05 ( ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_int @ ( removeAll_int @ X2 @ Xs ) ) @ ( size_size_list_int @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3033_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.05 ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ ( removeAll_VEBT_VEBT @ X2 @ Xs ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3034_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: $o,Xs: list_o] :
% 9.00/5.05 ( ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_o @ ( removeAll_o @ X2 @ Xs ) ) @ ( size_size_list_o @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3035_length__removeAll__less,axiom,
% 9.00/5.05 ! [X2: nat,Xs: list_nat] :
% 9.00/5.05 ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 9.00/5.05 => ( ord_less_nat @ ( size_size_list_nat @ ( removeAll_nat @ X2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3036_power__diff__power__eq,axiom,
% 9.00/5.05 ! [A2: int,N: nat,M: nat] :
% 9.00/5.05 ( ( A2 != zero_zero_int )
% 9.00/5.05 => ( ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.05 => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 9.00/5.05 = ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.00/5.05 & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.00/5.05 => ( ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 9.00/5.05 = ( divide_divide_int @ one_one_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3037_power__diff__power__eq,axiom,
% 9.00/5.05 ! [A2: nat,N: nat,M: nat] :
% 9.00/5.05 ( ( A2 != zero_zero_nat )
% 9.00/5.05 => ( ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.05 => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 9.00/5.05 = ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) ) ) )
% 9.00/5.05 & ( ~ ( ord_less_eq_nat @ N @ M )
% 9.00/5.05 => ( ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 9.00/5.05 = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ M ) ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3038_nat__intermed__int__val,axiom,
% 9.00/5.05 ! [M: nat,N: nat,F: nat > int,K: int] :
% 9.00/5.05 ( ! [I2: nat] :
% 9.00/5.05 ( ( ( ord_less_eq_nat @ M @ I2 )
% 9.00/5.05 & ( ord_less_nat @ I2 @ N ) )
% 9.00/5.05 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.00/5.05 => ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 9.00/5.05 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.00/5.05 => ? [I2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ M @ I2 )
% 9.00/5.05 & ( ord_less_eq_nat @ I2 @ N )
% 9.00/5.05 & ( ( F @ I2 )
% 9.00/5.05 = K ) ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3039_of__nat__zero__less__power__iff,axiom,
% 9.00/5.05 ! [X2: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X2 ) @ N ) )
% 9.00/5.05 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3040_of__nat__zero__less__power__iff,axiom,
% 9.00/5.05 ! [X2: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ N ) )
% 9.00/5.05 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3041_of__nat__zero__less__power__iff,axiom,
% 9.00/5.05 ! [X2: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X2 ) @ N ) )
% 9.00/5.05 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3042_of__nat__zero__less__power__iff,axiom,
% 9.00/5.05 ! [X2: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X2 ) @ N ) )
% 9.00/5.05 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3043_power__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: real,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.05 => ( ( ord_less_real @ B3 @ one_one_real )
% 9.00/5.05 => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3044_power__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: rat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.05 => ( ( ord_less_rat @ B3 @ one_one_rat )
% 9.00/5.05 => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3045_power__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: nat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ B3 @ one_one_nat )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3046_power__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: int,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.05 => ( ( ord_less_int @ B3 @ one_one_int )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3047_zero__less__power__abs__iff,axiom,
% 9.00/5.05 ! [A2: code_integer,N: nat] :
% 9.00/5.05 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) )
% 9.00/5.05 = ( ( A2 != zero_z3403309356797280102nteger )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3048_zero__less__power__abs__iff,axiom,
% 9.00/5.05 ! [A2: real,N: nat] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
% 9.00/5.05 = ( ( A2 != zero_zero_real )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3049_zero__less__power__abs__iff,axiom,
% 9.00/5.05 ! [A2: rat,N: nat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) )
% 9.00/5.05 = ( ( A2 != zero_zero_rat )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3050_zero__less__power__abs__iff,axiom,
% 9.00/5.05 ! [A2: int,N: nat] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) )
% 9.00/5.05 = ( ( A2 != zero_zero_int )
% 9.00/5.05 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3051_power__mono__iff,axiom,
% 9.00/5.05 ! [A2: real,B3: real,N: nat] :
% 9.00/5.05 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.05 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3052_power__mono__iff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat,N: nat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3053_power__mono__iff,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.05 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3054_power__mono__iff,axiom,
% 9.00/5.05 ! [A2: int,B3: int,N: nat] :
% 9.00/5.05 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.05 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3055_power__increasing__iff,axiom,
% 9.00/5.05 ! [B3: real,X2: nat,Y3: nat] :
% 9.00/5.05 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3056_power__increasing__iff,axiom,
% 9.00/5.05 ! [B3: rat,X2: nat,Y3: nat] :
% 9.00/5.05 ( ( ord_less_rat @ one_one_rat @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3057_power__increasing__iff,axiom,
% 9.00/5.05 ! [B3: nat,X2: nat,Y3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ one_one_nat @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3058_power__increasing__iff,axiom,
% 9.00/5.05 ! [B3: int,X2: nat,Y3: nat] :
% 9.00/5.05 ( ( ord_less_int @ one_one_int @ B3 )
% 9.00/5.05 => ( ( ord_less_eq_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3059_power__strict__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: real,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.05 => ( ( ord_less_real @ B3 @ one_one_real )
% 9.00/5.05 => ( ( ord_less_real @ ( power_power_real @ B3 @ M ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3060_power__strict__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: rat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.05 => ( ( ord_less_rat @ B3 @ one_one_rat )
% 9.00/5.05 => ( ( ord_less_rat @ ( power_power_rat @ B3 @ M ) @ ( power_power_rat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3061_power__strict__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: nat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.05 => ( ( ord_less_nat @ B3 @ one_one_nat )
% 9.00/5.05 => ( ( ord_less_nat @ ( power_power_nat @ B3 @ M ) @ ( power_power_nat @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3062_power__strict__decreasing__iff,axiom,
% 9.00/5.05 ! [B3: int,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.05 => ( ( ord_less_int @ B3 @ one_one_int )
% 9.00/5.05 => ( ( ord_less_int @ ( power_power_int @ B3 @ M ) @ ( power_power_int @ B3 @ N ) )
% 9.00/5.05 = ( ord_less_nat @ N @ M ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3063_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3064_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3065_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3066_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3067_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.00/5.05 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3068_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.00/5.05 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3069_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.00/5.05 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3070_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.00/5.05 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3071_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3072_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3073_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3074_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.05 ! [X2: nat,B3: nat,W2: nat] :
% 9.00/5.05 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) )
% 9.00/5.05 = ( ord_less_nat @ X2 @ ( power_power_nat @ B3 @ W2 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3075_even__odd__cases,axiom,
% 9.00/5.05 ! [X2: nat] :
% 9.00/5.05 ( ! [N3: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( plus_plus_nat @ N3 @ N3 ) )
% 9.00/5.05 => ~ ! [N3: nat] :
% 9.00/5.05 ( X2
% 9.00/5.05 != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3076_local_Opower__def,axiom,
% 9.00/5.05 ( vEBT_VEBT_power
% 9.00/5.05 = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3077_power__shift,axiom,
% 9.00/5.05 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.05 ( ( ( power_power_nat @ X2 @ Y3 )
% 9.00/5.05 = Z )
% 9.00/5.05 = ( ( vEBT_VEBT_power @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 9.00/5.05 = ( some_nat @ Z ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3078_add__left__cancel,axiom,
% 9.00/5.05 ! [A2: real,B3: real,C: real] :
% 9.00/5.05 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.05 = ( plus_plus_real @ A2 @ C ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3079_add__left__cancel,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.05 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.05 = ( plus_plus_rat @ A2 @ C ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3080_add__left__cancel,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ A2 @ B3 )
% 9.00/5.05 = ( plus_plus_nat @ A2 @ C ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3081_add__left__cancel,axiom,
% 9.00/5.05 ! [A2: int,B3: int,C: int] :
% 9.00/5.05 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.05 = ( plus_plus_int @ A2 @ C ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3082_add__right__cancel,axiom,
% 9.00/5.05 ! [B3: real,A2: real,C: real] :
% 9.00/5.05 ( ( ( plus_plus_real @ B3 @ A2 )
% 9.00/5.05 = ( plus_plus_real @ C @ A2 ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3083_add__right__cancel,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.05 ( ( ( plus_plus_rat @ B3 @ A2 )
% 9.00/5.05 = ( plus_plus_rat @ C @ A2 ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3084_add__right__cancel,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ B3 @ A2 )
% 9.00/5.05 = ( plus_plus_nat @ C @ A2 ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3085_add__right__cancel,axiom,
% 9.00/5.05 ! [B3: int,A2: int,C: int] :
% 9.00/5.05 ( ( ( plus_plus_int @ B3 @ A2 )
% 9.00/5.05 = ( plus_plus_int @ C @ A2 ) )
% 9.00/5.05 = ( B3 = C ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3086_add__le__cancel__right,axiom,
% 9.00/5.05 ! [A2: real,C: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3087_add__le__cancel__right,axiom,
% 9.00/5.05 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3088_add__le__cancel__right,axiom,
% 9.00/5.05 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3089_add__le__cancel__right,axiom,
% 9.00/5.05 ! [A2: int,C: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3090_add__le__cancel__left,axiom,
% 9.00/5.05 ! [C: real,A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3091_add__le__cancel__left,axiom,
% 9.00/5.05 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3092_add__le__cancel__left,axiom,
% 9.00/5.05 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3093_add__le__cancel__left,axiom,
% 9.00/5.05 ! [C: int,A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3094_double__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ( plus_plus_real @ A2 @ A2 )
% 9.00/5.05 = zero_zero_real )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3095_double__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ( plus_plus_rat @ A2 @ A2 )
% 9.00/5.05 = zero_zero_rat )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3096_double__eq__0__iff,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ( plus_plus_int @ A2 @ A2 )
% 9.00/5.05 = zero_zero_int )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3097_add__0,axiom,
% 9.00/5.05 ! [A2: literal] :
% 9.00/5.05 ( ( plus_plus_literal @ zero_zero_literal @ A2 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3098_add__0,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3099_add__0,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3100_add__0,axiom,
% 9.00/5.05 ! [A2: nat] :
% 9.00/5.05 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3101_add__0,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3102_zero__eq__add__iff__both__eq__0,axiom,
% 9.00/5.05 ! [X2: nat,Y3: nat] :
% 9.00/5.05 ( ( zero_zero_nat
% 9.00/5.05 = ( plus_plus_nat @ X2 @ Y3 ) )
% 9.00/5.05 = ( ( X2 = zero_zero_nat )
% 9.00/5.05 & ( Y3 = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3103_add__eq__0__iff__both__eq__0,axiom,
% 9.00/5.05 ! [X2: nat,Y3: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 9.00/5.05 = zero_zero_nat )
% 9.00/5.05 = ( ( X2 = zero_zero_nat )
% 9.00/5.05 & ( Y3 = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3104_add__cancel__right__right,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.05 = ( B3 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3105_add__cancel__right__right,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( B3 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3106_add__cancel__right__right,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_nat @ A2 @ B3 ) )
% 9.00/5.05 = ( B3 = zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3107_add__cancel__right__right,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.05 = ( B3 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3108_add__cancel__right__left,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_real @ B3 @ A2 ) )
% 9.00/5.05 = ( B3 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3109_add__cancel__right__left,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_rat @ B3 @ A2 ) )
% 9.00/5.05 = ( B3 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3110_add__cancel__right__left,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_nat @ B3 @ A2 ) )
% 9.00/5.05 = ( B3 = zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3111_add__cancel__right__left,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( A2
% 9.00/5.05 = ( plus_plus_int @ B3 @ A2 ) )
% 9.00/5.05 = ( B3 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3112_add__cancel__left__right,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3113_add__cancel__left__right,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3114_add__cancel__left__right,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ A2 @ B3 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3115_add__cancel__left__right,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3116_add__cancel__left__left,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( ( plus_plus_real @ B3 @ A2 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3117_add__cancel__left__left,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( ( plus_plus_rat @ B3 @ A2 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3118_add__cancel__left__left,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ B3 @ A2 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3119_add__cancel__left__left,axiom,
% 9.00/5.05 ! [B3: int,A2: int] :
% 9.00/5.05 ( ( ( plus_plus_int @ B3 @ A2 )
% 9.00/5.05 = A2 )
% 9.00/5.05 = ( B3 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3120_double__zero__sym,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( zero_zero_real
% 9.00/5.05 = ( plus_plus_real @ A2 @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3121_double__zero__sym,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( zero_zero_rat
% 9.00/5.05 = ( plus_plus_rat @ A2 @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3122_double__zero__sym,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( zero_zero_int
% 9.00/5.05 = ( plus_plus_int @ A2 @ A2 ) )
% 9.00/5.05 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3123_add_Oright__neutral,axiom,
% 9.00/5.05 ! [A2: literal] :
% 9.00/5.05 ( ( plus_plus_literal @ A2 @ zero_zero_literal )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3124_add_Oright__neutral,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3125_add_Oright__neutral,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3126_add_Oright__neutral,axiom,
% 9.00/5.05 ! [A2: nat] :
% 9.00/5.05 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3127_add_Oright__neutral,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3128_add__less__cancel__left,axiom,
% 9.00/5.05 ! [C: real,A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3129_add__less__cancel__left,axiom,
% 9.00/5.05 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3130_add__less__cancel__left,axiom,
% 9.00/5.05 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3131_add__less__cancel__left,axiom,
% 9.00/5.05 ! [C: int,A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 9.00/5.05 = ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3132_add__less__cancel__right,axiom,
% 9.00/5.05 ! [A2: real,C: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3133_add__less__cancel__right,axiom,
% 9.00/5.05 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3134_add__less__cancel__right,axiom,
% 9.00/5.05 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3135_add__less__cancel__right,axiom,
% 9.00/5.05 ! [A2: int,C: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.05 = ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3136_add__diff__cancel,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3137_add__diff__cancel,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3138_add__diff__cancel,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3139_diff__add__cancel,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3140_diff__add__cancel,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3141_diff__add__cancel,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3142_add__diff__cancel__left,axiom,
% 9.00/5.05 ! [C: real,A2: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 9.00/5.05 = ( minus_minus_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3143_add__diff__cancel__left,axiom,
% 9.00/5.05 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 9.00/5.05 = ( minus_minus_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3144_add__diff__cancel__left,axiom,
% 9.00/5.05 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 9.00/5.05 = ( minus_minus_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3145_add__diff__cancel__left,axiom,
% 9.00/5.05 ! [C: int,A2: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 9.00/5.05 = ( minus_minus_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3146_add__diff__cancel__left_H,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ A2 )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3147_add__diff__cancel__left_H,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ A2 )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3148_add__diff__cancel__left_H,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ A2 )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3149_add__diff__cancel__left_H,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ A2 )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3150_add__diff__cancel__right,axiom,
% 9.00/5.05 ! [A2: real,C: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.05 = ( minus_minus_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3151_add__diff__cancel__right,axiom,
% 9.00/5.05 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.05 = ( minus_minus_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3152_add__diff__cancel__right,axiom,
% 9.00/5.05 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 9.00/5.05 = ( minus_minus_nat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3153_add__diff__cancel__right,axiom,
% 9.00/5.05 ! [A2: int,C: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.05 = ( minus_minus_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3154_add__diff__cancel__right_H,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3155_add__diff__cancel__right_H,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3156_add__diff__cancel__right_H,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3157_add__diff__cancel__right_H,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3158_add__minus__cancel,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3159_add__minus__cancel,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( plus_plus_real @ A2 @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3160_add__minus__cancel,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ A2 @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3161_add__minus__cancel,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( plus_p5714425477246183910nteger @ A2 @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3162_add__minus__cancel,axiom,
% 9.00/5.05 ! [A2: complex,B3: complex] :
% 9.00/5.05 ( ( plus_plus_complex @ A2 @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3163_minus__add__cancel,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3164_minus__add__cancel,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3165_minus__add__cancel,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3166_minus__add__cancel,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3167_minus__add__cancel,axiom,
% 9.00/5.05 ! [A2: complex,B3: complex] :
% 9.00/5.05 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( plus_plus_complex @ A2 @ B3 ) )
% 9.00/5.05 = B3 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3168_minus__add__distrib,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.05 = ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3169_minus__add__distrib,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.05 = ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( uminus_uminus_real @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3170_minus__add__distrib,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( uminus_uminus_rat @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3171_minus__add__distrib,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 9.00/5.05 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ ( uminus1351360451143612070nteger @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3172_minus__add__distrib,axiom,
% 9.00/5.05 ! [A2: complex,B3: complex] :
% 9.00/5.05 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
% 9.00/5.05 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( uminus1482373934393186551omplex @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3173_abs__add__abs,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) )
% 9.00/5.05 = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3174_abs__add__abs,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) )
% 9.00/5.05 = ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3175_abs__add__abs,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) )
% 9.00/5.05 = ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3176_abs__add__abs,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) )
% 9.00/5.05 = ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3177_add__Suc__right,axiom,
% 9.00/5.05 ! [M: nat,N: nat] :
% 9.00/5.05 ( ( plus_plus_nat @ M @ ( suc @ N ) )
% 9.00/5.05 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3178_Nat_Oadd__0__right,axiom,
% 9.00/5.05 ! [M: nat] :
% 9.00/5.05 ( ( plus_plus_nat @ M @ zero_zero_nat )
% 9.00/5.05 = M ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3179_add__is__0,axiom,
% 9.00/5.05 ! [M: nat,N: nat] :
% 9.00/5.05 ( ( ( plus_plus_nat @ M @ N )
% 9.00/5.05 = zero_zero_nat )
% 9.00/5.05 = ( ( M = zero_zero_nat )
% 9.00/5.05 & ( N = zero_zero_nat ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3180_nat__add__left__cancel__less,axiom,
% 9.00/5.05 ! [K: nat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.00/5.05 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3181_nat__add__left__cancel__le,axiom,
% 9.00/5.05 ! [K: nat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.00/5.05 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3182_diff__diff__left,axiom,
% 9.00/5.05 ! [I: nat,J: nat,K: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.00/5.05 = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3183_add__le__same__cancel1,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3184_add__le__same__cancel1,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3185_add__le__same__cancel1,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3186_add__le__same__cancel1,axiom,
% 9.00/5.05 ! [B3: int,A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3187_add__le__same__cancel2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3188_add__le__same__cancel2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3189_add__le__same__cancel2,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3190_add__le__same__cancel2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3191_le__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3192_le__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3193_le__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3194_le__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3195_le__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3196_le__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3197_le__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3198_le__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3199_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
% 9.00/5.05 = ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3200_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 9.00/5.05 = ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3201_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 9.00/5.05 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3202_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3203_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3204_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3205_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3206_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3207_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
% 9.00/5.05 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3208_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( ord_less_real @ ( plus_plus_real @ A2 @ A2 ) @ zero_zero_real )
% 9.00/5.05 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3209_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ A2 ) @ zero_zero_rat )
% 9.00/5.05 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3210_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
% 9.00/5.05 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3211_less__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ A2 @ ( plus_plus_real @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3212_less__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3213_less__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3214_less__add__same__cancel2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ A2 @ ( plus_plus_int @ B3 @ A2 ) )
% 9.00/5.05 = ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3215_less__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_real @ zero_zero_real @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3216_less__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_rat @ zero_zero_rat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3217_less__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_nat @ zero_zero_nat @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3218_less__add__same__cancel1,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.05 = ( ord_less_int @ zero_zero_int @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3219_add__less__same__cancel2,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3220_add__less__same__cancel2,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3221_add__less__same__cancel2,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3222_add__less__same__cancel2,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3223_add__less__same__cancel1,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( ord_less_real @ ( plus_plus_real @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3224_add__less__same__cancel1,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( ord_less_rat @ ( plus_plus_rat @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3225_add__less__same__cancel1,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat] :
% 9.00/5.05 ( ( ord_less_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3226_add__less__same__cancel1,axiom,
% 9.00/5.05 ! [B3: int,A2: int] :
% 9.00/5.05 ( ( ord_less_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 9.00/5.05 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3227_le__add__diff__inverse2,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3228_le__add__diff__inverse2,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3229_le__add__diff__inverse2,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3230_le__add__diff__inverse2,axiom,
% 9.00/5.05 ! [B3: int,A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3231_le__add__diff__inverse,axiom,
% 9.00/5.05 ! [B3: real,A2: real] :
% 9.00/5.05 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3232_le__add__diff__inverse,axiom,
% 9.00/5.05 ! [B3: rat,A2: rat] :
% 9.00/5.05 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3233_le__add__diff__inverse,axiom,
% 9.00/5.05 ! [B3: nat,A2: nat] :
% 9.00/5.05 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3234_le__add__diff__inverse,axiom,
% 9.00/5.05 ! [B3: int,A2: int] :
% 9.00/5.05 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.05 => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.05 = A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3235_diff__add__zero,axiom,
% 9.00/5.05 ! [A2: nat,B3: nat] :
% 9.00/5.05 ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B3 ) )
% 9.00/5.05 = zero_zero_nat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3236_add_Oright__inverse,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
% 9.00/5.05 = zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3237_add_Oright__inverse,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( plus_plus_real @ A2 @ ( uminus_uminus_real @ A2 ) )
% 9.00/5.05 = zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3238_add_Oright__inverse,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ A2 @ ( uminus_uminus_rat @ A2 ) )
% 9.00/5.05 = zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3239_add_Oright__inverse,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( plus_p5714425477246183910nteger @ A2 @ ( uminus1351360451143612070nteger @ A2 ) )
% 9.00/5.05 = zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3240_add_Oright__inverse,axiom,
% 9.00/5.05 ! [A2: complex] :
% 9.00/5.05 ( ( plus_plus_complex @ A2 @ ( uminus1482373934393186551omplex @ A2 ) )
% 9.00/5.05 = zero_zero_complex ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3241_ab__left__minus,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 9.00/5.05 = zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3242_ab__left__minus,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 9.00/5.05 = zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3243_ab__left__minus,axiom,
% 9.00/5.05 ! [A2: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 9.00/5.05 = zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3244_ab__left__minus,axiom,
% 9.00/5.05 ! [A2: code_integer] :
% 9.00/5.05 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 9.00/5.05 = zero_z3403309356797280102nteger ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3245_ab__left__minus,axiom,
% 9.00/5.05 ! [A2: complex] :
% 9.00/5.05 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
% 9.00/5.05 = zero_zero_complex ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3246_power__inject__exp,axiom,
% 9.00/5.05 ! [A2: real,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.05 => ( ( ( power_power_real @ A2 @ M )
% 9.00/5.05 = ( power_power_real @ A2 @ N ) )
% 9.00/5.05 = ( M = N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3247_power__inject__exp,axiom,
% 9.00/5.05 ! [A2: rat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.05 => ( ( ( power_power_rat @ A2 @ M )
% 9.00/5.05 = ( power_power_rat @ A2 @ N ) )
% 9.00/5.05 = ( M = N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3248_power__inject__exp,axiom,
% 9.00/5.05 ! [A2: nat,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.05 => ( ( ( power_power_nat @ A2 @ M )
% 9.00/5.05 = ( power_power_nat @ A2 @ N ) )
% 9.00/5.05 = ( M = N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3249_power__inject__exp,axiom,
% 9.00/5.05 ! [A2: int,M: nat,N: nat] :
% 9.00/5.05 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.05 => ( ( ( power_power_int @ A2 @ M )
% 9.00/5.05 = ( power_power_int @ A2 @ N ) )
% 9.00/5.05 = ( M = N ) ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3250_power__0__Suc,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
% 9.00/5.05 = zero_zero_rat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3251_power__0__Suc,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
% 9.00/5.05 = zero_zero_int ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3252_power__0__Suc,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.00/5.05 = zero_zero_nat ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3253_power__0__Suc,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
% 9.00/5.05 = zero_zero_real ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3254_power__0__Suc,axiom,
% 9.00/5.05 ! [N: nat] :
% 9.00/5.05 ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
% 9.00/5.05 = zero_zero_complex ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3255_diff__minus__eq__add,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B3 ) )
% 9.00/5.05 = ( plus_plus_int @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3256_diff__minus__eq__add,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( minus_minus_real @ A2 @ ( uminus_uminus_real @ B3 ) )
% 9.00/5.05 = ( plus_plus_real @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3257_diff__minus__eq__add,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( minus_minus_rat @ A2 @ ( uminus_uminus_rat @ B3 ) )
% 9.00/5.05 = ( plus_plus_rat @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3258_diff__minus__eq__add,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( minus_8373710615458151222nteger @ A2 @ ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.05 = ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3259_diff__minus__eq__add,axiom,
% 9.00/5.05 ! [A2: complex,B3: complex] :
% 9.00/5.05 ( ( minus_minus_complex @ A2 @ ( uminus1482373934393186551omplex @ B3 ) )
% 9.00/5.05 = ( plus_plus_complex @ A2 @ B3 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3260_uminus__add__conv__diff,axiom,
% 9.00/5.05 ! [A2: int,B3: int] :
% 9.00/5.05 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B3 )
% 9.00/5.05 = ( minus_minus_int @ B3 @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3261_uminus__add__conv__diff,axiom,
% 9.00/5.05 ! [A2: real,B3: real] :
% 9.00/5.05 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ B3 )
% 9.00/5.05 = ( minus_minus_real @ B3 @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3262_uminus__add__conv__diff,axiom,
% 9.00/5.05 ! [A2: rat,B3: rat] :
% 9.00/5.05 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ B3 )
% 9.00/5.05 = ( minus_minus_rat @ B3 @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3263_uminus__add__conv__diff,axiom,
% 9.00/5.05 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.05 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ B3 )
% 9.00/5.05 = ( minus_8373710615458151222nteger @ B3 @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3264_uminus__add__conv__diff,axiom,
% 9.00/5.05 ! [A2: complex,B3: complex] :
% 9.00/5.05 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ B3 )
% 9.00/5.05 = ( minus_minus_complex @ B3 @ A2 ) ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3265_power__Suc0__right,axiom,
% 9.00/5.05 ! [A2: int] :
% 9.00/5.05 ( ( power_power_int @ A2 @ ( suc @ zero_zero_nat ) )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3266_power__Suc0__right,axiom,
% 9.00/5.05 ! [A2: nat] :
% 9.00/5.05 ( ( power_power_nat @ A2 @ ( suc @ zero_zero_nat ) )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3267_power__Suc0__right,axiom,
% 9.00/5.05 ! [A2: real] :
% 9.00/5.05 ( ( power_power_real @ A2 @ ( suc @ zero_zero_nat ) )
% 9.00/5.05 = A2 ) ).
% 9.00/5.05
% 9.00/5.05 thf(fact_3268_power__Suc0__right,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( power_power_complex @ A2 @ ( suc @ zero_zero_nat ) )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3269_of__nat__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3270_of__nat__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3271_of__nat__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3272_of__nat__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3273_add__gr__0,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.06 | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3274_power__Suc__0,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.06 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3275_nat__power__eq__Suc__0__iff,axiom,
% 9.00/5.06 ! [X2: nat,M: nat] :
% 9.00/5.06 ( ( ( power_power_nat @ X2 @ M )
% 9.00/5.06 = ( suc @ zero_zero_nat ) )
% 9.00/5.06 = ( ( M = zero_zero_nat )
% 9.00/5.06 | ( X2
% 9.00/5.06 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3276_nat__zero__less__power__iff,axiom,
% 9.00/5.06 ! [X2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X2 @ N ) )
% 9.00/5.06 = ( ( ord_less_nat @ zero_zero_nat @ X2 )
% 9.00/5.06 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3277_Nat_Odiff__diff__right,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3278_Nat_Oadd__diff__assoc2,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3279_Nat_Oadd__diff__assoc,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3280_add__neg__numeral__special_I8_J,axiom,
% 9.00/5.06 ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3281_add__neg__numeral__special_I8_J,axiom,
% 9.00/5.06 ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3282_add__neg__numeral__special_I8_J,axiom,
% 9.00/5.06 ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.00/5.06 = zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3283_add__neg__numeral__special_I8_J,axiom,
% 9.00/5.06 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.00/5.06 = zero_z3403309356797280102nteger ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3284_add__neg__numeral__special_I8_J,axiom,
% 9.00/5.06 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.00/5.06 = zero_zero_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3285_add__neg__numeral__special_I7_J,axiom,
% 9.00/5.06 ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3286_add__neg__numeral__special_I7_J,axiom,
% 9.00/5.06 ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3287_add__neg__numeral__special_I7_J,axiom,
% 9.00/5.06 ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.00/5.06 = zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3288_add__neg__numeral__special_I7_J,axiom,
% 9.00/5.06 ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.00/5.06 = zero_z3403309356797280102nteger ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3289_add__neg__numeral__special_I7_J,axiom,
% 9.00/5.06 ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.00/5.06 = zero_zero_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3290_power__strict__increasing__iff,axiom,
% 9.00/5.06 ! [B3: real,X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ ( power_power_real @ B3 @ X2 ) @ ( power_power_real @ B3 @ Y3 ) )
% 9.00/5.06 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3291_power__strict__increasing__iff,axiom,
% 9.00/5.06 ! [B3: rat,X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ ( power_power_rat @ B3 @ X2 ) @ ( power_power_rat @ B3 @ Y3 ) )
% 9.00/5.06 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3292_power__strict__increasing__iff,axiom,
% 9.00/5.06 ! [B3: nat,X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ ( power_power_nat @ B3 @ X2 ) @ ( power_power_nat @ B3 @ Y3 ) )
% 9.00/5.06 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3293_power__strict__increasing__iff,axiom,
% 9.00/5.06 ! [B3: int,X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ ( power_power_int @ B3 @ X2 ) @ ( power_power_int @ B3 @ Y3 ) )
% 9.00/5.06 = ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3294_power__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ( power_power_rat @ A2 @ N )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( ( A2 = zero_zero_rat )
% 9.00/5.06 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3295_power__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ( power_power_int @ A2 @ N )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ( A2 = zero_zero_int )
% 9.00/5.06 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3296_power__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ( power_power_nat @ A2 @ N )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 = ( ( A2 = zero_zero_nat )
% 9.00/5.06 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3297_power__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ( power_power_real @ A2 @ N )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( ( A2 = zero_zero_real )
% 9.00/5.06 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3298_power__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: complex,N: nat] :
% 9.00/5.06 ( ( ( power_power_complex @ A2 @ N )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 = ( ( A2 = zero_zero_complex )
% 9.00/5.06 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3299_of__nat__Suc,axiom,
% 9.00/5.06 ! [M: nat] :
% 9.00/5.06 ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 9.00/5.06 = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3300_of__nat__Suc,axiom,
% 9.00/5.06 ! [M: nat] :
% 9.00/5.06 ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 9.00/5.06 = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3301_of__nat__Suc,axiom,
% 9.00/5.06 ! [M: nat] :
% 9.00/5.06 ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 9.00/5.06 = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3302_of__nat__Suc,axiom,
% 9.00/5.06 ! [M: nat] :
% 9.00/5.06 ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 9.00/5.06 = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3303_of__nat__Suc,axiom,
% 9.00/5.06 ! [M: nat] :
% 9.00/5.06 ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 9.00/5.06 = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3304_diff__Suc__diff__eq2,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 9.00/5.06 = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3305_diff__Suc__diff__eq1,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3306_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.00/5.06 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.06 ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B3 ) @ W2 ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 9.00/5.06 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3307_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.00/5.06 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B3 ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 9.00/5.06 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3308_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.00/5.06 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.06 ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B3 ) @ W2 ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 9.00/5.06 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3309_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.00/5.06 ! [B3: nat,W2: nat,X2: nat] :
% 9.00/5.06 ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B3 ) @ W2 ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 9.00/5.06 = ( ord_less_nat @ ( power_power_nat @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3310_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3311_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3312_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3313_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3314_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ( plus_plus_real @ I @ K )
% 9.00/5.06 = ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3315_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ( plus_plus_rat @ I @ K )
% 9.00/5.06 = ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3316_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ( plus_plus_nat @ I @ K )
% 9.00/5.06 = ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3317_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ( plus_plus_int @ I @ K )
% 9.00/5.06 = ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3318_group__cancel_Oadd1,axiom,
% 9.00/5.06 ! [A3: real,K: real,A2: real,B3: real] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_real @ K @ A2 ) )
% 9.00/5.06 => ( ( plus_plus_real @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3319_group__cancel_Oadd1,axiom,
% 9.00/5.06 ! [A3: rat,K: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_rat @ K @ A2 ) )
% 9.00/5.06 => ( ( plus_plus_rat @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3320_group__cancel_Oadd1,axiom,
% 9.00/5.06 ! [A3: nat,K: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_nat @ K @ A2 ) )
% 9.00/5.06 => ( ( plus_plus_nat @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3321_group__cancel_Oadd1,axiom,
% 9.00/5.06 ! [A3: int,K: int,A2: int,B3: int] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_int @ K @ A2 ) )
% 9.00/5.06 => ( ( plus_plus_int @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3322_group__cancel_Oadd2,axiom,
% 9.00/5.06 ! [B2: real,K: real,B3: real,A2: real] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_real @ K @ B3 ) )
% 9.00/5.06 => ( ( plus_plus_real @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_real @ K @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3323_group__cancel_Oadd2,axiom,
% 9.00/5.06 ! [B2: rat,K: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_rat @ K @ B3 ) )
% 9.00/5.06 => ( ( plus_plus_rat @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_rat @ K @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3324_group__cancel_Oadd2,axiom,
% 9.00/5.06 ! [B2: nat,K: nat,B3: nat,A2: nat] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_nat @ K @ B3 ) )
% 9.00/5.06 => ( ( plus_plus_nat @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3325_group__cancel_Oadd2,axiom,
% 9.00/5.06 ! [B2: int,K: int,B3: int,A2: int] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_int @ K @ B3 ) )
% 9.00/5.06 => ( ( plus_plus_int @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3326_add_Oassoc,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( plus_plus_real @ ( plus_plus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3327_add_Oassoc,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3328_add_Oassoc,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3329_add_Oassoc,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( plus_plus_int @ ( plus_plus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3330_add_Oleft__cancel,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_real @ A2 @ C ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3331_add_Oleft__cancel,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_rat @ A2 @ C ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3332_add_Oleft__cancel,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_int @ A2 @ C ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3333_add_Oright__cancel,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_real @ C @ A2 ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3334_add_Oright__cancel,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_rat @ C @ A2 ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3335_add_Oright__cancel,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_int @ C @ A2 ) )
% 9.00/5.06 = ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3336_add_Ocommute,axiom,
% 9.00/5.06 ( plus_plus_real
% 9.00/5.06 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3337_add_Ocommute,axiom,
% 9.00/5.06 ( plus_plus_rat
% 9.00/5.06 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3338_add_Ocommute,axiom,
% 9.00/5.06 ( plus_plus_nat
% 9.00/5.06 = ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3339_add_Ocommute,axiom,
% 9.00/5.06 ( plus_plus_int
% 9.00/5.06 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3340_add_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( plus_plus_real @ B3 @ ( plus_plus_real @ A2 @ C ) )
% 9.00/5.06 = ( plus_plus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3341_add_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) )
% 9.00/5.06 = ( plus_plus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3342_add_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) )
% 9.00/5.06 = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3343_add_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( plus_plus_int @ B3 @ ( plus_plus_int @ A2 @ C ) )
% 9.00/5.06 = ( plus_plus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3344_add__left__imp__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_real @ A2 @ C ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3345_add__left__imp__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_rat @ A2 @ C ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3346_add__left__imp__eq,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ A2 @ C ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3347_add__left__imp__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = ( plus_plus_int @ A2 @ C ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3348_add__right__imp__eq,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_real @ C @ A2 ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3349_add__right__imp__eq,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_rat @ C @ A2 ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3350_add__right__imp__eq,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_nat @ C @ A2 ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3351_add__right__imp__eq,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ B3 @ A2 )
% 9.00/5.06 = ( plus_plus_int @ C @ A2 ) )
% 9.00/5.06 => ( B3 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3352_add__le__imp__le__right,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3353_add__le__imp__le__right,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3354_add__le__imp__le__right,axiom,
% 9.00/5.06 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3355_add__le__imp__le__right,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3356_add__le__imp__le__left,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_eq_real @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3357_add__le__imp__le__left,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3358_add__le__imp__le__left,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3359_add__le__imp__le__left,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3360_le__iff__add,axiom,
% 9.00/5.06 ( ord_less_eq_nat
% 9.00/5.06 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.06 ? [C5: nat] :
% 9.00/5.06 ( B4
% 9.00/5.06 = ( plus_plus_nat @ A4 @ C5 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3361_add__right__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3362_add__right__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3363_add__right__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3364_add__right__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3365_less__eqE,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ~ ! [C3: nat] :
% 9.00/5.06 ( B3
% 9.00/5.06 != ( plus_plus_nat @ A2 @ C3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3366_add__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3367_add__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3368_add__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3369_add__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3370_add__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ D )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3371_add__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ D )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3372_add__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ D )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3373_add__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ D )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3374_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_eq_real @ I @ J )
% 9.00/5.06 & ( ord_less_eq_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3375_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_eq_rat @ I @ J )
% 9.00/5.06 & ( ord_less_eq_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3376_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 & ( ord_less_eq_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3377_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_eq_int @ I @ J )
% 9.00/5.06 & ( ord_less_eq_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3378_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_eq_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3379_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_eq_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3380_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_eq_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3381_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_eq_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3382_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_eq_real @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3383_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_eq_rat @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3384_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3385_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_eq_int @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3386_verit__sum__simplify,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3387_verit__sum__simplify,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3388_verit__sum__simplify,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3389_verit__sum__simplify,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3390_add_Ogroup__left__neutral,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3391_add_Ogroup__left__neutral,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3392_add_Ogroup__left__neutral,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3393_add_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( plus_plus_real @ A2 @ zero_zero_real )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3394_add_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3395_add_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3396_add_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( plus_plus_int @ A2 @ zero_zero_int )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3397_comm__monoid__add__class_Oadd__0,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( plus_plus_real @ zero_zero_real @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3398_comm__monoid__add__class_Oadd__0,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3399_comm__monoid__add__class_Oadd__0,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3400_comm__monoid__add__class_Oadd__0,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( plus_plus_int @ zero_zero_int @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3401_add__mono__thms__linordered__field_I5_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_real @ I @ J )
% 9.00/5.06 & ( ord_less_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3402_add__mono__thms__linordered__field_I5_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_rat @ I @ J )
% 9.00/5.06 & ( ord_less_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3403_add__mono__thms__linordered__field_I5_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_nat @ I @ J )
% 9.00/5.06 & ( ord_less_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3404_add__mono__thms__linordered__field_I5_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_int @ I @ J )
% 9.00/5.06 & ( ord_less_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3405_add__mono__thms__linordered__field_I2_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3406_add__mono__thms__linordered__field_I2_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3407_add__mono__thms__linordered__field_I2_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3408_add__mono__thms__linordered__field_I2_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( I = J )
% 9.00/5.06 & ( ord_less_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3409_add__mono__thms__linordered__field_I1_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_real @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3410_add__mono__thms__linordered__field_I1_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_rat @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3411_add__mono__thms__linordered__field_I1_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_nat @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3412_add__mono__thms__linordered__field_I1_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_int @ I @ J )
% 9.00/5.06 & ( K = L ) )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3413_add__strict__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ C @ D )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3414_add__strict__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ C @ D )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3415_add__strict__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ C @ D )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3416_add__strict__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ C @ D )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3417_add__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3418_add__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3419_add__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3420_add__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3421_add__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3422_add__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3423_add__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3424_add__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3425_add__less__imp__less__left,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( plus_plus_real @ C @ A2 ) @ ( plus_plus_real @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3426_add__less__imp__less__left,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( plus_plus_rat @ C @ A2 ) @ ( plus_plus_rat @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3427_add__less__imp__less__left,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_nat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3428_add__less__imp__less__left,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B3 ) )
% 9.00/5.06 => ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3429_add__less__imp__less__right,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3430_add__less__imp__less__right,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3431_add__less__imp__less__right,axiom,
% 9.00/5.06 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_nat @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3432_add__less__imp__less__right,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.06 => ( ord_less_int @ A2 @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3433_group__cancel_Osub1,axiom,
% 9.00/5.06 ! [A3: real,K: real,A2: real,B3: real] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_real @ K @ A2 ) )
% 9.00/5.06 => ( ( minus_minus_real @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_real @ K @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3434_group__cancel_Osub1,axiom,
% 9.00/5.06 ! [A3: rat,K: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_rat @ K @ A2 ) )
% 9.00/5.06 => ( ( minus_minus_rat @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_rat @ K @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3435_group__cancel_Osub1,axiom,
% 9.00/5.06 ! [A3: int,K: int,A2: int,B3: int] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_int @ K @ A2 ) )
% 9.00/5.06 => ( ( minus_minus_int @ A3 @ B3 )
% 9.00/5.06 = ( plus_plus_int @ K @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3436_diff__eq__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ( minus_minus_real @ A2 @ B3 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( A2
% 9.00/5.06 = ( plus_plus_real @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3437_diff__eq__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ( minus_minus_rat @ A2 @ B3 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( A2
% 9.00/5.06 = ( plus_plus_rat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3438_diff__eq__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ( minus_minus_int @ A2 @ B3 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( A2
% 9.00/5.06 = ( plus_plus_int @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3439_eq__diff__eq,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( minus_minus_real @ C @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3440_eq__diff__eq,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( minus_minus_rat @ C @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3441_eq__diff__eq,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( minus_minus_int @ C @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3442_add__diff__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( plus_plus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3443_add__diff__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3444_add__diff__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( plus_plus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3445_diff__diff__eq2,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( minus_minus_real @ A2 @ ( minus_minus_real @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3446_diff__diff__eq2,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( minus_minus_rat @ A2 @ ( minus_minus_rat @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3447_diff__diff__eq2,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( minus_minus_int @ A2 @ ( minus_minus_int @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3448_diff__add__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( plus_plus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3449_diff__add__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3450_diff__add__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3451_diff__add__eq__diff__diff__swap,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_real @ ( minus_minus_real @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3452_diff__add__eq__diff__diff__swap,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_rat @ ( minus_minus_rat @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3453_diff__add__eq__diff__diff__swap,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) )
% 9.00/5.06 = ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3454_add__implies__diff,axiom,
% 9.00/5.06 ! [C: real,B3: real,A2: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ C @ B3 )
% 9.00/5.06 = A2 )
% 9.00/5.06 => ( C
% 9.00/5.06 = ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3455_add__implies__diff,axiom,
% 9.00/5.06 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ C @ B3 )
% 9.00/5.06 = A2 )
% 9.00/5.06 => ( C
% 9.00/5.06 = ( minus_minus_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3456_add__implies__diff,axiom,
% 9.00/5.06 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ C @ B3 )
% 9.00/5.06 = A2 )
% 9.00/5.06 => ( C
% 9.00/5.06 = ( minus_minus_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3457_add__implies__diff,axiom,
% 9.00/5.06 ! [C: int,B3: int,A2: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ C @ B3 )
% 9.00/5.06 = A2 )
% 9.00/5.06 => ( C
% 9.00/5.06 = ( minus_minus_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3458_diff__diff__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( minus_minus_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_real @ A2 @ ( plus_plus_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3459_diff__diff__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( minus_minus_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_rat @ A2 @ ( plus_plus_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3460_diff__diff__eq,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3461_diff__diff__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( minus_minus_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( minus_minus_int @ A2 @ ( plus_plus_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3462_group__cancel_Oneg1,axiom,
% 9.00/5.06 ! [A3: int,K: int,A2: int] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_int @ K @ A2 ) )
% 9.00/5.06 => ( ( uminus_uminus_int @ A3 )
% 9.00/5.06 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3463_group__cancel_Oneg1,axiom,
% 9.00/5.06 ! [A3: real,K: real,A2: real] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_real @ K @ A2 ) )
% 9.00/5.06 => ( ( uminus_uminus_real @ A3 )
% 9.00/5.06 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3464_group__cancel_Oneg1,axiom,
% 9.00/5.06 ! [A3: rat,K: rat,A2: rat] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_rat @ K @ A2 ) )
% 9.00/5.06 => ( ( uminus_uminus_rat @ A3 )
% 9.00/5.06 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3465_group__cancel_Oneg1,axiom,
% 9.00/5.06 ! [A3: code_integer,K: code_integer,A2: code_integer] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_p5714425477246183910nteger @ K @ A2 ) )
% 9.00/5.06 => ( ( uminus1351360451143612070nteger @ A3 )
% 9.00/5.06 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3466_group__cancel_Oneg1,axiom,
% 9.00/5.06 ! [A3: complex,K: complex,A2: complex] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_complex @ K @ A2 ) )
% 9.00/5.06 => ( ( uminus1482373934393186551omplex @ A3 )
% 9.00/5.06 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3467_add_Oinverse__distrib__swap,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B3 ) )
% 9.00/5.06 = ( plus_plus_int @ ( uminus_uminus_int @ B3 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3468_add_Oinverse__distrib__swap,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( uminus_uminus_real @ ( plus_plus_real @ A2 @ B3 ) )
% 9.00/5.06 = ( plus_plus_real @ ( uminus_uminus_real @ B3 ) @ ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3469_add_Oinverse__distrib__swap,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( uminus_uminus_rat @ ( plus_plus_rat @ A2 @ B3 ) )
% 9.00/5.06 = ( plus_plus_rat @ ( uminus_uminus_rat @ B3 ) @ ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3470_add_Oinverse__distrib__swap,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.06 ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) )
% 9.00/5.06 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B3 ) @ ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3471_add_Oinverse__distrib__swap,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A2 @ B3 ) )
% 9.00/5.06 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B3 ) @ ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3472_nat__arith_Osuc1,axiom,
% 9.00/5.06 ! [A3: nat,K: nat,A2: nat] :
% 9.00/5.06 ( ( A3
% 9.00/5.06 = ( plus_plus_nat @ K @ A2 ) )
% 9.00/5.06 => ( ( suc @ A3 )
% 9.00/5.06 = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3473_add__Suc,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.00/5.06 = ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3474_add__Suc__shift,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ ( suc @ M ) @ N )
% 9.00/5.06 = ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3475_add__eq__self__zero,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ M @ N )
% 9.00/5.06 = M )
% 9.00/5.06 => ( N = zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3476_plus__nat_Oadd__0,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( plus_plus_nat @ zero_zero_nat @ N )
% 9.00/5.06 = N ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3477_add__lessD1,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.00/5.06 => ( ord_less_nat @ I @ K ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3478_add__less__mono,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ J )
% 9.00/5.06 => ( ( ord_less_nat @ K @ L )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3479_not__add__less1,axiom,
% 9.00/5.06 ! [I: nat,J: nat] :
% 9.00/5.06 ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3480_not__add__less2,axiom,
% 9.00/5.06 ! [J: nat,I: nat] :
% 9.00/5.06 ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3481_add__less__mono1,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ J )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3482_trans__less__add1,axiom,
% 9.00/5.06 ! [I: nat,J: nat,M: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ J )
% 9.00/5.06 => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3483_trans__less__add2,axiom,
% 9.00/5.06 ! [I: nat,J: nat,M: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ J )
% 9.00/5.06 => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3484_less__add__eq__less,axiom,
% 9.00/5.06 ! [K: nat,L: nat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ K @ L )
% 9.00/5.06 => ( ( ( plus_plus_nat @ M @ L )
% 9.00/5.06 = ( plus_plus_nat @ K @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3485_nat__le__iff__add,axiom,
% 9.00/5.06 ( ord_less_eq_nat
% 9.00/5.06 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.06 ? [K3: nat] :
% 9.00/5.06 ( N2
% 9.00/5.06 = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3486_trans__le__add2,axiom,
% 9.00/5.06 ! [I: nat,J: nat,M: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3487_trans__le__add1,axiom,
% 9.00/5.06 ! [I: nat,J: nat,M: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3488_add__le__mono1,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3489_add__le__mono,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 => ( ( ord_less_eq_nat @ K @ L )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3490_le__Suc__ex,axiom,
% 9.00/5.06 ! [K: nat,L: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ L )
% 9.00/5.06 => ? [N3: nat] :
% 9.00/5.06 ( L
% 9.00/5.06 = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3491_add__leD2,axiom,
% 9.00/5.06 ! [M: nat,K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_nat @ K @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3492_add__leD1,axiom,
% 9.00/5.06 ! [M: nat,K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3493_le__add2,axiom,
% 9.00/5.06 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3494_le__add1,axiom,
% 9.00/5.06 ! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3495_add__leE,axiom,
% 9.00/5.06 ! [M: nat,K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
% 9.00/5.06 => ~ ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.06 => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3496_Nat_Odiff__cancel,axiom,
% 9.00/5.06 ! [K: nat,M: nat,N: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
% 9.00/5.06 = ( minus_minus_nat @ M @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3497_diff__cancel2,axiom,
% 9.00/5.06 ! [M: nat,K: nat,N: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( minus_minus_nat @ M @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3498_diff__add__inverse,axiom,
% 9.00/5.06 ! [N: nat,M: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
% 9.00/5.06 = M ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3499_diff__add__inverse2,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
% 9.00/5.06 = M ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3500_nat__power__less__imp__less,axiom,
% 9.00/5.06 ! [I: nat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ I )
% 9.00/5.06 => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3501_abs__real__def,axiom,
% 9.00/5.06 ( abs_abs_real
% 9.00/5.06 = ( ^ [A4: real] : ( if_real @ ( ord_less_real @ A4 @ zero_zero_real ) @ ( uminus_uminus_real @ A4 ) @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3502_add__decreasing,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3503_add__decreasing,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3504_add__decreasing,axiom,
% 9.00/5.06 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3505_add__decreasing,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3506_add__increasing,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ C )
% 9.00/5.06 => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3507_add__increasing,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3508_add__increasing,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3509_add__increasing,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.06 => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3510_add__decreasing2,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3511_add__decreasing2,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3512_add__decreasing2,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3513_add__decreasing2,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3514_add__increasing2,axiom,
% 9.00/5.06 ! [C: real,B3: real,A2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.06 => ( ord_less_eq_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3515_add__increasing2,axiom,
% 9.00/5.06 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.06 => ( ord_less_eq_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3516_add__increasing2,axiom,
% 9.00/5.06 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.06 => ( ord_less_eq_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3517_add__increasing2,axiom,
% 9.00/5.06 ! [C: int,B3: int,A2: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.06 => ( ord_less_eq_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3518_add__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3519_add__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3520_add__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3521_add__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3522_add__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3523_add__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3524_add__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3525_add__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3526_add__nonneg__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( ( X2 = zero_zero_real )
% 9.00/5.06 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3527_add__nonneg__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.06 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( ( X2 = zero_zero_rat )
% 9.00/5.06 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3528_add__nonneg__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.00/5.06 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 = ( ( X2 = zero_zero_nat )
% 9.00/5.06 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3529_add__nonneg__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: int,Y3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.06 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ( X2 = zero_zero_int )
% 9.00/5.06 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3530_add__nonpos__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.00/5.06 => ( ( ( plus_plus_real @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( ( X2 = zero_zero_real )
% 9.00/5.06 & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3531_add__nonpos__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ X2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.00/5.06 => ( ( ( plus_plus_rat @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( ( X2 = zero_zero_rat )
% 9.00/5.06 & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3532_add__nonpos__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: nat,Y3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ X2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 9.00/5.06 => ( ( ( plus_plus_nat @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 = ( ( X2 = zero_zero_nat )
% 9.00/5.06 & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3533_add__nonpos__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: int,Y3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.00/5.06 => ( ( ( plus_plus_int @ X2 @ Y3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ( X2 = zero_zero_int )
% 9.00/5.06 & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3534_add__less__le__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ D )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3535_add__less__le__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ D )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3536_add__less__le__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ D )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3537_add__less__le__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ D )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3538_add__le__less__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ C @ D )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ C ) @ ( plus_plus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3539_add__le__less__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ C @ D )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ C ) @ ( plus_plus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3540_add__le__less__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ C @ D )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3541_add__le__less__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ C @ D )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3542_add__mono__thms__linordered__field_I3_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_real @ I @ J )
% 9.00/5.06 & ( ord_less_eq_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3543_add__mono__thms__linordered__field_I3_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_rat @ I @ J )
% 9.00/5.06 & ( ord_less_eq_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3544_add__mono__thms__linordered__field_I3_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_nat @ I @ J )
% 9.00/5.06 & ( ord_less_eq_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3545_add__mono__thms__linordered__field_I3_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_int @ I @ J )
% 9.00/5.06 & ( ord_less_eq_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3546_add__mono__thms__linordered__field_I4_J,axiom,
% 9.00/5.06 ! [I: real,J: real,K: real,L: real] :
% 9.00/5.06 ( ( ( ord_less_eq_real @ I @ J )
% 9.00/5.06 & ( ord_less_real @ K @ L ) )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3547_add__mono__thms__linordered__field_I4_J,axiom,
% 9.00/5.06 ! [I: rat,J: rat,K: rat,L: rat] :
% 9.00/5.06 ( ( ( ord_less_eq_rat @ I @ J )
% 9.00/5.06 & ( ord_less_rat @ K @ L ) )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3548_add__mono__thms__linordered__field_I4_J,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.06 ( ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 & ( ord_less_nat @ K @ L ) )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3549_add__mono__thms__linordered__field_I4_J,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int,L: int] :
% 9.00/5.06 ( ( ( ord_less_eq_int @ I @ J )
% 9.00/5.06 & ( ord_less_int @ K @ L ) )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3550_pos__add__strict,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.06 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3551_pos__add__strict,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.06 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3552_pos__add__strict,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.06 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3553_pos__add__strict,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.06 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3554_canonically__ordered__monoid__add__class_OlessE,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ~ ! [C3: nat] :
% 9.00/5.06 ( ( B3
% 9.00/5.06 = ( plus_plus_nat @ A2 @ C3 ) )
% 9.00/5.06 => ( C3 = zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3555_add__pos__pos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3556_add__pos__pos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3557_add__pos__pos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3558_add__pos__pos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3559_add__neg__neg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3560_add__neg__neg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3561_add__neg__neg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3562_add__neg__neg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3563_add__less__zeroD,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.06 | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3564_add__less__zeroD,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ X2 @ zero_zero_rat )
% 9.00/5.06 | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3565_add__less__zeroD,axiom,
% 9.00/5.06 ! [X2: int,Y3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( plus_plus_int @ X2 @ Y3 ) @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ X2 @ zero_zero_int )
% 9.00/5.06 | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3566_power__gt__expt,axiom,
% 9.00/5.06 ! [N: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.06 => ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3567_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ( minus_minus_nat @ B3 @ A2 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( plus_plus_nat @ C @ A2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3568_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_nat @ A2 @ ( minus_minus_nat @ B3 @ A2 ) )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3569_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3570_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 )
% 9.00/5.06 = ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3571_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ C )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3572_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 )
% 9.00/5.06 = ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3573_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 9.00/5.06 = ( minus_minus_nat @ ( plus_plus_nat @ C @ B3 ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3574_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B3 @ A2 ) )
% 9.00/5.06 = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3575_le__add__diff,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3576_diff__add,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_nat @ ( minus_minus_nat @ B3 @ A2 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3577_le__diff__eq,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3578_le__diff__eq,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_eq_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3579_le__diff__eq,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3580_diff__le__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_eq_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3581_diff__le__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_eq_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3582_diff__le__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_eq_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3583_add__le__add__imp__diff__le,axiom,
% 9.00/5.06 ! [I: real,K: real,N: real,J: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K ) )
% 9.00/5.06 => ( ord_less_eq_real @ ( minus_minus_real @ N @ K ) @ J ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3584_add__le__add__imp__diff__le,axiom,
% 9.00/5.06 ! [I: rat,K: rat,N: rat,J: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.00/5.06 => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K ) )
% 9.00/5.06 => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K ) @ J ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3585_add__le__add__imp__diff__le,axiom,
% 9.00/5.06 ! [I: nat,K: nat,N: nat,J: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.00/5.06 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3586_add__le__add__imp__diff__le,axiom,
% 9.00/5.06 ! [I: int,K: int,N: int,J: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.00/5.06 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 9.00/5.06 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3587_add__le__imp__le__diff,axiom,
% 9.00/5.06 ! [I: real,K: real,N: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3588_add__le__imp__le__diff,axiom,
% 9.00/5.06 ! [I: rat,K: rat,N: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3589_add__le__imp__le__diff,axiom,
% 9.00/5.06 ! [I: nat,K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3590_add__le__imp__le__diff,axiom,
% 9.00/5.06 ! [I: int,K: int,N: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 9.00/5.06 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3591_less__add__one,axiom,
% 9.00/5.06 ! [A2: real] : ( ord_less_real @ A2 @ ( plus_plus_real @ A2 @ one_one_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3592_less__add__one,axiom,
% 9.00/5.06 ! [A2: rat] : ( ord_less_rat @ A2 @ ( plus_plus_rat @ A2 @ one_one_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3593_less__add__one,axiom,
% 9.00/5.06 ! [A2: nat] : ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3594_less__add__one,axiom,
% 9.00/5.06 ! [A2: int] : ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ one_one_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3595_add__mono1,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ one_one_real ) @ ( plus_plus_real @ B3 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3596_add__mono1,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ one_one_rat ) @ ( plus_plus_rat @ B3 @ one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3597_add__mono1,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) @ ( plus_plus_nat @ B3 @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3598_add__mono1,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ one_one_int ) @ ( plus_plus_int @ B3 @ one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3599_nat__one__le__power,axiom,
% 9.00/5.06 ! [I: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 9.00/5.06 => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3600_less__diff__eq,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ ( minus_minus_real @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3601_less__diff__eq,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ ( minus_minus_rat @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3602_less__diff__eq,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ ( minus_minus_int @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3603_diff__less__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ ( minus_minus_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_real @ A2 @ ( plus_plus_real @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3604_diff__less__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_rat @ A2 @ ( plus_plus_rat @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3605_diff__less__eq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ ( minus_minus_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( ord_less_int @ A2 @ ( plus_plus_int @ C @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3606_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ~ ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_real @ B3 @ ( minus_minus_real @ A2 @ B3 ) )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3607_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ~ ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_rat @ B3 @ ( minus_minus_rat @ A2 @ B3 ) )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3608_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ~ ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_nat @ B3 @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3609_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ~ ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( plus_plus_int @ B3 @ ( minus_minus_int @ A2 @ B3 ) )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3610_add__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( uminus_uminus_int @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3611_add__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( uminus_uminus_real @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3612_add__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( uminus_uminus_rat @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3613_add__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.06 ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 9.00/5.06 = zero_z3403309356797280102nteger )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( uminus1351360451143612070nteger @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3614_add__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( plus_plus_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( uminus1482373934393186551omplex @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3615_ab__group__add__class_Oab__left__minus,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3616_ab__group__add__class_Oab__left__minus,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ A2 )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3617_ab__group__add__class_Oab__left__minus,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ A2 )
% 9.00/5.06 = zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3618_ab__group__add__class_Oab__left__minus,axiom,
% 9.00/5.06 ! [A2: code_integer] :
% 9.00/5.06 ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A2 ) @ A2 )
% 9.00/5.06 = zero_z3403309356797280102nteger ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3619_ab__group__add__class_Oab__left__minus,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ A2 )
% 9.00/5.06 = zero_zero_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3620_add_Oinverse__unique,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 => ( ( uminus_uminus_int @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3621_add_Oinverse__unique,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 => ( ( uminus_uminus_real @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3622_add_Oinverse__unique,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 => ( ( uminus_uminus_rat @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3623_add_Oinverse__unique,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.06 ( ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 9.00/5.06 = zero_z3403309356797280102nteger )
% 9.00/5.06 => ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3624_add_Oinverse__unique,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( plus_plus_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 => ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3625_eq__neg__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( uminus_uminus_int @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3626_eq__neg__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( uminus_uminus_real @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3627_eq__neg__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( uminus_uminus_rat @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3628_eq__neg__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( uminus1351360451143612070nteger @ B3 ) )
% 9.00/5.06 = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 9.00/5.06 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3629_eq__neg__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( uminus1482373934393186551omplex @ B3 ) )
% 9.00/5.06 = ( ( plus_plus_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3630_neg__eq__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( uminus_uminus_int @ A2 )
% 9.00/5.06 = B3 )
% 9.00/5.06 = ( ( plus_plus_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3631_neg__eq__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( uminus_uminus_real @ A2 )
% 9.00/5.06 = B3 )
% 9.00/5.06 = ( ( plus_plus_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3632_neg__eq__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( uminus_uminus_rat @ A2 )
% 9.00/5.06 = B3 )
% 9.00/5.06 = ( ( plus_plus_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3633_neg__eq__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.06 ( ( ( uminus1351360451143612070nteger @ A2 )
% 9.00/5.06 = B3 )
% 9.00/5.06 = ( ( plus_p5714425477246183910nteger @ A2 @ B3 )
% 9.00/5.06 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3634_neg__eq__iff__add__eq__0,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.06 = B3 )
% 9.00/5.06 = ( ( plus_plus_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3635_group__cancel_Osub2,axiom,
% 9.00/5.06 ! [B2: int,K: int,B3: int,A2: int] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_int @ K @ B3 ) )
% 9.00/5.06 => ( ( minus_minus_int @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3636_group__cancel_Osub2,axiom,
% 9.00/5.06 ! [B2: real,K: real,B3: real,A2: real] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_real @ K @ B3 ) )
% 9.00/5.06 => ( ( minus_minus_real @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3637_group__cancel_Osub2,axiom,
% 9.00/5.06 ! [B2: rat,K: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_rat @ K @ B3 ) )
% 9.00/5.06 => ( ( minus_minus_rat @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3638_group__cancel_Osub2,axiom,
% 9.00/5.06 ! [B2: code_integer,K: code_integer,B3: code_integer,A2: code_integer] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_p5714425477246183910nteger @ K @ B3 ) )
% 9.00/5.06 => ( ( minus_8373710615458151222nteger @ A2 @ B2 )
% 9.00/5.06 = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3639_group__cancel_Osub2,axiom,
% 9.00/5.06 ! [B2: complex,K: complex,B3: complex,A2: complex] :
% 9.00/5.06 ( ( B2
% 9.00/5.06 = ( plus_plus_complex @ K @ B3 ) )
% 9.00/5.06 => ( ( minus_minus_complex @ A2 @ B2 )
% 9.00/5.06 = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3640_diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_int
% 9.00/5.06 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3641_diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_real
% 9.00/5.06 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3642_diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_rat
% 9.00/5.06 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3643_diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_8373710615458151222nteger
% 9.00/5.06 = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3644_diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_complex
% 9.00/5.06 = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3645_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_int
% 9.00/5.06 = ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3646_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_real
% 9.00/5.06 = ( ^ [A4: real,B4: real] : ( plus_plus_real @ A4 @ ( uminus_uminus_real @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3647_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_rat
% 9.00/5.06 = ( ^ [A4: rat,B4: rat] : ( plus_plus_rat @ A4 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_8373710615458151222nteger
% 9.00/5.06 = ( ^ [A4: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A4 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3649_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.00/5.06 ( minus_minus_complex
% 9.00/5.06 = ( ^ [A4: complex,B4: complex] : ( plus_plus_complex @ A4 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3650_abs__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3651_abs__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3652_abs__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3653_abs__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3654_add__is__1,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ M @ N )
% 9.00/5.06 = ( suc @ zero_zero_nat ) )
% 9.00/5.06 = ( ( ( M
% 9.00/5.06 = ( suc @ zero_zero_nat ) )
% 9.00/5.06 & ( N = zero_zero_nat ) )
% 9.00/5.06 | ( ( M = zero_zero_nat )
% 9.00/5.06 & ( N
% 9.00/5.06 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3655_one__is__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ( suc @ zero_zero_nat )
% 9.00/5.06 = ( plus_plus_nat @ M @ N ) )
% 9.00/5.06 = ( ( ( M
% 9.00/5.06 = ( suc @ zero_zero_nat ) )
% 9.00/5.06 & ( N = zero_zero_nat ) )
% 9.00/5.06 | ( ( M = zero_zero_nat )
% 9.00/5.06 & ( N
% 9.00/5.06 = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3656_less__imp__Suc__add,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ N )
% 9.00/5.06 => ? [K2: nat] :
% 9.00/5.06 ( N
% 9.00/5.06 = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3657_less__iff__Suc__add,axiom,
% 9.00/5.06 ( ord_less_nat
% 9.00/5.06 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.06 ? [K3: nat] :
% 9.00/5.06 ( N2
% 9.00/5.06 = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3658_less__add__Suc2,axiom,
% 9.00/5.06 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3659_less__add__Suc1,axiom,
% 9.00/5.06 ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3660_less__natE,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ N )
% 9.00/5.06 => ~ ! [Q4: nat] :
% 9.00/5.06 ( N
% 9.00/5.06 != ( suc @ ( plus_plus_nat @ M @ Q4 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3661_real__arch__pow,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.06 => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X2 @ N3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3662_less__imp__add__positive,axiom,
% 9.00/5.06 ! [I: nat,J: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ J )
% 9.00/5.06 => ? [K2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 9.00/5.06 & ( ( plus_plus_nat @ I @ K2 )
% 9.00/5.06 = J ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3663_mono__nat__linear__lb,axiom,
% 9.00/5.06 ! [F: nat > nat,M: nat,K: nat] :
% 9.00/5.06 ( ! [M4: nat,N3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M4 @ N3 )
% 9.00/5.06 => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3664_diff__add__0,axiom,
% 9.00/5.06 ! [N: nat,M: nat] :
% 9.00/5.06 ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
% 9.00/5.06 = zero_zero_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3665_add__diff__inverse__nat,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ~ ( ord_less_nat @ M @ N )
% 9.00/5.06 => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = M ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3666_less__diff__conv,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.00/5.06 = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3667_Suc__eq__plus1,axiom,
% 9.00/5.06 ( suc
% 9.00/5.06 = ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3668_plus__1__eq__Suc,axiom,
% 9.00/5.06 ( ( plus_plus_nat @ one_one_nat )
% 9.00/5.06 = suc ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3669_Suc__eq__plus1__left,axiom,
% 9.00/5.06 ( suc
% 9.00/5.06 = ( plus_plus_nat @ one_one_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3670_Nat_Ole__imp__diff__is__add,axiom,
% 9.00/5.06 ! [I: nat,J: nat,K: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.06 => ( ( ( minus_minus_nat @ J @ I )
% 9.00/5.06 = K )
% 9.00/5.06 = ( J
% 9.00/5.06 = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3671_Nat_Odiff__add__assoc2,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 9.00/5.06 = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3672_Nat_Odiff__add__assoc,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.00/5.06 = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3673_Nat_Ole__diff__conv2,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.00/5.06 = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3674_le__diff__conv,axiom,
% 9.00/5.06 ! [J: nat,K: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.00/5.06 = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3675_add__strict__increasing2,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ C )
% 9.00/5.06 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3676_add__strict__increasing2,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ C )
% 9.00/5.06 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3677_add__strict__increasing2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ C )
% 9.00/5.06 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3678_add__strict__increasing2,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ C )
% 9.00/5.06 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3679_add__strict__increasing,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ C )
% 9.00/5.06 => ( ord_less_real @ B3 @ ( plus_plus_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3680_add__strict__increasing,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.06 => ( ord_less_rat @ B3 @ ( plus_plus_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3681_add__strict__increasing,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.06 => ( ord_less_nat @ B3 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3682_add__strict__increasing,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.06 => ( ord_less_int @ B3 @ ( plus_plus_int @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3683_add__pos__nonneg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3684_add__pos__nonneg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3685_add__pos__nonneg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3686_add__pos__nonneg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3687_add__nonpos__neg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3688_add__nonpos__neg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3689_add__nonpos__neg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3690_add__nonpos__neg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3691_add__nonneg__pos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3692_add__nonneg__pos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3693_add__nonneg__pos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3694_add__nonneg__pos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3695_add__neg__nonpos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( plus_plus_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3696_add__neg__nonpos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( plus_plus_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3697_add__neg__nonpos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3698_add__neg__nonpos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( plus_plus_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3699_field__le__epsilon,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ! [E: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ E )
% 9.00/5.06 => ( ord_less_eq_real @ X2 @ ( plus_plus_real @ Y3 @ E ) ) )
% 9.00/5.06 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3700_field__le__epsilon,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ! [E: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.00/5.06 => ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ Y3 @ E ) ) )
% 9.00/5.06 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3701_discrete,axiom,
% 9.00/5.06 ( ord_less_nat
% 9.00/5.06 = ( ^ [A4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A4 @ one_one_nat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3702_discrete,axiom,
% 9.00/5.06 ( ord_less_int
% 9.00/5.06 = ( ^ [A4: int] : ( ord_less_eq_int @ ( plus_plus_int @ A4 @ one_one_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3703_zero__less__two,axiom,
% 9.00/5.06 ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3704_zero__less__two,axiom,
% 9.00/5.06 ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3705_zero__less__two,axiom,
% 9.00/5.06 ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3706_zero__less__two,axiom,
% 9.00/5.06 ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3707_div__add__self2,axiom,
% 9.00/5.06 ! [B3: int,A2: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3708_div__add__self2,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3709_div__add__self1,axiom,
% 9.00/5.06 ! [B3: int,A2: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ B3 @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3710_div__add__self1,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ B3 @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ ( divide_divide_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3711_gt__half__sum,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3712_gt__half__sum,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3713_less__half__sum,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_rat @ A2 @ ( divide_divide_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3714_less__half__sum,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ord_less_real @ A2 @ ( divide_divide_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3715_abs__diff__le__iff,axiom,
% 9.00/5.06 ! [X2: code_integer,A2: code_integer,R2: code_integer] :
% 9.00/5.06 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_le3102999989581377725nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3716_abs__diff__le__iff,axiom,
% 9.00/5.06 ! [X2: real,A2: real,R2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_eq_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_eq_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3717_abs__diff__le__iff,axiom,
% 9.00/5.06 ! [X2: rat,A2: rat,R2: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3718_abs__diff__le__iff,axiom,
% 9.00/5.06 ! [X2: int,A2: int,R2: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_eq_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_eq_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3719_abs__diff__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A2 @ B3 ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3720_abs__diff__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3721_abs__diff__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A2 @ B3 ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3722_abs__diff__triangle__ineq,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A2 @ B3 ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B3 @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3723_abs__triangle__ineq4,axiom,
% 9.00/5.06 ! [A2: code_integer,B3: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A2 @ B3 ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3724_abs__triangle__ineq4,axiom,
% 9.00/5.06 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3725_abs__triangle__ineq4,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A2 @ B3 ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3726_abs__triangle__ineq4,axiom,
% 9.00/5.06 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B3 ) ) @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3727_abs__diff__less__iff,axiom,
% 9.00/5.06 ! [X2: code_integer,A2: code_integer,R2: code_integer] :
% 9.00/5.06 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_le6747313008572928689nteger @ X2 @ ( plus_p5714425477246183910nteger @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3728_abs__diff__less__iff,axiom,
% 9.00/5.06 ! [X2: real,A2: real,R2: real] :
% 9.00/5.06 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_real @ ( minus_minus_real @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_real @ X2 @ ( plus_plus_real @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3729_abs__diff__less__iff,axiom,
% 9.00/5.06 ! [X2: rat,A2: rat,R2: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_rat @ ( minus_minus_rat @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_rat @ X2 @ ( plus_plus_rat @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3730_abs__diff__less__iff,axiom,
% 9.00/5.06 ! [X2: int,A2: int,R2: int] :
% 9.00/5.06 ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ A2 ) ) @ R2 )
% 9.00/5.06 = ( ( ord_less_int @ ( minus_minus_int @ A2 @ R2 ) @ X2 )
% 9.00/5.06 & ( ord_less_int @ X2 @ ( plus_plus_int @ A2 @ R2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3731_real__arch__pow__inv,axiom,
% 9.00/5.06 ! [Y3: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.06 => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X2 @ N3 ) @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3732_nat__diff__split__asm,axiom,
% 9.00/5.06 ! [P: nat > $o,A2: nat,B3: nat] :
% 9.00/5.06 ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.06 = ( ~ ( ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 & ~ ( P @ zero_zero_nat ) )
% 9.00/5.06 | ? [D5: nat] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( plus_plus_nat @ B3 @ D5 ) )
% 9.00/5.06 & ~ ( P @ D5 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3733_nat__diff__split,axiom,
% 9.00/5.06 ! [P: nat > $o,A2: nat,B3: nat] :
% 9.00/5.06 ( ( P @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( P @ zero_zero_nat ) )
% 9.00/5.06 & ! [D5: nat] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( plus_plus_nat @ B3 @ D5 ) )
% 9.00/5.06 => ( P @ D5 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3734_less__diff__conv2,axiom,
% 9.00/5.06 ! [K: nat,J: nat,I: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ J )
% 9.00/5.06 => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.00/5.06 = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3735_abs__add__one__gt__zero,axiom,
% 9.00/5.06 ! [X2: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3736_abs__add__one__gt__zero,axiom,
% 9.00/5.06 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3737_abs__add__one__gt__zero,axiom,
% 9.00/5.06 ! [X2: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3738_abs__add__one__gt__zero,axiom,
% 9.00/5.06 ! [X2: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3739_add__eq__if,axiom,
% 9.00/5.06 ( plus_plus_nat
% 9.00/5.06 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3740_frac__add,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.00/5.06 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 9.00/5.06 & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.00/5.06 => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X2 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3741_frac__add,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.00/5.06 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
% 9.00/5.06 = ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 9.00/5.06 & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.00/5.06 => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X2 @ Y3 ) )
% 9.00/5.06 = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X2 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3742_option_Osize__gen_I2_J,axiom,
% 9.00/5.06 ! [X2: nat > nat,X22: nat] :
% 9.00/5.06 ( ( size_option_nat @ X2 @ ( some_nat @ X22 ) )
% 9.00/5.06 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3743_option_Osize__gen_I2_J,axiom,
% 9.00/5.06 ! [X2: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 9.00/5.06 ( ( size_o8335143837870341156at_nat @ X2 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.00/5.06 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3744_option_Osize__gen_I2_J,axiom,
% 9.00/5.06 ! [X2: num > nat,X22: num] :
% 9.00/5.06 ( ( size_option_num @ X2 @ ( some_num @ X22 ) )
% 9.00/5.06 = ( plus_plus_nat @ ( X2 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3745_power__not__zero,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_rat )
% 9.00/5.06 => ( ( power_power_rat @ A2 @ N )
% 9.00/5.06 != zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3746_power__not__zero,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_int )
% 9.00/5.06 => ( ( power_power_int @ A2 @ N )
% 9.00/5.06 != zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3747_power__not__zero,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_nat @ A2 @ N )
% 9.00/5.06 != zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3748_power__not__zero,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_real )
% 9.00/5.06 => ( ( power_power_real @ A2 @ N )
% 9.00/5.06 != zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3749_power__not__zero,axiom,
% 9.00/5.06 ! [A2: complex,N: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_complex )
% 9.00/5.06 => ( ( power_power_complex @ A2 @ N )
% 9.00/5.06 != zero_zero_complex ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3750_nat0__intermed__int__val,axiom,
% 9.00/5.06 ! [N: nat,F: nat > int,K: int] :
% 9.00/5.06 ( ! [I2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I2 @ N )
% 9.00/5.06 => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.00/5.06 => ( ( ord_less_eq_int @ K @ ( F @ N ) )
% 9.00/5.06 => ? [I2: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ I2 @ N )
% 9.00/5.06 & ( ( F @ I2 )
% 9.00/5.06 = K ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3751_power__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3752_power__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3753_power__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3754_power__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3755_zero__le__power,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3756_zero__le__power,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3757_zero__le__power,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3758_zero__le__power,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3759_zero__less__power,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3760_zero__less__power,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3761_zero__less__power,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3762_zero__less__power,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3763_one__le__power,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3764_one__le__power,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3765_one__le__power,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3766_one__le__power,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3767_power__0,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( power_power_rat @ A2 @ zero_zero_nat )
% 9.00/5.06 = one_one_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3768_power__0,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( power_power_int @ A2 @ zero_zero_nat )
% 9.00/5.06 = one_one_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3769_power__0,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( power_power_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 = one_one_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3770_power__0,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( power_power_real @ A2 @ zero_zero_nat )
% 9.00/5.06 = one_one_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3771_power__0,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( power_power_complex @ A2 @ zero_zero_nat )
% 9.00/5.06 = one_one_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3772_power__less__imp__less__base,axiom,
% 9.00/5.06 ! [A2: real,N: nat,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3773_power__less__imp__less__base,axiom,
% 9.00/5.06 ! [A2: rat,N: nat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3774_power__less__imp__less__base,axiom,
% 9.00/5.06 ! [A2: nat,N: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3775_power__less__imp__less__base,axiom,
% 9.00/5.06 ! [A2: int,N: nat,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3776_power__le__one,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3777_power__le__one,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3778_power__le__one,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3779_power__le__one,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3780_power__inject__base,axiom,
% 9.00/5.06 ! [A2: real,N: nat,B3: real] :
% 9.00/5.06 ( ( ( power_power_real @ A2 @ ( suc @ N ) )
% 9.00/5.06 = ( power_power_real @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3781_power__inject__base,axiom,
% 9.00/5.06 ! [A2: rat,N: nat,B3: rat] :
% 9.00/5.06 ( ( ( power_power_rat @ A2 @ ( suc @ N ) )
% 9.00/5.06 = ( power_power_rat @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3782_power__inject__base,axiom,
% 9.00/5.06 ! [A2: nat,N: nat,B3: nat] :
% 9.00/5.06 ( ( ( power_power_nat @ A2 @ ( suc @ N ) )
% 9.00/5.06 = ( power_power_nat @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3783_power__inject__base,axiom,
% 9.00/5.06 ! [A2: int,N: nat,B3: int] :
% 9.00/5.06 ( ( ( power_power_int @ A2 @ ( suc @ N ) )
% 9.00/5.06 = ( power_power_int @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3784_power__le__imp__le__base,axiom,
% 9.00/5.06 ! [A2: real,N: nat,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ ( power_power_real @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3785_power__le__imp__le__base,axiom,
% 9.00/5.06 ! [A2: rat,N: nat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ ( power_power_rat @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3786_power__le__imp__le__base,axiom,
% 9.00/5.06 ! [A2: nat,N: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ ( power_power_nat @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3787_power__le__imp__le__base,axiom,
% 9.00/5.06 ! [A2: int,N: nat,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ ( power_power_int @ B3 @ ( suc @ N ) ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3788_power__gt1,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ ( suc @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3789_power__gt1,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3790_power__gt1,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3791_power__gt1,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ ( suc @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3792_power__0__left,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ( N = zero_zero_nat )
% 9.00/5.06 => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.00/5.06 = one_one_rat ) )
% 9.00/5.06 & ( ( N != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.00/5.06 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3793_power__0__left,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ( N = zero_zero_nat )
% 9.00/5.06 => ( ( power_power_int @ zero_zero_int @ N )
% 9.00/5.06 = one_one_int ) )
% 9.00/5.06 & ( ( N != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_int @ zero_zero_int @ N )
% 9.00/5.06 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3794_power__0__left,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ( N = zero_zero_nat )
% 9.00/5.06 => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.00/5.06 = one_one_nat ) )
% 9.00/5.06 & ( ( N != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.00/5.06 = zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3795_power__0__left,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ( N = zero_zero_nat )
% 9.00/5.06 => ( ( power_power_real @ zero_zero_real @ N )
% 9.00/5.06 = one_one_real ) )
% 9.00/5.06 & ( ( N != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_real @ zero_zero_real @ N )
% 9.00/5.06 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3796_power__0__left,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ( N = zero_zero_nat )
% 9.00/5.06 => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.00/5.06 = one_one_complex ) )
% 9.00/5.06 & ( ( N != zero_zero_nat )
% 9.00/5.06 => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.00/5.06 = zero_zero_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3797_power__less__imp__less__exp,axiom,
% 9.00/5.06 ! [A2: real,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3798_power__less__imp__less__exp,axiom,
% 9.00/5.06 ! [A2: rat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3799_power__less__imp__less__exp,axiom,
% 9.00/5.06 ! [A2: nat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3800_power__less__imp__less__exp,axiom,
% 9.00/5.06 ! [A2: int,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3801_power__strict__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3802_power__strict__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: rat] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3803_power__strict__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3804_power__strict__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: int] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3805_zero__le__power__abs,axiom,
% 9.00/5.06 ! [A2: code_integer,N: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A2 ) @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3806_zero__le__power__abs,axiom,
% 9.00/5.06 ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3807_zero__le__power__abs,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A2 ) @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3808_zero__le__power__abs,axiom,
% 9.00/5.06 ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A2 ) @ N ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3809_power__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3810_power__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: rat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3811_power__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3812_power__increasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: int] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ A2 @ N6 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3813_zero__power,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_rat @ zero_zero_rat @ N )
% 9.00/5.06 = zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3814_zero__power,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_int @ zero_zero_int @ N )
% 9.00/5.06 = zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3815_zero__power,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_nat @ zero_zero_nat @ N )
% 9.00/5.06 = zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3816_zero__power,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_real @ zero_zero_real @ N )
% 9.00/5.06 = zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3817_zero__power,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_complex @ zero_zero_complex @ N )
% 9.00/5.06 = zero_zero_complex ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3818_power__Suc__le__self,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3819_power__Suc__le__self,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3820_power__Suc__le__self,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3821_power__Suc__le__self,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3822_power__Suc__less__one,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ A2 @ one_one_real )
% 9.00/5.06 => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ N ) ) @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3823_power__Suc__less__one,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 9.00/5.06 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ N ) ) @ one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3824_power__Suc__less__one,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 9.00/5.06 => ( ord_less_nat @ ( power_power_nat @ A2 @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3825_power__Suc__less__one,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ A2 @ one_one_int )
% 9.00/5.06 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3826_power__strict__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ A2 @ one_one_real )
% 9.00/5.06 => ( ord_less_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3827_power__strict__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: rat] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 9.00/5.06 => ( ord_less_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3828_power__strict__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 9.00/5.06 => ( ord_less_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3829_power__strict__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: int] :
% 9.00/5.06 ( ( ord_less_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ A2 @ one_one_int )
% 9.00/5.06 => ( ord_less_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3830_power__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ A2 @ one_one_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ A2 @ N6 ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3831_power__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: rat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( power_power_rat @ A2 @ N6 ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3832_power__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( power_power_nat @ A2 @ N6 ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3833_power__decreasing,axiom,
% 9.00/5.06 ! [N: nat,N6: nat,A2: int] :
% 9.00/5.06 ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ A2 @ one_one_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( power_power_int @ A2 @ N6 ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3834_power__le__imp__le__exp,axiom,
% 9.00/5.06 ! [A2: real,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3835_power__le__imp__le__exp,axiom,
% 9.00/5.06 ! [A2: rat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3836_power__le__imp__le__exp,axiom,
% 9.00/5.06 ! [A2: nat,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3837_power__le__imp__le__exp,axiom,
% 9.00/5.06 ! [A2: int,M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) )
% 9.00/5.06 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3838_power__eq__imp__eq__base,axiom,
% 9.00/5.06 ! [A2: real,N: nat,B3: real] :
% 9.00/5.06 ( ( ( power_power_real @ A2 @ N )
% 9.00/5.06 = ( power_power_real @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3839_power__eq__imp__eq__base,axiom,
% 9.00/5.06 ! [A2: rat,N: nat,B3: rat] :
% 9.00/5.06 ( ( ( power_power_rat @ A2 @ N )
% 9.00/5.06 = ( power_power_rat @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3840_power__eq__imp__eq__base,axiom,
% 9.00/5.06 ! [A2: nat,N: nat,B3: nat] :
% 9.00/5.06 ( ( ( power_power_nat @ A2 @ N )
% 9.00/5.06 = ( power_power_nat @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3841_power__eq__imp__eq__base,axiom,
% 9.00/5.06 ! [A2: int,N: nat,B3: int] :
% 9.00/5.06 ( ( ( power_power_int @ A2 @ N )
% 9.00/5.06 = ( power_power_int @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3842_power__eq__iff__eq__base,axiom,
% 9.00/5.06 ! [N: nat,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ( ( power_power_real @ A2 @ N )
% 9.00/5.06 = ( power_power_real @ B3 @ N ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3843_power__eq__iff__eq__base,axiom,
% 9.00/5.06 ! [N: nat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ( ( power_power_rat @ A2 @ N )
% 9.00/5.06 = ( power_power_rat @ B3 @ N ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3844_power__eq__iff__eq__base,axiom,
% 9.00/5.06 ! [N: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ( ( power_power_nat @ A2 @ N )
% 9.00/5.06 = ( power_power_nat @ B3 @ N ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3845_power__eq__iff__eq__base,axiom,
% 9.00/5.06 ! [N: nat,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ( ( power_power_int @ A2 @ N )
% 9.00/5.06 = ( power_power_int @ B3 @ N ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3846_self__le__power,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3847_self__le__power,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3848_self__le__power,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3849_self__le__power,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3850_one__less__power,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_real @ one_one_real @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3851_one__less__power,axiom,
% 9.00/5.06 ! [A2: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3852_one__less__power,axiom,
% 9.00/5.06 ! [A2: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3853_one__less__power,axiom,
% 9.00/5.06 ! [A2: int,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_int @ one_one_int @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3854_power__diff,axiom,
% 9.00/5.06 ! [A2: rat,N: nat,M: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.06 => ( ( power_power_rat @ A2 @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = ( divide_divide_rat @ ( power_power_rat @ A2 @ M ) @ ( power_power_rat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3855_power__diff,axiom,
% 9.00/5.06 ! [A2: int,N: nat,M: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.06 => ( ( power_power_int @ A2 @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = ( divide_divide_int @ ( power_power_int @ A2 @ M ) @ ( power_power_int @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3856_power__diff,axiom,
% 9.00/5.06 ! [A2: nat,N: nat,M: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.06 => ( ( power_power_nat @ A2 @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = ( divide_divide_nat @ ( power_power_nat @ A2 @ M ) @ ( power_power_nat @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3857_power__diff,axiom,
% 9.00/5.06 ! [A2: real,N: nat,M: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.06 => ( ( power_power_real @ A2 @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = ( divide_divide_real @ ( power_power_real @ A2 @ M ) @ ( power_power_real @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3858_power__diff,axiom,
% 9.00/5.06 ! [A2: complex,N: nat,M: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_complex )
% 9.00/5.06 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.06 => ( ( power_power_complex @ A2 @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ ( power_power_complex @ A2 @ M ) @ ( power_power_complex @ A2 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3859_power__strict__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( power_power_real @ B3 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3860_power__strict__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,N: nat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( power_power_rat @ B3 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3861_power__strict__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( power_power_nat @ B3 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3862_power__strict__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,N: nat] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( power_power_int @ B3 @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3863_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.00/5.06 ! [K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.06 => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3864_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.00/5.06 ! [K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.06 => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3865_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.00/5.06 ! [K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.06 => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3866_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.00/5.06 ! [K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.06 => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3867_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.00/5.06 ! [K: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.06 => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K ) )
% 9.00/5.06 = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3868_add__shift,axiom,
% 9.00/5.06 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.06 ( ( ( plus_plus_nat @ X2 @ Y3 )
% 9.00/5.06 = Z )
% 9.00/5.06 = ( ( vEBT_VEBT_add @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 9.00/5.06 = ( some_nat @ Z ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3869_lemma__interval,axiom,
% 9.00/5.06 ! [A2: real,X2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ B3 )
% 9.00/5.06 => ? [D6: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.06 & ! [Y5: real] :
% 9.00/5.06 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
% 9.00/5.06 => ( ( ord_less_eq_real @ A2 @ Y5 )
% 9.00/5.06 & ( ord_less_eq_real @ Y5 @ B3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3870_realpow__pos__nth__unique,axiom,
% 9.00/5.06 ! [N: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ? [X5: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X5 )
% 9.00/5.06 & ( ( power_power_real @ X5 @ N )
% 9.00/5.06 = A2 )
% 9.00/5.06 & ! [Y5: real] :
% 9.00/5.06 ( ( ( ord_less_real @ zero_zero_real @ Y5 )
% 9.00/5.06 & ( ( power_power_real @ Y5 @ N )
% 9.00/5.06 = A2 ) )
% 9.00/5.06 => ( Y5 = X5 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3871_realpow__pos__nth,axiom,
% 9.00/5.06 ! [N: nat,A2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ? [R3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.00/5.06 & ( ( power_power_real @ R3 @ N )
% 9.00/5.06 = A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3872_lemma__interval__lt,axiom,
% 9.00/5.06 ! [A2: real,X2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ B3 )
% 9.00/5.06 => ? [D6: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.06 & ! [Y5: real] :
% 9.00/5.06 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y5 ) ) @ D6 )
% 9.00/5.06 => ( ( ord_less_real @ A2 @ Y5 )
% 9.00/5.06 & ( ord_less_real @ Y5 @ B3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3873_realpow__pos__nth2,axiom,
% 9.00/5.06 ! [A2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ? [R3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.00/5.06 & ( ( power_power_real @ R3 @ ( suc @ N ) )
% 9.00/5.06 = A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3874_add__def,axiom,
% 9.00/5.06 ( vEBT_VEBT_add
% 9.00/5.06 = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3875_nth__enumerate__eq,axiom,
% 9.00/5.06 ! [M: nat,Xs: list_int,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs ) @ M )
% 9.00/5.06 = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3876_nth__enumerate__eq,axiom,
% 9.00/5.06 ! [M: nat,Xs: list_VEBT_VEBT,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs ) @ M )
% 9.00/5.06 = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3877_nth__enumerate__eq,axiom,
% 9.00/5.06 ! [M: nat,Xs: list_o,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 9.00/5.06 => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs ) @ M )
% 9.00/5.06 = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3878_nth__enumerate__eq,axiom,
% 9.00/5.06 ! [M: nat,Xs: list_nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 9.00/5.06 => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M )
% 9.00/5.06 = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3879_zle__add1__eq__le,axiom,
% 9.00/5.06 ! [W2: int,Z: int] :
% 9.00/5.06 ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.00/5.06 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3880_real__0__less__add__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3881_real__add__less__0__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3882_real__add__le__0__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( plus_plus_real @ X2 @ Y3 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_eq_real @ Y3 @ ( uminus_uminus_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3883_real__0__le__add__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3884_int__ge__induct,axiom,
% 9.00/5.06 ! [K: int,I: int,P: int > $o] :
% 9.00/5.06 ( ( ord_less_eq_int @ K @ I )
% 9.00/5.06 => ( ( P @ K )
% 9.00/5.06 => ( ! [I2: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ K @ I2 )
% 9.00/5.06 => ( ( P @ I2 )
% 9.00/5.06 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.06 => ( P @ I ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3885_zless__add1__eq,axiom,
% 9.00/5.06 ! [W2: int,Z: int] :
% 9.00/5.06 ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z @ one_one_int ) )
% 9.00/5.06 = ( ( ord_less_int @ W2 @ Z )
% 9.00/5.06 | ( W2 = Z ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3886_int__gr__induct,axiom,
% 9.00/5.06 ! [K: int,I: int,P: int > $o] :
% 9.00/5.06 ( ( ord_less_int @ K @ I )
% 9.00/5.06 => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 9.00/5.06 => ( ! [I2: int] :
% 9.00/5.06 ( ( ord_less_int @ K @ I2 )
% 9.00/5.06 => ( ( P @ I2 )
% 9.00/5.06 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.06 => ( P @ I ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3887_zle__iff__zadd,axiom,
% 9.00/5.06 ( ord_less_eq_int
% 9.00/5.06 = ( ^ [W3: int,Z2: int] :
% 9.00/5.06 ? [N2: nat] :
% 9.00/5.06 ( Z2
% 9.00/5.06 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3888_zless__iff__Suc__zadd,axiom,
% 9.00/5.06 ( ord_less_int
% 9.00/5.06 = ( ^ [W3: int,Z2: int] :
% 9.00/5.06 ? [N2: nat] :
% 9.00/5.06 ( Z2
% 9.00/5.06 = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3889_odd__less__0__iff,axiom,
% 9.00/5.06 ! [Z: int] :
% 9.00/5.06 ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
% 9.00/5.06 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3890_add1__zle__eq,axiom,
% 9.00/5.06 ! [W2: int,Z: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z )
% 9.00/5.06 = ( ord_less_int @ W2 @ Z ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3891_zless__imp__add1__zle,axiom,
% 9.00/5.06 ! [W2: int,Z: int] :
% 9.00/5.06 ( ( ord_less_int @ W2 @ Z )
% 9.00/5.06 => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3892_int__induct,axiom,
% 9.00/5.06 ! [P: int > $o,K: int,I: int] :
% 9.00/5.06 ( ( P @ K )
% 9.00/5.06 => ( ! [I2: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ K @ I2 )
% 9.00/5.06 => ( ( P @ I2 )
% 9.00/5.06 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.06 => ( ! [I2: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ I2 @ K )
% 9.00/5.06 => ( ( P @ I2 )
% 9.00/5.06 => ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
% 9.00/5.06 => ( P @ I ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3893_nat__less__real__le,axiom,
% 9.00/5.06 ( ord_less_nat
% 9.00/5.06 = ( ^ [N2: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3894_nat__le__real__less,axiom,
% 9.00/5.06 ( ord_less_eq_nat
% 9.00/5.06 = ( ^ [N2: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3895_le__imp__0__less,axiom,
% 9.00/5.06 ! [Z: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3896_div__pos__neg__trivial,axiom,
% 9.00/5.06 ! [K: int,L: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.06 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ K @ L )
% 9.00/5.06 = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3897_div__pos__geq,axiom,
% 9.00/5.06 ! [L: int,K: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.06 => ( ( ord_less_eq_int @ L @ K )
% 9.00/5.06 => ( ( divide_divide_int @ K @ L )
% 9.00/5.06 = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3898_sin__bound__lemma,axiom,
% 9.00/5.06 ! [X2: real,Y3: real,U: real,V: real] :
% 9.00/5.06 ( ( X2 = Y3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
% 9.00/5.06 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X2 @ U ) @ Y3 ) ) @ V ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3899_Euclid__induct,axiom,
% 9.00/5.06 ! [P: nat > nat > $o,A2: nat,B3: nat] :
% 9.00/5.06 ( ! [A: nat,B: nat] :
% 9.00/5.06 ( ( P @ A @ B )
% 9.00/5.06 = ( P @ B @ A ) )
% 9.00/5.06 => ( ! [A: nat] : ( P @ A @ zero_zero_nat )
% 9.00/5.06 => ( ! [A: nat,B: nat] :
% 9.00/5.06 ( ( P @ A @ B )
% 9.00/5.06 => ( P @ A @ ( plus_plus_nat @ A @ B ) ) )
% 9.00/5.06 => ( P @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3900_add__0__iff,axiom,
% 9.00/5.06 ! [B3: real,A2: real] :
% 9.00/5.06 ( ( B3
% 9.00/5.06 = ( plus_plus_real @ B3 @ A2 ) )
% 9.00/5.06 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3901_add__0__iff,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat] :
% 9.00/5.06 ( ( B3
% 9.00/5.06 = ( plus_plus_rat @ B3 @ A2 ) )
% 9.00/5.06 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3902_add__0__iff,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat] :
% 9.00/5.06 ( ( B3
% 9.00/5.06 = ( plus_plus_nat @ B3 @ A2 ) )
% 9.00/5.06 = ( A2 = zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3903_add__0__iff,axiom,
% 9.00/5.06 ! [B3: int,A2: int] :
% 9.00/5.06 ( ( B3
% 9.00/5.06 = ( plus_plus_int @ B3 @ A2 ) )
% 9.00/5.06 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3904_exp__ge__one__minus__x__over__n__power__n,axiom,
% 9.00/5.06 ! [X2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3905_frac__unique__iff,axiom,
% 9.00/5.06 ! [X2: real,A2: real] :
% 9.00/5.06 ( ( ( archim2898591450579166408c_real @ X2 )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( ( member_real @ ( minus_minus_real @ X2 @ A2 ) @ ring_1_Ints_real )
% 9.00/5.06 & ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_real @ A2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3906_frac__unique__iff,axiom,
% 9.00/5.06 ! [X2: rat,A2: rat] :
% 9.00/5.06 ( ( ( archimedean_frac_rat @ X2 )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( ( member_rat @ ( minus_minus_rat @ X2 @ A2 ) @ ring_1_Ints_rat )
% 9.00/5.06 & ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_rat @ A2 @ one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3907_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_int,Ys2: list_int] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr4439495888332055232nt_int @ ( zip_int_int @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( product_Pair_int_int @ ( nth_int @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3908_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr3474266648193625910T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3909_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_int,Ys2: list_o] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr7514405829937366042_int_o @ ( zip_int_o @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( product_Pair_int_o @ ( nth_int @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3910_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_int,Ys2: list_nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr8617346907841251940nt_nat @ ( zip_int_nat @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( product_Pair_int_nat @ ( nth_int @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3911_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3912_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3913_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3914_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3915_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_o,Ys2: list_int] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3916_nth__zip,axiom,
% 9.00/5.06 ! [I: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
% 9.00/5.06 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.06 => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.00/5.06 => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys2 ) @ I )
% 9.00/5.06 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys2 @ I ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3917_exp__less__mono,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.06 => ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3918_exp__less__cancel__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3919_exp__le__cancel__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3920_exp__zero,axiom,
% 9.00/5.06 ( ( exp_complex @ zero_zero_complex )
% 9.00/5.06 = one_one_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3921_exp__zero,axiom,
% 9.00/5.06 ( ( exp_real @ zero_zero_real )
% 9.00/5.06 = one_one_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3922_frac__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ( archim2898591450579166408c_real @ X2 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( member_real @ X2 @ ring_1_Ints_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3923_frac__eq__0__iff,axiom,
% 9.00/5.06 ! [X2: rat] :
% 9.00/5.06 ( ( ( archimedean_frac_rat @ X2 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( member_rat @ X2 @ ring_1_Ints_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3924_exp__less__one__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ ( exp_real @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3925_one__less__exp__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) )
% 9.00/5.06 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3926_exp__le__one__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( exp_real @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3927_one__le__exp__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3928_frac__gt__0__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X2 ) )
% 9.00/5.06 = ( ~ ( member_real @ X2 @ ring_1_Ints_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3929_frac__gt__0__iff,axiom,
% 9.00/5.06 ! [X2: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X2 ) )
% 9.00/5.06 = ( ~ ( member_rat @ X2 @ ring_1_Ints_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3930_exp__not__eq__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( exp_real @ X2 )
% 9.00/5.06 != zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3931_exp__less__cancel,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( exp_real @ X2 ) @ ( exp_real @ Y3 ) )
% 9.00/5.06 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3932_Ints__0,axiom,
% 9.00/5.06 member_real @ zero_zero_real @ ring_1_Ints_real ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3933_Ints__0,axiom,
% 9.00/5.06 member_rat @ zero_zero_rat @ ring_1_Ints_rat ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3934_Ints__0,axiom,
% 9.00/5.06 member_int @ zero_zero_int @ ring_1_Ints_int ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3935_not__exp__less__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ~ ( ord_less_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3936_exp__gt__zero,axiom,
% 9.00/5.06 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3937_exp__total,axiom,
% 9.00/5.06 ! [Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ? [X5: real] :
% 9.00/5.06 ( ( exp_real @ X5 )
% 9.00/5.06 = Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3938_exp__ge__zero,axiom,
% 9.00/5.06 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3939_not__exp__le__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ~ ( ord_less_eq_real @ ( exp_real @ X2 ) @ zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3940_exp__ge__add__one__self,axiom,
% 9.00/5.06 ! [X2: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3941_exp__gt__one,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_real @ one_one_real @ ( exp_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3942_exp__ge__add__one__self__aux,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( exp_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3943_Ints__double__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( ( plus_plus_real @ A2 @ A2 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( A2 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3944_Ints__double__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( ( plus_plus_rat @ A2 @ A2 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( A2 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3945_Ints__double__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( member_int @ A2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( ( plus_plus_int @ A2 @ A2 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( A2 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3946_lemma__exp__total,axiom,
% 9.00/5.06 ! [Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ one_one_real @ Y3 )
% 9.00/5.06 => ? [X5: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.00/5.06 & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y3 @ one_one_real ) )
% 9.00/5.06 & ( ( exp_real @ X5 )
% 9.00/5.06 = Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3947_Ints__odd__nonzero,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( member_complex @ A2 @ ring_1_Ints_complex )
% 9.00/5.06 => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A2 ) @ A2 )
% 9.00/5.06 != zero_zero_complex ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3948_Ints__odd__nonzero,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 )
% 9.00/5.06 != zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3949_Ints__odd__nonzero,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 )
% 9.00/5.06 != zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3950_Ints__odd__nonzero,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( member_int @ A2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 )
% 9.00/5.06 != zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3951_exp__divide__power__eq,axiom,
% 9.00/5.06 ! [N: nat,X2: complex] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X2 @ ( semiri8010041392384452111omplex @ N ) ) ) @ N )
% 9.00/5.06 = ( exp_complex @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3952_exp__divide__power__eq,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
% 9.00/5.06 = ( exp_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3953_Ints__odd__less__0,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A2 ) @ A2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ A2 @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3954_Ints__odd__less__0,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A2 ) @ A2 ) @ zero_zero_rat )
% 9.00/5.06 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3955_Ints__odd__less__0,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( member_int @ A2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A2 ) @ A2 ) @ zero_zero_int )
% 9.00/5.06 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3956_Ints__nonzero__abs__ge1,axiom,
% 9.00/5.06 ! [X2: code_integer] :
% 9.00/5.06 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 9.00/5.06 => ( ( X2 != zero_z3403309356797280102nteger )
% 9.00/5.06 => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3957_Ints__nonzero__abs__ge1,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( member_real @ X2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( X2 != zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3958_Ints__nonzero__abs__ge1,axiom,
% 9.00/5.06 ! [X2: rat] :
% 9.00/5.06 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( X2 != zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3959_Ints__nonzero__abs__ge1,axiom,
% 9.00/5.06 ! [X2: int] :
% 9.00/5.06 ( ( member_int @ X2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( X2 != zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3960_Ints__nonzero__abs__less1,axiom,
% 9.00/5.06 ! [X2: code_integer] :
% 9.00/5.06 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 9.00/5.06 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer )
% 9.00/5.06 => ( X2 = zero_z3403309356797280102nteger ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3961_Ints__nonzero__abs__less1,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( member_real @ X2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.06 => ( X2 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3962_Ints__nonzero__abs__less1,axiom,
% 9.00/5.06 ! [X2: rat] :
% 9.00/5.06 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat )
% 9.00/5.06 => ( X2 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3963_Ints__nonzero__abs__less1,axiom,
% 9.00/5.06 ! [X2: int] :
% 9.00/5.06 ( ( member_int @ X2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int )
% 9.00/5.06 => ( X2 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3964_Ints__eq__abs__less1,axiom,
% 9.00/5.06 ! [X2: code_integer,Y3: code_integer] :
% 9.00/5.06 ( ( member_Code_integer @ X2 @ ring_11222124179247155820nteger )
% 9.00/5.06 => ( ( member_Code_integer @ Y3 @ ring_11222124179247155820nteger )
% 9.00/5.06 => ( ( X2 = Y3 )
% 9.00/5.06 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X2 @ Y3 ) ) @ one_one_Code_integer ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3965_Ints__eq__abs__less1,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( member_real @ X2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( member_real @ Y3 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( X2 = Y3 )
% 9.00/5.06 = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3966_Ints__eq__abs__less1,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( member_rat @ Y3 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( X2 = Y3 )
% 9.00/5.06 = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3967_Ints__eq__abs__less1,axiom,
% 9.00/5.06 ! [X2: int,Y3: int] :
% 9.00/5.06 ( ( member_int @ X2 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( member_int @ Y3 @ ring_1_Ints_int )
% 9.00/5.06 => ( ( X2 = Y3 )
% 9.00/5.06 = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3968_frac__neg,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ( member_real @ X2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.06 = zero_zero_real ) )
% 9.00/5.06 & ( ~ ( member_real @ X2 @ ring_1_Ints_real )
% 9.00/5.06 => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.06 = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3969_frac__neg,axiom,
% 9.00/5.06 ! [X2: rat] :
% 9.00/5.06 ( ( ( member_rat @ X2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
% 9.00/5.06 = zero_zero_rat ) )
% 9.00/5.06 & ( ~ ( member_rat @ X2 @ ring_1_Ints_rat )
% 9.00/5.06 => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X2 ) )
% 9.00/5.06 = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3970_exp__ge__one__plus__x__over__n__power__n,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X2 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3971_arcosh__1,axiom,
% 9.00/5.06 ( ( arcosh_real @ one_one_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3972_artanh__0,axiom,
% 9.00/5.06 ( ( artanh_real @ zero_zero_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3973_arsinh__0,axiom,
% 9.00/5.06 ( ( arsinh_real @ zero_zero_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3974_artanh__minus__real,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.06 => ( ( artanh_real @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.06 = ( uminus_uminus_real @ ( artanh_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3975_ln__one,axiom,
% 9.00/5.06 ( ( ln_ln_real @ one_one_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3976_sinh__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ( sinh_real @ X2 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( member_real @ ( exp_real @ X2 ) @ ( insert_real @ one_one_real @ ( insert_real @ ( uminus_uminus_real @ one_one_real ) @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3977_sinh__zero__iff,axiom,
% 9.00/5.06 ! [X2: complex] :
% 9.00/5.06 ( ( ( sinh_complex @ X2 )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 = ( member_complex @ ( exp_complex @ X2 ) @ ( insert_complex @ one_one_complex @ ( insert_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ bot_bot_set_complex ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3978_Gcd__remove0__nat,axiom,
% 9.00/5.06 ! [M5: set_nat] :
% 9.00/5.06 ( ( finite_finite_nat @ M5 )
% 9.00/5.06 => ( ( gcd_Gcd_nat @ M5 )
% 9.00/5.06 = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3979_log__of__power__le,axiom,
% 9.00/5.06 ! [M: nat,B3: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.06 => ( ord_less_eq_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3980_sinh__real__less__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3981_sinh__real__le__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( sinh_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3982_ln__less__cancel__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3983_ln__inj__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ( ln_ln_real @ X2 )
% 9.00/5.06 = ( ln_ln_real @ Y3 ) )
% 9.00/5.06 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3984_sinh__real__pos__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 9.00/5.06 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3985_sinh__real__neg__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3986_sinh__real__nonpos__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( sinh_real @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3987_sinh__real__nonneg__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3988_sinh__0,axiom,
% 9.00/5.06 ( ( sinh_real @ zero_zero_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3989_ln__le__cancel__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3990_ln__less__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3991_ln__gt__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 = ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3992_ln__eq__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ( ln_ln_real @ X2 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( X2 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3993_Gcd__empty,axiom,
% 9.00/5.06 ( ( gcd_Gcd_nat @ bot_bot_set_nat )
% 9.00/5.06 = zero_zero_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3994_Gcd__empty,axiom,
% 9.00/5.06 ( ( gcd_Gcd_int @ bot_bot_set_int )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3995_exp__ln,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( exp_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 = X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3996_exp__ln__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ( exp_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 = X2 )
% 9.00/5.06 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3997_ln__ge__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3998_ln__le__zero__iff,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_3999_zero__less__log__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
% 9.00/5.06 = ( ord_less_real @ one_one_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4000_log__less__zero__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4001_one__less__log__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ ( log @ A2 @ X2 ) )
% 9.00/5.06 = ( ord_less_real @ A2 @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4002_log__less__one__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4003_log__less__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ord_less_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4004_log__eq__one,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( A2 != one_one_real )
% 9.00/5.06 => ( ( log @ A2 @ A2 )
% 9.00/5.06 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4005_Gcd__0__iff,axiom,
% 9.00/5.06 ! [A3: set_nat] :
% 9.00/5.06 ( ( ( gcd_Gcd_nat @ A3 )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 = ( ord_less_eq_set_nat @ A3 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4006_Gcd__0__iff,axiom,
% 9.00/5.06 ! [A3: set_int] :
% 9.00/5.06 ( ( ( gcd_Gcd_int @ A3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ord_less_eq_set_int @ A3 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4007_log__le__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4008_log__le__one__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4009_one__le__log__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ one_one_real @ ( log @ A2 @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ A2 @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4010_log__le__zero__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( log @ A2 @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4011_zero__le__log__cancel__iff,axiom,
% 9.00/5.06 ! [A2: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A2 @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4012_log__pow__cancel,axiom,
% 9.00/5.06 ! [A2: real,B3: nat] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( A2 != one_one_real )
% 9.00/5.06 => ( ( log @ A2 @ ( power_power_real @ A2 @ B3 ) )
% 9.00/5.06 = ( semiri5074537144036343181t_real @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4013_ln__less__self,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4014_ln__bound,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4015_ln__gt__zero__imp__gt__one,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4016_ln__less__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.06 => ( ord_less_real @ ( ln_ln_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4017_ln__gt__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4018_ln__ge__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4019_ln__ge__zero__imp__ge__one,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4020_ln__add__one__self__le__self,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4021_ln__eq__minus__one,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ( ln_ln_real @ X2 )
% 9.00/5.06 = ( minus_minus_real @ X2 @ one_one_real ) )
% 9.00/5.06 => ( X2 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4022_ln__div,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( ln_ln_real @ ( divide_divide_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4023_log__base__change,axiom,
% 9.00/5.06 ! [A2: real,B3: real,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( A2 != one_one_real )
% 9.00/5.06 => ( ( log @ B3 @ X2 )
% 9.00/5.06 = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4024_ln__ge__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X2 ) )
% 9.00/5.06 = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4025_ln__x__over__x__mono,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y3 ) @ Y3 ) @ ( divide_divide_real @ ( ln_ln_real @ X2 ) @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4026_less__log__of__power,axiom,
% 9.00/5.06 ! [B3: real,N: nat,M: real] :
% 9.00/5.06 ( ( ord_less_real @ ( power_power_real @ B3 @ N ) @ M )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4027_log__of__power__eq,axiom,
% 9.00/5.06 ! [M: nat,B3: real,N: nat] :
% 9.00/5.06 ( ( ( semiri5074537144036343181t_real @ M )
% 9.00/5.06 = ( power_power_real @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ( semiri5074537144036343181t_real @ N )
% 9.00/5.06 = ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4028_ln__le__minus__one,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4029_ln__diff__le,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X2 @ Y3 ) @ Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4030_ln__add__one__self__le__self2,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4031_log__divide,axiom,
% 9.00/5.06 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( A2 != one_one_real )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.06 => ( ( log @ A2 @ ( divide_divide_real @ X2 @ Y3 ) )
% 9.00/5.06 = ( minus_minus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4032_le__log__of__power,axiom,
% 9.00/5.06 ! [B3: real,N: nat,M: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( power_power_real @ B3 @ N ) @ M )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ M ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4033_log__base__pow,axiom,
% 9.00/5.06 ! [A2: real,N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( log @ ( power_power_real @ A2 @ N ) @ X2 )
% 9.00/5.06 = ( divide_divide_real @ ( log @ A2 @ X2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4034_ln__one__minus__pos__upper__bound,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) @ ( uminus_uminus_real @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4035_log__of__power__less,axiom,
% 9.00/5.06 ! [M: nat,B3: real,N: nat] :
% 9.00/5.06 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B3 @ N ) )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.06 => ( ord_less_real @ ( log @ B3 @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4036_log__root,axiom,
% 9.00/5.06 ! [N: nat,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( log @ B3 @ ( root @ N @ A2 ) )
% 9.00/5.06 = ( divide_divide_real @ ( log @ B3 @ A2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4037_decr__lemma,axiom,
% 9.00/5.06 ! [D: int,X2: int,Z: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.06 => ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4038_incr__lemma,axiom,
% 9.00/5.06 ! [D: int,Z: int,X2: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.06 => ( ord_less_int @ Z @ ( plus_plus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4039_Bernoulli__inequality,axiom,
% 9.00/5.06 ! [X2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4040_linear__plus__1__le__power,axiom,
% 9.00/5.06 ! [X2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X2 @ one_one_real ) @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4041_find__Some__iff,axiom,
% 9.00/5.06 ! [P: int > $o,Xs: list_int,X2: int] :
% 9.00/5.06 ( ( ( find_int @ P @ Xs )
% 9.00/5.06 = ( some_int @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_int @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_int @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4042_find__Some__iff,axiom,
% 9.00/5.06 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X2: product_prod_nat_nat] :
% 9.00/5.06 ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.00/5.06 = ( some_P7363390416028606310at_nat @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4043_find__Some__iff,axiom,
% 9.00/5.06 ! [P: num > $o,Xs: list_num,X2: num] :
% 9.00/5.06 ( ( ( find_num @ P @ Xs )
% 9.00/5.06 = ( some_num @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_num @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_num @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4044_find__Some__iff,axiom,
% 9.00/5.06 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 9.00/5.06 ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.00/5.06 = ( some_VEBT_VEBT @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4045_find__Some__iff,axiom,
% 9.00/5.06 ! [P: $o > $o,Xs: list_o,X2: $o] :
% 9.00/5.06 ( ( ( find_o @ P @ Xs )
% 9.00/5.06 = ( some_o @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_o @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_o @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4046_find__Some__iff,axiom,
% 9.00/5.06 ! [P: nat > $o,Xs: list_nat,X2: nat] :
% 9.00/5.06 ( ( ( find_nat @ P @ Xs )
% 9.00/5.06 = ( some_nat @ X2 ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_nat @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_nat @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4047_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: int,P: int > $o,Xs: list_int] :
% 9.00/5.06 ( ( ( some_int @ X2 )
% 9.00/5.06 = ( find_int @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_int @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_int @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4048_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.00/5.06 ( ( ( some_P7363390416028606310at_nat @ X2 )
% 9.00/5.06 = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4049_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: num,P: num > $o,Xs: list_num] :
% 9.00/5.06 ( ( ( some_num @ X2 )
% 9.00/5.06 = ( find_num @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_num @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_num @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_num @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4050_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.00/5.06 ( ( ( some_VEBT_VEBT @ X2 )
% 9.00/5.06 = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_VEBT_VEBT @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4051_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: $o,P: $o > $o,Xs: list_o] :
% 9.00/5.06 ( ( ( some_o @ X2 )
% 9.00/5.06 = ( find_o @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_o @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_o @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4052_find__Some__iff2,axiom,
% 9.00/5.06 ! [X2: nat,P: nat > $o,Xs: list_nat] :
% 9.00/5.06 ( ( ( some_nat @ X2 )
% 9.00/5.06 = ( find_nat @ P @ Xs ) )
% 9.00/5.06 = ( ? [I4: nat] :
% 9.00/5.06 ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
% 9.00/5.06 & ( P @ ( nth_nat @ Xs @ I4 ) )
% 9.00/5.06 & ( X2
% 9.00/5.06 = ( nth_nat @ Xs @ I4 ) )
% 9.00/5.06 & ! [J3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ J3 @ I4 )
% 9.00/5.06 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4053_mult__cancel__right,axiom,
% 9.00/5.06 ! [A2: complex,C: complex,B3: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = ( times_times_complex @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4054_mult__cancel__right,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4055_mult__cancel__right,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4056_mult__cancel__right,axiom,
% 9.00/5.06 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.06 ( ( ( times_times_nat @ A2 @ C )
% 9.00/5.06 = ( times_times_nat @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_nat )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4057_mult__cancel__right,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ( times_times_int @ A2 @ C )
% 9.00/5.06 = ( times_times_int @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4058_mult__cancel__left,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ C @ A2 )
% 9.00/5.06 = ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4059_mult__cancel__left,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ( times_times_real @ C @ A2 )
% 9.00/5.06 = ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4060_mult__cancel__left,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ C @ A2 )
% 9.00/5.06 = ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4061_mult__cancel__left,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( times_times_nat @ C @ A2 )
% 9.00/5.06 = ( times_times_nat @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_nat )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4062_mult__cancel__left,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ( times_times_int @ C @ A2 )
% 9.00/5.06 = ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4063_mult__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 = ( ( A2 = zero_zero_complex )
% 9.00/5.06 | ( B3 = zero_zero_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4064_mult__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( times_times_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( ( A2 = zero_zero_real )
% 9.00/5.06 | ( B3 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4065_mult__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( ( A2 = zero_zero_rat )
% 9.00/5.06 | ( B3 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4066_mult__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( times_times_nat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 = ( ( A2 = zero_zero_nat )
% 9.00/5.06 | ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4067_mult__eq__0__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( times_times_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ( A2 = zero_zero_int )
% 9.00/5.06 | ( B3 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4068_mult__zero__right,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ A2 @ zero_zero_complex )
% 9.00/5.06 = zero_zero_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4069_mult__zero__right,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ A2 @ zero_zero_real )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4070_mult__zero__right,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 = zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4071_mult__zero__right,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 = zero_zero_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4072_mult__zero__right,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ A2 @ zero_zero_int )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4073_mult__zero__left,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ zero_zero_complex @ A2 )
% 9.00/5.06 = zero_zero_complex ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4074_mult__zero__left,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ zero_zero_real @ A2 )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4075_mult__zero__left,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 = zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4076_mult__zero__left,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 = zero_zero_nat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4077_mult__zero__left,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ zero_zero_int @ A2 )
% 9.00/5.06 = zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4078_mult_Oright__neutral,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ A2 @ one_one_complex )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4079_mult_Oright__neutral,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ A2 @ one_one_real )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4080_mult_Oright__neutral,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ A2 @ one_one_rat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4081_mult_Oright__neutral,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ A2 @ one_one_nat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4082_mult_Oright__neutral,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ A2 @ one_one_int )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4083_mult__1,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ one_one_complex @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4084_mult__1,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ one_one_real @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4085_mult__1,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ one_one_rat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4086_mult__1,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ one_one_nat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4087_mult__1,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ one_one_int @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4088_of__nat__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N ) )
% 9.00/5.06 = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4089_of__nat__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N ) )
% 9.00/5.06 = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4090_of__nat__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N ) )
% 9.00/5.06 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4091_of__nat__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N ) )
% 9.00/5.06 = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4092_of__nat__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N ) )
% 9.00/5.06 = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4093_sum__squares__eq__zero__iff,axiom,
% 9.00/5.06 ! [X2: real,Y3: real] :
% 9.00/5.06 ( ( ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( ( X2 = zero_zero_real )
% 9.00/5.06 & ( Y3 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4094_sum__squares__eq__zero__iff,axiom,
% 9.00/5.06 ! [X2: rat,Y3: rat] :
% 9.00/5.06 ( ( ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 = ( ( X2 = zero_zero_rat )
% 9.00/5.06 & ( Y3 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4095_sum__squares__eq__zero__iff,axiom,
% 9.00/5.06 ! [X2: int,Y3: int] :
% 9.00/5.06 ( ( ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 = ( ( X2 = zero_zero_int )
% 9.00/5.06 & ( Y3 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4096_mult__cancel__right2,axiom,
% 9.00/5.06 ! [A2: complex,C: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( A2 = one_one_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4097_mult__cancel__right2,axiom,
% 9.00/5.06 ! [A2: real,C: real] :
% 9.00/5.06 ( ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( A2 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4098_mult__cancel__right2,axiom,
% 9.00/5.06 ! [A2: rat,C: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( A2 = one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4099_mult__cancel__right2,axiom,
% 9.00/5.06 ! [A2: int,C: int] :
% 9.00/5.06 ( ( ( times_times_int @ A2 @ C )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( A2 = one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4100_mult__cancel__right1,axiom,
% 9.00/5.06 ! [C: complex,B3: complex] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_complex @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( B3 = one_one_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4101_mult__cancel__right1,axiom,
% 9.00/5.06 ! [C: real,B3: real] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( B3 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4102_mult__cancel__right1,axiom,
% 9.00/5.06 ! [C: rat,B3: rat] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( B3 = one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4103_mult__cancel__right1,axiom,
% 9.00/5.06 ! [C: int,B3: int] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_int @ B3 @ C ) )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( B3 = one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4104_mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: complex,A2: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ C @ A2 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( A2 = one_one_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4105_mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: real,A2: real] :
% 9.00/5.06 ( ( ( times_times_real @ C @ A2 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( A2 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4106_mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: rat,A2: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ C @ A2 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( A2 = one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4107_mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: int,A2: int] :
% 9.00/5.06 ( ( ( times_times_int @ C @ A2 )
% 9.00/5.06 = C )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( A2 = one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4108_mult__cancel__left1,axiom,
% 9.00/5.06 ! [C: complex,B3: complex] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_complex )
% 9.00/5.06 | ( B3 = one_one_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4109_mult__cancel__left1,axiom,
% 9.00/5.06 ! [C: real,B3: real] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_real )
% 9.00/5.06 | ( B3 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4110_mult__cancel__left1,axiom,
% 9.00/5.06 ! [C: rat,B3: rat] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_rat )
% 9.00/5.06 | ( B3 = one_one_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4111_mult__cancel__left1,axiom,
% 9.00/5.06 ! [C: int,B3: int] :
% 9.00/5.06 ( ( C
% 9.00/5.06 = ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( ( C = zero_zero_int )
% 9.00/5.06 | ( B3 = one_one_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4112_div__mult__mult1,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( C != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4113_div__mult__mult1,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( C != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4114_div__mult__mult2,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( C != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4115_div__mult__mult2,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( C != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4116_div__mult__mult1__if,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ( C = zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = zero_zero_int ) )
% 9.00/5.06 & ( ( C != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4117_div__mult__mult1__if,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( C = zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 9.00/5.06 = zero_zero_nat ) )
% 9.00/5.06 & ( ( C != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4118_nonzero__mult__div__cancel__left,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( A2 != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4119_nonzero__mult__div__cancel__left,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( A2 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4120_nonzero__mult__div__cancel__left,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4121_nonzero__mult__div__cancel__left,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( A2 != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4122_nonzero__mult__div__cancel__left,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( A2 != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ A2 )
% 9.00/5.06 = B3 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4123_nonzero__mult__div__cancel__right,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat] :
% 9.00/5.06 ( ( B3 != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4124_nonzero__mult__div__cancel__right,axiom,
% 9.00/5.06 ! [B3: int,A2: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4125_nonzero__mult__div__cancel__right,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4126_nonzero__mult__div__cancel__right,axiom,
% 9.00/5.06 ! [B3: real,A2: real] :
% 9.00/5.06 ( ( B3 != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4127_nonzero__mult__div__cancel__right,axiom,
% 9.00/5.06 ! [B3: complex,A2: complex] :
% 9.00/5.06 ( ( B3 != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ B3 ) @ B3 )
% 9.00/5.06 = A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4128_mult__divide__mult__cancel__left__if,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( C = zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = zero_zero_rat ) )
% 9.00/5.06 & ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4129_mult__divide__mult__cancel__left__if,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ( C = zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = zero_zero_real ) )
% 9.00/5.06 & ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4130_mult__divide__mult__cancel__left__if,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( C = zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = zero_zero_complex ) )
% 9.00/5.06 & ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4131_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4132_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4133_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4134_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4135_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4136_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A2 ) @ ( times_times_complex @ B3 @ C ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4137_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4138_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4139_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ B3 @ C ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4140_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4141_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( divide_divide_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4142_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A2 @ C ) @ ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4143_not__real__square__gt__zero,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X2 @ X2 ) ) )
% 9.00/5.06 = ( X2 = zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4144_real__root__Suc__0,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( root @ ( suc @ zero_zero_nat ) @ X2 )
% 9.00/5.06 = X2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4145_real__root__eq__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ( root @ N @ X2 )
% 9.00/5.06 = ( root @ N @ Y3 ) )
% 9.00/5.06 = ( X2 = Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4146_root__0,axiom,
% 9.00/5.06 ! [X2: real] :
% 9.00/5.06 ( ( root @ zero_zero_nat @ X2 )
% 9.00/5.06 = zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4147_div__mult__self1,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ C @ B3 ) ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4148_div__mult__self1,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ C @ B3 ) ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4149_div__mult__self2,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ A2 @ ( times_times_int @ B3 @ C ) ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4150_div__mult__self2,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4151_div__mult__self3,axiom,
% 9.00/5.06 ! [B3: int,C: int,A2: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B3 ) @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4152_div__mult__self3,axiom,
% 9.00/5.06 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B3 ) @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4153_div__mult__self4,axiom,
% 9.00/5.06 ! [B3: int,C: int,A2: int] :
% 9.00/5.06 ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B3 @ C ) @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_int @ C @ ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4154_div__mult__self4,axiom,
% 9.00/5.06 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.06 ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B3 @ C ) @ A2 ) @ B3 )
% 9.00/5.06 = ( plus_plus_nat @ C @ ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4155_nonzero__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat] :
% 9.00/5.06 ( ( B3 != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ B3 @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.06 = ( divide_divide_rat @ one_one_rat @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4156_nonzero__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [B3: real,A2: real] :
% 9.00/5.06 ( ( B3 != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ B3 @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.06 = ( divide_divide_real @ one_one_real @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4157_nonzero__divide__mult__cancel__right,axiom,
% 9.00/5.06 ! [B3: complex,A2: complex] :
% 9.00/5.06 ( ( B3 != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ B3 @ ( times_times_complex @ A2 @ B3 ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ one_one_complex @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4158_nonzero__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( A2 != zero_zero_rat )
% 9.00/5.06 => ( ( divide_divide_rat @ A2 @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.06 = ( divide_divide_rat @ one_one_rat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4159_nonzero__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( A2 != zero_zero_real )
% 9.00/5.06 => ( ( divide_divide_real @ A2 @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.06 = ( divide_divide_real @ one_one_real @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4160_nonzero__divide__mult__cancel__left,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( A2 != zero_zero_complex )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ A2 @ ( times_times_complex @ A2 @ B3 ) )
% 9.00/5.06 = ( divide1717551699836669952omplex @ one_one_complex @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4161_real__root__eq__0__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ( root @ N @ X2 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 = ( X2 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4162_real__root__less__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4163_real__root__le__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4164_real__root__eq__1__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ( root @ N @ X2 )
% 9.00/5.06 = one_one_real )
% 9.00/5.06 = ( X2 = one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4165_real__root__one,axiom,
% 9.00/5.06 ! [N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( root @ N @ one_one_real )
% 9.00/5.06 = one_one_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4166_real__root__lt__0__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4167_real__root__gt__0__iff,axiom,
% 9.00/5.06 ! [N: nat,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4168_real__root__le__0__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ zero_zero_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4169_real__root__ge__0__iff,axiom,
% 9.00/5.06 ! [N: nat,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4170_real__root__lt__1__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ ( root @ N @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4171_real__root__gt__1__iff,axiom,
% 9.00/5.06 ! [N: nat,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4172_real__root__le__1__iff,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ ( root @ N @ X2 ) @ one_one_real )
% 9.00/5.06 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4173_real__root__ge__1__iff,axiom,
% 9.00/5.06 ! [N: nat,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y3 ) )
% 9.00/5.06 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4174_real__root__pow__pos2,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 9.00/5.06 = X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4175_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex,C: complex] :
% 9.00/5.06 ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4176_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4177_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4178_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4179_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4180_mult_Oassoc,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex,C: complex] :
% 9.00/5.06 ( ( times_times_complex @ ( times_times_complex @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4181_mult_Oassoc,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( times_times_real @ ( times_times_real @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4182_mult_Oassoc,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( times_times_rat @ ( times_times_rat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4183_mult_Oassoc,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( times_times_nat @ ( times_times_nat @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4184_mult_Oassoc,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( times_times_int @ ( times_times_int @ A2 @ B3 ) @ C )
% 9.00/5.06 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4185_mult_Ocommute,axiom,
% 9.00/5.06 ( times_times_complex
% 9.00/5.06 = ( ^ [A4: complex,B4: complex] : ( times_times_complex @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4186_mult_Ocommute,axiom,
% 9.00/5.06 ( times_times_real
% 9.00/5.06 = ( ^ [A4: real,B4: real] : ( times_times_real @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4187_mult_Ocommute,axiom,
% 9.00/5.06 ( times_times_rat
% 9.00/5.06 = ( ^ [A4: rat,B4: rat] : ( times_times_rat @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4188_mult_Ocommute,axiom,
% 9.00/5.06 ( times_times_nat
% 9.00/5.06 = ( ^ [A4: nat,B4: nat] : ( times_times_nat @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4189_mult_Ocommute,axiom,
% 9.00/5.06 ( times_times_int
% 9.00/5.06 = ( ^ [A4: int,B4: int] : ( times_times_int @ B4 @ A4 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4190_mult_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: complex,A2: complex,C: complex] :
% 9.00/5.06 ( ( times_times_complex @ B3 @ ( times_times_complex @ A2 @ C ) )
% 9.00/5.06 = ( times_times_complex @ A2 @ ( times_times_complex @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4191_mult_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( times_times_real @ B3 @ ( times_times_real @ A2 @ C ) )
% 9.00/5.06 = ( times_times_real @ A2 @ ( times_times_real @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4192_mult_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( times_times_rat @ B3 @ ( times_times_rat @ A2 @ C ) )
% 9.00/5.06 = ( times_times_rat @ A2 @ ( times_times_rat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4193_mult_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.06 ( ( times_times_nat @ B3 @ ( times_times_nat @ A2 @ C ) )
% 9.00/5.06 = ( times_times_nat @ A2 @ ( times_times_nat @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4194_mult_Oleft__commute,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( times_times_int @ B3 @ ( times_times_int @ A2 @ C ) )
% 9.00/5.06 = ( times_times_int @ A2 @ ( times_times_int @ B3 @ C ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4195_mult__right__cancel,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = ( times_times_complex @ B3 @ C ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4196_mult__right__cancel,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4197_mult__right__cancel,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4198_mult__right__cancel,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( C != zero_zero_nat )
% 9.00/5.06 => ( ( ( times_times_nat @ A2 @ C )
% 9.00/5.06 = ( times_times_nat @ B3 @ C ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4199_mult__right__cancel,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( C != zero_zero_int )
% 9.00/5.06 => ( ( ( times_times_int @ A2 @ C )
% 9.00/5.06 = ( times_times_int @ B3 @ C ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4200_mult__left__cancel,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( ( times_times_complex @ C @ A2 )
% 9.00/5.06 = ( times_times_complex @ C @ B3 ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4201_mult__left__cancel,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( ( times_times_real @ C @ A2 )
% 9.00/5.06 = ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4202_mult__left__cancel,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( ( times_times_rat @ C @ A2 )
% 9.00/5.06 = ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4203_mult__left__cancel,axiom,
% 9.00/5.06 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.06 ( ( C != zero_zero_nat )
% 9.00/5.06 => ( ( ( times_times_nat @ C @ A2 )
% 9.00/5.06 = ( times_times_nat @ C @ B3 ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4204_mult__left__cancel,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( C != zero_zero_int )
% 9.00/5.06 => ( ( ( times_times_int @ C @ A2 )
% 9.00/5.06 = ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( A2 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4205_no__zero__divisors,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( A2 != zero_zero_complex )
% 9.00/5.06 => ( ( B3 != zero_zero_complex )
% 9.00/5.06 => ( ( times_times_complex @ A2 @ B3 )
% 9.00/5.06 != zero_zero_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4206_no__zero__divisors,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( A2 != zero_zero_real )
% 9.00/5.06 => ( ( B3 != zero_zero_real )
% 9.00/5.06 => ( ( times_times_real @ A2 @ B3 )
% 9.00/5.06 != zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4207_no__zero__divisors,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( A2 != zero_zero_rat )
% 9.00/5.06 => ( ( B3 != zero_zero_rat )
% 9.00/5.06 => ( ( times_times_rat @ A2 @ B3 )
% 9.00/5.06 != zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4208_no__zero__divisors,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( A2 != zero_zero_nat )
% 9.00/5.06 => ( ( B3 != zero_zero_nat )
% 9.00/5.06 => ( ( times_times_nat @ A2 @ B3 )
% 9.00/5.06 != zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4209_no__zero__divisors,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( A2 != zero_zero_int )
% 9.00/5.06 => ( ( B3 != zero_zero_int )
% 9.00/5.06 => ( ( times_times_int @ A2 @ B3 )
% 9.00/5.06 != zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4210_divisors__zero,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ A2 @ B3 )
% 9.00/5.06 = zero_zero_complex )
% 9.00/5.06 => ( ( A2 = zero_zero_complex )
% 9.00/5.06 | ( B3 = zero_zero_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4211_divisors__zero,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( times_times_real @ A2 @ B3 )
% 9.00/5.06 = zero_zero_real )
% 9.00/5.06 => ( ( A2 = zero_zero_real )
% 9.00/5.06 | ( B3 = zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4212_divisors__zero,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_rat )
% 9.00/5.06 => ( ( A2 = zero_zero_rat )
% 9.00/5.06 | ( B3 = zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4213_divisors__zero,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( times_times_nat @ A2 @ B3 )
% 9.00/5.06 = zero_zero_nat )
% 9.00/5.06 => ( ( A2 = zero_zero_nat )
% 9.00/5.06 | ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4214_divisors__zero,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( times_times_int @ A2 @ B3 )
% 9.00/5.06 = zero_zero_int )
% 9.00/5.06 => ( ( A2 = zero_zero_int )
% 9.00/5.06 | ( B3 = zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4215_mult__not__zero,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex] :
% 9.00/5.06 ( ( ( times_times_complex @ A2 @ B3 )
% 9.00/5.06 != zero_zero_complex )
% 9.00/5.06 => ( ( A2 != zero_zero_complex )
% 9.00/5.06 & ( B3 != zero_zero_complex ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4216_mult__not__zero,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( times_times_real @ A2 @ B3 )
% 9.00/5.06 != zero_zero_real )
% 9.00/5.06 => ( ( A2 != zero_zero_real )
% 9.00/5.06 & ( B3 != zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4217_mult__not__zero,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( times_times_rat @ A2 @ B3 )
% 9.00/5.06 != zero_zero_rat )
% 9.00/5.06 => ( ( A2 != zero_zero_rat )
% 9.00/5.06 & ( B3 != zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4218_mult__not__zero,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( times_times_nat @ A2 @ B3 )
% 9.00/5.06 != zero_zero_nat )
% 9.00/5.06 => ( ( A2 != zero_zero_nat )
% 9.00/5.06 & ( B3 != zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4219_mult__not__zero,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( times_times_int @ A2 @ B3 )
% 9.00/5.06 != zero_zero_int )
% 9.00/5.06 => ( ( A2 != zero_zero_int )
% 9.00/5.06 & ( B3 != zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4220_comm__monoid__mult__class_Omult__1,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ one_one_complex @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4221_comm__monoid__mult__class_Omult__1,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ one_one_real @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4222_comm__monoid__mult__class_Omult__1,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ one_one_rat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4223_comm__monoid__mult__class_Omult__1,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ one_one_nat @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4224_comm__monoid__mult__class_Omult__1,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ one_one_int @ A2 )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4225_mult_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: complex] :
% 9.00/5.06 ( ( times_times_complex @ A2 @ one_one_complex )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4226_mult_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ( ( times_times_real @ A2 @ one_one_real )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4227_mult_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ( ( times_times_rat @ A2 @ one_one_rat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4228_mult_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: nat] :
% 9.00/5.06 ( ( times_times_nat @ A2 @ one_one_nat )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4229_mult_Ocomm__neutral,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ( ( times_times_int @ A2 @ one_one_int )
% 9.00/5.06 = A2 ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4230_mult__of__nat__commute,axiom,
% 9.00/5.06 ! [X2: nat,Y3: complex] :
% 9.00/5.06 ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X2 ) @ Y3 )
% 9.00/5.06 = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4231_mult__of__nat__commute,axiom,
% 9.00/5.06 ! [X2: nat,Y3: rat] :
% 9.00/5.06 ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X2 ) @ Y3 )
% 9.00/5.06 = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4232_mult__of__nat__commute,axiom,
% 9.00/5.06 ! [X2: nat,Y3: nat] :
% 9.00/5.06 ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ Y3 )
% 9.00/5.06 = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4233_mult__of__nat__commute,axiom,
% 9.00/5.06 ! [X2: nat,Y3: int] :
% 9.00/5.06 ( ( times_times_int @ ( semiri1314217659103216013at_int @ X2 ) @ Y3 )
% 9.00/5.06 = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4234_mult__of__nat__commute,axiom,
% 9.00/5.06 ! [X2: nat,Y3: real] :
% 9.00/5.06 ( ( times_times_real @ ( semiri5074537144036343181t_real @ X2 ) @ Y3 )
% 9.00/5.06 = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4235_real__root__pos__pos__le,axiom,
% 9.00/5.06 ! [X2: real,N: nat] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4236_mult__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4237_mult__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4238_mult__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4239_mult__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4240_mult__mono_H,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4241_mult__mono_H,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4242_mult__mono_H,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4243_mult__mono_H,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4244_zero__le__square,axiom,
% 9.00/5.06 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4245_zero__le__square,axiom,
% 9.00/5.06 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4246_zero__le__square,axiom,
% 9.00/5.06 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ A2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4247_split__mult__pos__le,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4248_split__mult__pos__le,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
% 9.00/5.06 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4249_split__mult__pos__le,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
% 9.00/5.06 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4250_mult__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4251_mult__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4252_mult__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4253_mult__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4254_mult__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4255_mult__nonpos__nonpos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4256_mult__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4257_mult__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4258_mult__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4259_mult__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4260_mult__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4261_mult__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4262_mult__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4263_mult__right__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4264_mult__right__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4265_mult__right__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4266_mult__right__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4267_mult__le__0__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
% 9.00/5.06 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 9.00/5.06 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4268_mult__le__0__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.06 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 9.00/5.06 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4269_mult__le__0__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
% 9.00/5.06 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
% 9.00/5.06 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4270_split__mult__neg__le,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_eq_real @ B3 @ zero_zero_real ) )
% 9.00/5.06 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4271_split__mult__neg__le,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) )
% 9.00/5.06 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) ) )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4272_split__mult__neg__le,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 & ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
% 9.00/5.06 | ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 & ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4273_split__mult__neg__le,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_eq_int @ B3 @ zero_zero_int ) )
% 9.00/5.06 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_eq_int @ zero_zero_int @ B3 ) ) )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4274_mult__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4275_mult__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4276_mult__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4277_mult__nonneg__nonneg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4278_mult__nonneg__nonpos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4279_mult__nonneg__nonpos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4280_mult__nonneg__nonpos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4281_mult__nonneg__nonpos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4282_mult__nonpos__nonneg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4283_mult__nonpos__nonneg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4284_mult__nonpos__nonneg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4285_mult__nonpos__nonneg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4286_mult__nonneg__nonpos2,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4287_mult__nonneg__nonpos2,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4288_mult__nonneg__nonpos2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4289_mult__nonneg__nonpos2,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4290_zero__le__mult__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_eq_real @ zero_zero_real @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4291_zero__le__mult__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_eq_rat @ zero_zero_rat @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4292_zero__le__mult__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_eq_int @ zero_zero_int @ B3 ) )
% 9.00/5.06 | ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_eq_int @ B3 @ zero_zero_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4293_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4294_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4295_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4296_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4297_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4298_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4299_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4300_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4301_mult__less__cancel__right__disj,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.06 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 & ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.06 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4302_mult__less__cancel__right__disj,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.06 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 & ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4303_mult__less__cancel__right__disj,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.06 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 & ( ord_less_int @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.06 & ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4304_mult__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4305_mult__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4306_mult__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4307_mult__strict__right__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4308_mult__strict__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4309_mult__strict__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4310_mult__strict__right__mono__neg,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4311_mult__less__cancel__left__disj,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 & ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.06 & ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4312_mult__less__cancel__left__disj,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 & ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4313_mult__less__cancel__left__disj,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 & ( ord_less_int @ A2 @ B3 ) )
% 9.00/5.06 | ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.06 & ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4314_mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4315_mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4316_mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4317_mult__strict__left__mono,axiom,
% 9.00/5.06 ! [A2: int,B3: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4318_mult__strict__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: real,A2: real,C: real] :
% 9.00/5.06 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4319_mult__strict__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.06 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4320_mult__strict__left__mono__neg,axiom,
% 9.00/5.06 ! [B3: int,A2: int,C: int] :
% 9.00/5.06 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4321_mult__less__cancel__left__pos,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4322_mult__less__cancel__left__pos,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4323_mult__less__cancel__left__pos,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.06 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4324_mult__less__cancel__left__neg,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_real @ B3 @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4325_mult__less__cancel__left__neg,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4326_mult__less__cancel__left__neg,axiom,
% 9.00/5.06 ! [C: int,A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.06 = ( ord_less_int @ B3 @ A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4327_zero__less__mult__pos2,axiom,
% 9.00/5.06 ! [B3: real,A2: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A2 ) )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4328_zero__less__mult__pos2,axiom,
% 9.00/5.06 ! [B3: rat,A2: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B3 @ A2 ) )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4329_zero__less__mult__pos2,axiom,
% 9.00/5.06 ! [B3: nat,A2: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A2 ) )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4330_zero__less__mult__pos2,axiom,
% 9.00/5.06 ! [B3: int,A2: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B3 @ A2 ) )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4331_zero__less__mult__pos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4332_zero__less__mult__pos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4333_zero__less__mult__pos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4334_zero__less__mult__pos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4335_zero__less__mult__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_real @ zero_zero_real @ B3 ) )
% 9.00/5.06 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4336_zero__less__mult__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_rat @ zero_zero_rat @ B3 ) )
% 9.00/5.06 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_rat @ B3 @ zero_zero_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4337_zero__less__mult__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) )
% 9.00/5.06 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_int @ zero_zero_int @ B3 ) )
% 9.00/5.06 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_int @ B3 @ zero_zero_int ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4338_mult__pos__neg2,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ B3 @ A2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4339_mult__pos__neg2,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ B3 @ A2 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4340_mult__pos__neg2,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ B3 @ A2 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4341_mult__pos__neg2,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ B3 @ A2 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4342_mult__pos__pos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4343_mult__pos__pos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4344_mult__pos__pos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4345_mult__pos__pos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4346_mult__pos__neg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4347_mult__pos__neg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4348_mult__pos__neg,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.06 => ( ( ord_less_nat @ B3 @ zero_zero_nat )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4349_mult__pos__neg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4350_mult__neg__pos,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4351_mult__neg__pos,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4352_mult__neg__pos,axiom,
% 9.00/5.06 ! [A2: nat,B3: nat] :
% 9.00/5.06 ( ( ord_less_nat @ A2 @ zero_zero_nat )
% 9.00/5.06 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.06 => ( ord_less_nat @ ( times_times_nat @ A2 @ B3 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4353_mult__neg__pos,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4354_mult__less__0__iff,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ ( times_times_real @ A2 @ B3 ) @ zero_zero_real )
% 9.00/5.06 = ( ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.06 & ( ord_less_real @ B3 @ zero_zero_real ) )
% 9.00/5.06 | ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 & ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4355_mult__less__0__iff,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( times_times_rat @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.06 = ( ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 & ( ord_less_rat @ B3 @ zero_zero_rat ) )
% 9.00/5.06 | ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 & ( ord_less_rat @ zero_zero_rat @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4356_mult__less__0__iff,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ ( times_times_int @ A2 @ B3 ) @ zero_zero_int )
% 9.00/5.06 = ( ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.06 & ( ord_less_int @ B3 @ zero_zero_int ) )
% 9.00/5.06 | ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 & ( ord_less_int @ zero_zero_int @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4357_not__square__less__zero,axiom,
% 9.00/5.06 ! [A2: real] :
% 9.00/5.06 ~ ( ord_less_real @ ( times_times_real @ A2 @ A2 ) @ zero_zero_real ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4358_not__square__less__zero,axiom,
% 9.00/5.06 ! [A2: rat] :
% 9.00/5.06 ~ ( ord_less_rat @ ( times_times_rat @ A2 @ A2 ) @ zero_zero_rat ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4359_not__square__less__zero,axiom,
% 9.00/5.06 ! [A2: int] :
% 9.00/5.06 ~ ( ord_less_int @ ( times_times_int @ A2 @ A2 ) @ zero_zero_int ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4360_mult__neg__neg,axiom,
% 9.00/5.06 ! [A2: real,B3: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.06 => ( ( ord_less_real @ B3 @ zero_zero_real )
% 9.00/5.06 => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4361_mult__neg__neg,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.06 => ( ( ord_less_rat @ B3 @ zero_zero_rat )
% 9.00/5.06 => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4362_mult__neg__neg,axiom,
% 9.00/5.06 ! [A2: int,B3: int] :
% 9.00/5.06 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.06 => ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.06 => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4363_add__scale__eq__noteq,axiom,
% 9.00/5.06 ! [R2: complex,A2: complex,B3: complex,C: complex,D: complex] :
% 9.00/5.06 ( ( R2 != zero_zero_complex )
% 9.00/5.06 => ( ( ( A2 = B3 )
% 9.00/5.06 & ( C != D ) )
% 9.00/5.06 => ( ( plus_plus_complex @ A2 @ ( times_times_complex @ R2 @ C ) )
% 9.00/5.06 != ( plus_plus_complex @ B3 @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4364_add__scale__eq__noteq,axiom,
% 9.00/5.06 ! [R2: real,A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( R2 != zero_zero_real )
% 9.00/5.06 => ( ( ( A2 = B3 )
% 9.00/5.06 & ( C != D ) )
% 9.00/5.06 => ( ( plus_plus_real @ A2 @ ( times_times_real @ R2 @ C ) )
% 9.00/5.06 != ( plus_plus_real @ B3 @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4365_add__scale__eq__noteq,axiom,
% 9.00/5.06 ! [R2: rat,A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( R2 != zero_zero_rat )
% 9.00/5.06 => ( ( ( A2 = B3 )
% 9.00/5.06 & ( C != D ) )
% 9.00/5.06 => ( ( plus_plus_rat @ A2 @ ( times_times_rat @ R2 @ C ) )
% 9.00/5.06 != ( plus_plus_rat @ B3 @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4366_add__scale__eq__noteq,axiom,
% 9.00/5.06 ! [R2: nat,A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.06 ( ( R2 != zero_zero_nat )
% 9.00/5.06 => ( ( ( A2 = B3 )
% 9.00/5.06 & ( C != D ) )
% 9.00/5.06 => ( ( plus_plus_nat @ A2 @ ( times_times_nat @ R2 @ C ) )
% 9.00/5.06 != ( plus_plus_nat @ B3 @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4367_add__scale__eq__noteq,axiom,
% 9.00/5.06 ! [R2: int,A2: int,B3: int,C: int,D: int] :
% 9.00/5.06 ( ( R2 != zero_zero_int )
% 9.00/5.06 => ( ( ( A2 = B3 )
% 9.00/5.06 & ( C != D ) )
% 9.00/5.06 => ( ( plus_plus_int @ A2 @ ( times_times_int @ R2 @ C ) )
% 9.00/5.06 != ( plus_plus_int @ B3 @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4368_less__1__mult,axiom,
% 9.00/5.06 ! [M: real,N: real] :
% 9.00/5.06 ( ( ord_less_real @ one_one_real @ M )
% 9.00/5.06 => ( ( ord_less_real @ one_one_real @ N )
% 9.00/5.06 => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4369_less__1__mult,axiom,
% 9.00/5.06 ! [M: rat,N: rat] :
% 9.00/5.06 ( ( ord_less_rat @ one_one_rat @ M )
% 9.00/5.06 => ( ( ord_less_rat @ one_one_rat @ N )
% 9.00/5.06 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4370_less__1__mult,axiom,
% 9.00/5.06 ! [M: nat,N: nat] :
% 9.00/5.06 ( ( ord_less_nat @ one_one_nat @ M )
% 9.00/5.06 => ( ( ord_less_nat @ one_one_nat @ N )
% 9.00/5.06 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4371_less__1__mult,axiom,
% 9.00/5.06 ! [M: int,N: int] :
% 9.00/5.06 ( ( ord_less_int @ one_one_int @ M )
% 9.00/5.06 => ( ( ord_less_int @ one_one_int @ N )
% 9.00/5.06 => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4372_frac__eq__eq,axiom,
% 9.00/5.06 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 9.00/5.06 ( ( Y3 != zero_zero_rat )
% 9.00/5.06 => ( ( Z != zero_zero_rat )
% 9.00/5.06 => ( ( ( divide_divide_rat @ X2 @ Y3 )
% 9.00/5.06 = ( divide_divide_rat @ W2 @ Z ) )
% 9.00/5.06 = ( ( times_times_rat @ X2 @ Z )
% 9.00/5.06 = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4373_frac__eq__eq,axiom,
% 9.00/5.06 ! [Y3: real,Z: real,X2: real,W2: real] :
% 9.00/5.06 ( ( Y3 != zero_zero_real )
% 9.00/5.06 => ( ( Z != zero_zero_real )
% 9.00/5.06 => ( ( ( divide_divide_real @ X2 @ Y3 )
% 9.00/5.06 = ( divide_divide_real @ W2 @ Z ) )
% 9.00/5.06 = ( ( times_times_real @ X2 @ Z )
% 9.00/5.06 = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4374_frac__eq__eq,axiom,
% 9.00/5.06 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 9.00/5.06 ( ( Y3 != zero_zero_complex )
% 9.00/5.06 => ( ( Z != zero_zero_complex )
% 9.00/5.06 => ( ( ( divide1717551699836669952omplex @ X2 @ Y3 )
% 9.00/5.06 = ( divide1717551699836669952omplex @ W2 @ Z ) )
% 9.00/5.06 = ( ( times_times_complex @ X2 @ Z )
% 9.00/5.06 = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4375_divide__eq__eq,axiom,
% 9.00/5.06 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.06 ( ( ( divide_divide_rat @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( ( ( C != zero_zero_rat )
% 9.00/5.06 => ( B3
% 9.00/5.06 = ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.06 & ( ( C = zero_zero_rat )
% 9.00/5.06 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4376_divide__eq__eq,axiom,
% 9.00/5.06 ! [B3: real,C: real,A2: real] :
% 9.00/5.06 ( ( ( divide_divide_real @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( ( ( C != zero_zero_real )
% 9.00/5.06 => ( B3
% 9.00/5.06 = ( times_times_real @ A2 @ C ) ) )
% 9.00/5.06 & ( ( C = zero_zero_real )
% 9.00/5.06 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4377_divide__eq__eq,axiom,
% 9.00/5.06 ! [B3: complex,C: complex,A2: complex] :
% 9.00/5.06 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( ( ( C != zero_zero_complex )
% 9.00/5.06 => ( B3
% 9.00/5.06 = ( times_times_complex @ A2 @ C ) ) )
% 9.00/5.06 & ( ( C = zero_zero_complex )
% 9.00/5.06 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4378_eq__divide__eq,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.06 = ( ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = B3 ) )
% 9.00/5.06 & ( ( C = zero_zero_rat )
% 9.00/5.06 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4379_eq__divide__eq,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( divide_divide_real @ B3 @ C ) )
% 9.00/5.06 = ( ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = B3 ) )
% 9.00/5.06 & ( ( C = zero_zero_real )
% 9.00/5.06 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4380_eq__divide__eq,axiom,
% 9.00/5.06 ! [A2: complex,B3: complex,C: complex] :
% 9.00/5.06 ( ( A2
% 9.00/5.06 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.06 = ( ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = B3 ) )
% 9.00/5.06 & ( ( C = zero_zero_complex )
% 9.00/5.06 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4381_divide__eq__imp,axiom,
% 9.00/5.06 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( B3
% 9.00/5.06 = ( times_times_rat @ A2 @ C ) )
% 9.00/5.06 => ( ( divide_divide_rat @ B3 @ C )
% 9.00/5.06 = A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4382_divide__eq__imp,axiom,
% 9.00/5.06 ! [C: real,B3: real,A2: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( B3
% 9.00/5.06 = ( times_times_real @ A2 @ C ) )
% 9.00/5.06 => ( ( divide_divide_real @ B3 @ C )
% 9.00/5.06 = A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4383_divide__eq__imp,axiom,
% 9.00/5.06 ! [C: complex,B3: complex,A2: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( B3
% 9.00/5.06 = ( times_times_complex @ A2 @ C ) )
% 9.00/5.06 => ( ( divide1717551699836669952omplex @ B3 @ C )
% 9.00/5.06 = A2 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4384_eq__divide__imp,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = B3 )
% 9.00/5.06 => ( A2
% 9.00/5.06 = ( divide_divide_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4385_eq__divide__imp,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = B3 )
% 9.00/5.06 => ( A2
% 9.00/5.06 = ( divide_divide_real @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4386_eq__divide__imp,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = B3 )
% 9.00/5.06 => ( A2
% 9.00/5.06 = ( divide1717551699836669952omplex @ B3 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4387_nonzero__divide__eq__eq,axiom,
% 9.00/5.06 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( ( divide_divide_rat @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4388_nonzero__divide__eq__eq,axiom,
% 9.00/5.06 ! [C: real,B3: real,A2: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( ( divide_divide_real @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4389_nonzero__divide__eq__eq,axiom,
% 9.00/5.06 ! [C: complex,B3: complex,A2: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 9.00/5.06 = A2 )
% 9.00/5.06 = ( B3
% 9.00/5.06 = ( times_times_complex @ A2 @ C ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4390_nonzero__eq__divide__eq,axiom,
% 9.00/5.06 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.06 ( ( C != zero_zero_rat )
% 9.00/5.06 => ( ( A2
% 9.00/5.06 = ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.06 = ( ( times_times_rat @ A2 @ C )
% 9.00/5.06 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4391_nonzero__eq__divide__eq,axiom,
% 9.00/5.06 ! [C: real,A2: real,B3: real] :
% 9.00/5.06 ( ( C != zero_zero_real )
% 9.00/5.06 => ( ( A2
% 9.00/5.06 = ( divide_divide_real @ B3 @ C ) )
% 9.00/5.06 = ( ( times_times_real @ A2 @ C )
% 9.00/5.06 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4392_nonzero__eq__divide__eq,axiom,
% 9.00/5.06 ! [C: complex,A2: complex,B3: complex] :
% 9.00/5.06 ( ( C != zero_zero_complex )
% 9.00/5.06 => ( ( A2
% 9.00/5.06 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.06 = ( ( times_times_complex @ A2 @ C )
% 9.00/5.06 = B3 ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4393_abs__mult__less,axiom,
% 9.00/5.06 ! [A2: code_integer,C: code_integer,B3: code_integer,D: code_integer] :
% 9.00/5.06 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A2 ) @ C )
% 9.00/5.06 => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B3 ) @ D )
% 9.00/5.06 => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4394_abs__mult__less,axiom,
% 9.00/5.06 ! [A2: real,C: real,B3: real,D: real] :
% 9.00/5.06 ( ( ord_less_real @ ( abs_abs_real @ A2 ) @ C )
% 9.00/5.06 => ( ( ord_less_real @ ( abs_abs_real @ B3 ) @ D )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4395_abs__mult__less,axiom,
% 9.00/5.06 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_rat @ ( abs_abs_rat @ A2 ) @ C )
% 9.00/5.06 => ( ( ord_less_rat @ ( abs_abs_rat @ B3 ) @ D )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4396_abs__mult__less,axiom,
% 9.00/5.06 ! [A2: int,C: int,B3: int,D: int] :
% 9.00/5.06 ( ( ord_less_int @ ( abs_abs_int @ A2 ) @ C )
% 9.00/5.06 => ( ( ord_less_int @ ( abs_abs_int @ B3 ) @ D )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4397_zmult__zless__mono2,axiom,
% 9.00/5.06 ! [I: int,J: int,K: int] :
% 9.00/5.06 ( ( ord_less_int @ I @ J )
% 9.00/5.06 => ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.06 => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4398_real__minus__mult__self__le,axiom,
% 9.00/5.06 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X2 @ X2 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4399_Gcd__int__greater__eq__0,axiom,
% 9.00/5.06 ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K4 ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4400_real__root__less__mono,axiom,
% 9.00/5.06 ! [N: nat,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.06 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4401_real__root__le__mono,axiom,
% 9.00/5.06 ! [N: nat,X2: real,Y3: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.06 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N @ Y3 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4402_real__root__power,axiom,
% 9.00/5.06 ! [N: nat,X2: real,K: nat] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( root @ N @ ( power_power_real @ X2 @ K ) )
% 9.00/5.06 = ( power_power_real @ ( root @ N @ X2 ) @ K ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4403_real__root__abs,axiom,
% 9.00/5.06 ! [N: nat,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( root @ N @ ( abs_abs_real @ X2 ) )
% 9.00/5.06 = ( abs_abs_real @ ( root @ N @ X2 ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4404_log__base__root,axiom,
% 9.00/5.06 ! [N: nat,B3: real,X2: real] :
% 9.00/5.06 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.06 => ( ( log @ ( root @ N @ B3 ) @ X2 )
% 9.00/5.06 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4405_mult__less__le__imp__less,axiom,
% 9.00/5.06 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.06 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_real @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.06 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.06 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.06
% 9.00/5.06 thf(fact_4406_mult__less__le__imp__less,axiom,
% 9.00/5.06 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.06 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.06 => ( ( ord_less_eq_rat @ C @ D )
% 9.00/5.06 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.06 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.06 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4407_mult__less__le__imp__less,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.07 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4408_mult__less__le__imp__less,axiom,
% 9.00/5.07 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.07 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4409_mult__le__less__imp__less,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ C @ D )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4410_mult__le__less__imp__less,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_rat @ C @ D )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4411_mult__le__less__imp__less,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_nat @ C @ D )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4412_mult__le__less__imp__less,axiom,
% 9.00/5.07 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ C @ D )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4413_mult__right__le__imp__le,axiom,
% 9.00/5.07 ! [A2: real,C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4414_mult__right__le__imp__le,axiom,
% 9.00/5.07 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4415_mult__right__le__imp__le,axiom,
% 9.00/5.07 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4416_mult__right__le__imp__le,axiom,
% 9.00/5.07 ! [A2: int,C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4417_mult__left__le__imp__le,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4418_mult__left__le__imp__le,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4419_mult__left__le__imp__le,axiom,
% 9.00/5.07 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4420_mult__left__le__imp__le,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4421_mult__le__cancel__left__pos,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4422_mult__le__cancel__left__pos,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4423_mult__le__cancel__left__pos,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4424_mult__le__cancel__left__neg,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ B3 @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4425_mult__le__cancel__left__neg,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_rat @ B3 @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4426_mult__le__cancel__left__neg,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4427_mult__less__cancel__right,axiom,
% 9.00/5.07 ! [A2: real,C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4428_mult__less__cancel__right,axiom,
% 9.00/5.07 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4429_mult__less__cancel__right,axiom,
% 9.00/5.07 ! [A2: int,C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4430_mult__strict__mono_H,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4431_mult__strict__mono_H,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_rat @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4432_mult__strict__mono_H,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.07 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_nat @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4433_mult__strict__mono_H,axiom,
% 9.00/5.07 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.07 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ C @ D )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4434_mult__right__less__imp__less,axiom,
% 9.00/5.07 ! [A2: real,C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4435_mult__right__less__imp__less,axiom,
% 9.00/5.07 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4436_mult__right__less__imp__less,axiom,
% 9.00/5.07 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4437_mult__right__less__imp__less,axiom,
% 9.00/5.07 ! [A2: int,C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4438_mult__less__cancel__left,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4439_mult__less__cancel__left,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4440_mult__less__cancel__left,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4441_mult__strict__mono,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ C @ D )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4442_mult__strict__mono,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_rat @ C @ D )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4443_mult__strict__mono,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.07 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_nat @ C @ D )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ A2 @ C ) @ ( times_times_nat @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4444_mult__strict__mono,axiom,
% 9.00/5.07 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.07 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ C @ D )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4445_mult__left__less__imp__less,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4446_mult__left__less__imp__less,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4447_mult__left__less__imp__less,axiom,
% 9.00/5.07 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( times_times_nat @ C @ A2 ) @ ( times_times_nat @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ord_less_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4448_mult__left__less__imp__less,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4449_mult__le__cancel__right,axiom,
% 9.00/5.07 ! [A2: real,C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4450_mult__le__cancel__right,axiom,
% 9.00/5.07 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4451_mult__le__cancel__right,axiom,
% 9.00/5.07 ! [A2: int,C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4452_mult__le__cancel__left,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4453_mult__le__cancel__left,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4454_mult__le__cancel__left,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ B3 ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ B3 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4455_sum__squares__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 9.00/5.07 = ( ( X2 = zero_zero_real )
% 9.00/5.07 & ( Y3 = zero_zero_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4456_sum__squares__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 9.00/5.07 = ( ( X2 = zero_zero_rat )
% 9.00/5.07 & ( Y3 = zero_zero_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4457_sum__squares__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 9.00/5.07 = ( ( X2 = zero_zero_int )
% 9.00/5.07 & ( Y3 = zero_zero_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4458_sum__squares__ge__zero,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4459_sum__squares__ge__zero,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4460_sum__squares__ge__zero,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4461_mult__left__le__one__le,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4462_mult__left__le__one__le,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4463_mult__left__le__one__le,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.00/5.07 => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X2 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4464_mult__right__le__one__le,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4465_mult__right__le__one__le,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4466_mult__right__le__one__le,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.00/5.07 => ( ord_less_eq_int @ ( times_times_int @ X2 @ Y3 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4467_mult__le__one,axiom,
% 9.00/5.07 ! [A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ one_one_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ B3 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ B3 ) @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4468_mult__le__one,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ one_one_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ B3 @ one_one_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ B3 ) @ one_one_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4469_mult__le__one,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ A2 @ one_one_nat )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ B3 @ one_one_nat )
% 9.00/5.07 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ B3 ) @ one_one_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4470_mult__le__one,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ A2 @ one_one_int )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ B3 @ one_one_int )
% 9.00/5.07 => ( ord_less_eq_int @ ( times_times_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4471_mult__left__le,axiom,
% 9.00/5.07 ! [C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ C @ one_one_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4472_mult__left__le,axiom,
% 9.00/5.07 ! [C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ C @ one_one_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4473_mult__left__le,axiom,
% 9.00/5.07 ! [C: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ C @ one_one_nat )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ord_less_eq_nat @ ( times_times_nat @ A2 @ C ) @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4474_mult__left__le,axiom,
% 9.00/5.07 ! [C: int,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ C @ one_one_int )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ A2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4475_sum__squares__gt__zero__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 9.00/5.07 = ( ( X2 != zero_zero_real )
% 9.00/5.07 | ( Y3 != zero_zero_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4476_sum__squares__gt__zero__iff,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 9.00/5.07 = ( ( X2 != zero_zero_rat )
% 9.00/5.07 | ( Y3 != zero_zero_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4477_sum__squares__gt__zero__iff,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 9.00/5.07 = ( ( X2 != zero_zero_int )
% 9.00/5.07 | ( Y3 != zero_zero_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4478_not__sum__squares__lt__zero,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4479_not__sum__squares__lt__zero,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ X2 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4480_not__sum__squares__lt__zero,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X2 @ X2 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4481_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.00/5.07 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.07 => ( ( divide_divide_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
% 9.00/5.07 = ( divide_divide_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4482_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4483_divide__strict__left__mono__neg,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4484_divide__strict__left__mono__neg,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4485_divide__strict__left__mono,axiom,
% 9.00/5.07 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4486_divide__strict__left__mono,axiom,
% 9.00/5.07 ! [B3: real,A2: real,C: real] :
% 9.00/5.07 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4487_mult__imp__less__div__pos,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 9.00/5.07 => ( ord_less_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4488_mult__imp__less__div__pos,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 9.00/5.07 => ( ord_less_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4489_mult__imp__div__pos__less,axiom,
% 9.00/5.07 ! [Y3: rat,X2: rat,Z: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 9.00/5.07 => ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4490_mult__imp__div__pos__less,axiom,
% 9.00/5.07 ! [Y3: real,X2: real,Z: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 9.00/5.07 => ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4491_pos__less__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4492_pos__less__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4493_pos__divide__less__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4494_pos__divide__less__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4495_neg__less__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4496_neg__less__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4497_neg__divide__less__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4498_neg__divide__less__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4499_less__divide__eq,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4500_less__divide__eq,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4501_divide__less__eq,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4502_divide__less__eq,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4503_ordered__ring__class_Ole__add__iff1,axiom,
% 9.00/5.07 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4504_ordered__ring__class_Ole__add__iff1,axiom,
% 9.00/5.07 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4505_ordered__ring__class_Ole__add__iff1,axiom,
% 9.00/5.07 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4506_ordered__ring__class_Ole__add__iff2,axiom,
% 9.00/5.07 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4507_ordered__ring__class_Ole__add__iff2,axiom,
% 9.00/5.07 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4508_ordered__ring__class_Ole__add__iff2,axiom,
% 9.00/5.07 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4509_add__divide__eq__if__simps_I2_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4510_add__divide__eq__if__simps_I2_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4511_add__divide__eq__if__simps_I2_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4512_add__divide__eq__if__simps_I1_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4513_add__divide__eq__if__simps_I1_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4514_add__divide__eq__if__simps_I1_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4515_add__frac__eq,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4516_add__frac__eq,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real,W2: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4517_add__frac__eq,axiom,
% 9.00/5.07 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 9.00/5.07 ( ( Y3 != zero_zero_complex )
% 9.00/5.07 => ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4518_add__frac__num,axiom,
% 9.00/5.07 ! [Y3: rat,X2: rat,Z: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4519_add__frac__num,axiom,
% 9.00/5.07 ! [Y3: real,X2: real,Z: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4520_add__frac__num,axiom,
% 9.00/5.07 ! [Y3: complex,X2: complex,Z: complex] :
% 9.00/5.07 ( ( Y3 != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ Z )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4521_add__num__frac,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4522_add__num__frac,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4523_add__num__frac,axiom,
% 9.00/5.07 ! [Y3: complex,Z: complex,X2: complex] :
% 9.00/5.07 ( ( Y3 != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X2 @ Y3 ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Z @ Y3 ) ) @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4524_add__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4525_add__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4526_add__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4527_divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4528_divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4529_divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4530_less__add__iff1,axiom,
% 9.00/5.07 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 9.00/5.07 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4531_less__add__iff1,axiom,
% 9.00/5.07 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4532_less__add__iff1,axiom,
% 9.00/5.07 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 9.00/5.07 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A2 @ B3 ) @ E2 ) @ C ) @ D ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4533_less__add__iff2,axiom,
% 9.00/5.07 ! [A2: real,E2: real,C: real,B3: real,D: real] :
% 9.00/5.07 ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A2 @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4534_less__add__iff2,axiom,
% 9.00/5.07 ! [A2: rat,E2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A2 @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4535_less__add__iff2,axiom,
% 9.00/5.07 ! [A2: int,E2: int,C: int,B3: int,D: int] :
% 9.00/5.07 ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A2 @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B3 @ E2 ) @ D ) )
% 9.00/5.07 = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B3 @ A2 ) @ E2 ) @ D ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4536_add__divide__eq__if__simps_I4_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ A2 @ ( divide_divide_rat @ B3 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4537_add__divide__eq__if__simps_I4_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ A2 @ ( divide_divide_real @ B3 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4538_add__divide__eq__if__simps_I4_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 9.00/5.07 = A2 ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ A2 @ ( divide1717551699836669952omplex @ B3 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A2 @ Z ) @ B3 ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4539_diff__frac__eq,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4540_diff__frac__eq,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real,W2: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4541_diff__frac__eq,axiom,
% 9.00/5.07 ! [Y3: complex,Z: complex,X2: complex,W2: complex] :
% 9.00/5.07 ( ( Y3 != zero_zero_complex )
% 9.00/5.07 => ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4542_diff__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ X2 @ ( divide_divide_rat @ Y3 @ Z ) )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4543_diff__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ X2 @ ( divide_divide_real @ Y3 @ Z ) )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4544_diff__divide__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ X2 @ ( divide1717551699836669952omplex @ Y3 @ Z ) )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X2 @ Z ) @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4545_divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( divide_divide_rat @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4546_divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( divide_divide_real @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4547_divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X2 @ Z ) @ Y3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4548_ex__less__of__nat__mult,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4549_ex__less__of__nat__mult,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4550_eq__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( C != zero_zero_real )
% 9.00/5.07 => ( ( times_times_real @ A2 @ C )
% 9.00/5.07 = ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ( C = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4551_eq__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( C != zero_zero_rat )
% 9.00/5.07 => ( ( times_times_rat @ A2 @ C )
% 9.00/5.07 = ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ( C = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4552_eq__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: complex,B3: complex,C: complex] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( C != zero_zero_complex )
% 9.00/5.07 => ( ( times_times_complex @ A2 @ C )
% 9.00/5.07 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 9.00/5.07 & ( ( C = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4553_minus__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( C != zero_zero_real )
% 9.00/5.07 => ( ( uminus_uminus_real @ B3 )
% 9.00/5.07 = ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ( C = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4554_minus__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( C != zero_zero_rat )
% 9.00/5.07 => ( ( uminus_uminus_rat @ B3 )
% 9.00/5.07 = ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ( C = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4555_minus__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: complex,C: complex,A2: complex] :
% 9.00/5.07 ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( C != zero_zero_complex )
% 9.00/5.07 => ( ( uminus1482373934393186551omplex @ B3 )
% 9.00/5.07 = ( times_times_complex @ A2 @ C ) ) )
% 9.00/5.07 & ( ( C = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4556_nonzero__neg__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: real,A2: real,C: real] :
% 9.00/5.07 ( ( B3 != zero_zero_real )
% 9.00/5.07 => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.07 = C )
% 9.00/5.07 = ( ( uminus_uminus_real @ A2 )
% 9.00/5.07 = ( times_times_real @ C @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4557_nonzero__neg__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.07 ( ( B3 != zero_zero_rat )
% 9.00/5.07 => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) )
% 9.00/5.07 = C )
% 9.00/5.07 = ( ( uminus_uminus_rat @ A2 )
% 9.00/5.07 = ( times_times_rat @ C @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4558_nonzero__neg__divide__eq__eq,axiom,
% 9.00/5.07 ! [B3: complex,A2: complex,C: complex] :
% 9.00/5.07 ( ( B3 != zero_zero_complex )
% 9.00/5.07 => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 9.00/5.07 = C )
% 9.00/5.07 = ( ( uminus1482373934393186551omplex @ A2 )
% 9.00/5.07 = ( times_times_complex @ C @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4559_nonzero__neg__divide__eq__eq2,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( B3 != zero_zero_real )
% 9.00/5.07 => ( ( C
% 9.00/5.07 = ( uminus_uminus_real @ ( divide_divide_real @ A2 @ B3 ) ) )
% 9.00/5.07 = ( ( times_times_real @ C @ B3 )
% 9.00/5.07 = ( uminus_uminus_real @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4560_nonzero__neg__divide__eq__eq2,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( B3 != zero_zero_rat )
% 9.00/5.07 => ( ( C
% 9.00/5.07 = ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ B3 ) ) )
% 9.00/5.07 = ( ( times_times_rat @ C @ B3 )
% 9.00/5.07 = ( uminus_uminus_rat @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4561_nonzero__neg__divide__eq__eq2,axiom,
% 9.00/5.07 ! [B3: complex,C: complex,A2: complex] :
% 9.00/5.07 ( ( B3 != zero_zero_complex )
% 9.00/5.07 => ( ( C
% 9.00/5.07 = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.07 = ( ( times_times_complex @ C @ B3 )
% 9.00/5.07 = ( uminus1482373934393186551omplex @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4562_power__gt1__lemma,axiom,
% 9.00/5.07 ! [A2: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4563_power__gt1__lemma,axiom,
% 9.00/5.07 ! [A2: rat,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.07 => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4564_power__gt1__lemma,axiom,
% 9.00/5.07 ! [A2: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.07 => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4565_power__gt1__lemma,axiom,
% 9.00/5.07 ! [A2: int,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.07 => ( ord_less_int @ one_one_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4566_power__less__power__Suc,axiom,
% 9.00/5.07 ! [A2: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.07 => ( ord_less_real @ ( power_power_real @ A2 @ N ) @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4567_power__less__power__Suc,axiom,
% 9.00/5.07 ! [A2: rat,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ one_one_rat @ A2 )
% 9.00/5.07 => ( ord_less_rat @ ( power_power_rat @ A2 @ N ) @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4568_power__less__power__Suc,axiom,
% 9.00/5.07 ! [A2: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ one_one_nat @ A2 )
% 9.00/5.07 => ( ord_less_nat @ ( power_power_nat @ A2 @ N ) @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4569_power__less__power__Suc,axiom,
% 9.00/5.07 ! [A2: int,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ A2 )
% 9.00/5.07 => ( ord_less_int @ ( power_power_int @ A2 @ N ) @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4570_abs__mult__pos,axiom,
% 9.00/5.07 ! [X2: code_integer,Y3: code_integer] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 9.00/5.07 => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X2 )
% 9.00/5.07 = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4571_abs__mult__pos,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X2 )
% 9.00/5.07 = ( abs_abs_real @ ( times_times_real @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4572_abs__mult__pos,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X2 )
% 9.00/5.07 = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4573_abs__mult__pos,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X2 )
% 9.00/5.07 = ( abs_abs_int @ ( times_times_int @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4574_abs__eq__mult,axiom,
% 9.00/5.07 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.07 ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.07 | ( ord_le3102999989581377725nteger @ A2 @ zero_z3403309356797280102nteger ) )
% 9.00/5.07 & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.07 | ( ord_le3102999989581377725nteger @ B3 @ zero_z3403309356797280102nteger ) ) )
% 9.00/5.07 => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A2 @ B3 ) )
% 9.00/5.07 = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A2 ) @ ( abs_abs_Code_integer @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4575_abs__eq__mult,axiom,
% 9.00/5.07 ! [A2: real,B3: real] :
% 9.00/5.07 ( ( ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 | ( ord_less_eq_real @ A2 @ zero_zero_real ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.07 | ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
% 9.00/5.07 => ( ( abs_abs_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.07 = ( times_times_real @ ( abs_abs_real @ A2 ) @ ( abs_abs_real @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4576_abs__eq__mult,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 | ( ord_less_eq_rat @ A2 @ zero_zero_rat ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.07 | ( ord_less_eq_rat @ B3 @ zero_zero_rat ) ) )
% 9.00/5.07 => ( ( abs_abs_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.07 = ( times_times_rat @ ( abs_abs_rat @ A2 ) @ ( abs_abs_rat @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4577_abs__eq__mult,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.07 | ( ord_less_eq_int @ A2 @ zero_zero_int ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.07 | ( ord_less_eq_int @ B3 @ zero_zero_int ) ) )
% 9.00/5.07 => ( ( abs_abs_int @ ( times_times_int @ A2 @ B3 ) )
% 9.00/5.07 = ( times_times_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4578_ln__mult,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ln_ln_real @ ( times_times_real @ X2 @ Y3 ) )
% 9.00/5.07 = ( plus_plus_real @ ( ln_ln_real @ X2 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4579_reals__Archimedean3,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ! [Y5: real] :
% 9.00/5.07 ? [N3: nat] : ( ord_less_real @ Y5 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4580_pos__zmult__eq__1__iff,axiom,
% 9.00/5.07 ! [M: int,N: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ M )
% 9.00/5.07 => ( ( ( times_times_int @ M @ N )
% 9.00/5.07 = one_one_int )
% 9.00/5.07 = ( ( M = one_one_int )
% 9.00/5.07 & ( N = one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4581_minusinfinity,axiom,
% 9.00/5.07 ! [D: int,P1: int > $o,P: int > $o] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.07 => ( ! [X5: int,K2: int] :
% 9.00/5.07 ( ( P1 @ X5 )
% 9.00/5.07 = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.00/5.07 => ( ? [Z5: int] :
% 9.00/5.07 ! [X5: int] :
% 9.00/5.07 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.07 => ( ( P @ X5 )
% 9.00/5.07 = ( P1 @ X5 ) ) )
% 9.00/5.07 => ( ? [X_12: int] : ( P1 @ X_12 )
% 9.00/5.07 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4582_plusinfinity,axiom,
% 9.00/5.07 ! [D: int,P4: int > $o,P: int > $o] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.07 => ( ! [X5: int,K2: int] :
% 9.00/5.07 ( ( P4 @ X5 )
% 9.00/5.07 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.00/5.07 => ( ? [Z5: int] :
% 9.00/5.07 ! [X5: int] :
% 9.00/5.07 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.07 => ( ( P @ X5 )
% 9.00/5.07 = ( P4 @ X5 ) ) )
% 9.00/5.07 => ( ? [X_12: int] : ( P4 @ X_12 )
% 9.00/5.07 => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4583_zdiv__zmult2__eq,axiom,
% 9.00/5.07 ! [C: int,A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ( divide_divide_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( divide_divide_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4584_find__None__iff2,axiom,
% 9.00/5.07 ! [P: real > $o,Xs: list_real] :
% 9.00/5.07 ( ( none_real
% 9.00/5.07 = ( find_real @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: real] :
% 9.00/5.07 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4585_find__None__iff2,axiom,
% 9.00/5.07 ! [P: $o > $o,Xs: list_o] :
% 9.00/5.07 ( ( none_o
% 9.00/5.07 = ( find_o @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: $o] :
% 9.00/5.07 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4586_find__None__iff2,axiom,
% 9.00/5.07 ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.00/5.07 ( ( none_set_nat
% 9.00/5.07 = ( find_set_nat @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: set_nat] :
% 9.00/5.07 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4587_find__None__iff2,axiom,
% 9.00/5.07 ! [P: int > $o,Xs: list_int] :
% 9.00/5.07 ( ( none_int
% 9.00/5.07 = ( find_int @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: int] :
% 9.00/5.07 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4588_find__None__iff2,axiom,
% 9.00/5.07 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( none_VEBT_VEBT
% 9.00/5.07 = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: vEBT_VEBT] :
% 9.00/5.07 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4589_find__None__iff2,axiom,
% 9.00/5.07 ! [P: nat > $o,Xs: list_nat] :
% 9.00/5.07 ( ( none_nat
% 9.00/5.07 = ( find_nat @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: nat] :
% 9.00/5.07 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4590_find__None__iff2,axiom,
% 9.00/5.07 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.00/5.07 ( ( none_P5556105721700978146at_nat
% 9.00/5.07 = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: product_prod_nat_nat] :
% 9.00/5.07 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4591_find__None__iff2,axiom,
% 9.00/5.07 ! [P: num > $o,Xs: list_num] :
% 9.00/5.07 ( ( none_num
% 9.00/5.07 = ( find_num @ P @ Xs ) )
% 9.00/5.07 = ( ~ ? [X: num] :
% 9.00/5.07 ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4592_find__None__iff,axiom,
% 9.00/5.07 ! [P: real > $o,Xs: list_real] :
% 9.00/5.07 ( ( ( find_real @ P @ Xs )
% 9.00/5.07 = none_real )
% 9.00/5.07 = ( ~ ? [X: real] :
% 9.00/5.07 ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4593_find__None__iff,axiom,
% 9.00/5.07 ! [P: $o > $o,Xs: list_o] :
% 9.00/5.07 ( ( ( find_o @ P @ Xs )
% 9.00/5.07 = none_o )
% 9.00/5.07 = ( ~ ? [X: $o] :
% 9.00/5.07 ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4594_find__None__iff,axiom,
% 9.00/5.07 ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.00/5.07 ( ( ( find_set_nat @ P @ Xs )
% 9.00/5.07 = none_set_nat )
% 9.00/5.07 = ( ~ ? [X: set_nat] :
% 9.00/5.07 ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4595_find__None__iff,axiom,
% 9.00/5.07 ! [P: int > $o,Xs: list_int] :
% 9.00/5.07 ( ( ( find_int @ P @ Xs )
% 9.00/5.07 = none_int )
% 9.00/5.07 = ( ~ ? [X: int] :
% 9.00/5.07 ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4596_find__None__iff,axiom,
% 9.00/5.07 ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.00/5.07 = none_VEBT_VEBT )
% 9.00/5.07 = ( ~ ? [X: vEBT_VEBT] :
% 9.00/5.07 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4597_find__None__iff,axiom,
% 9.00/5.07 ! [P: nat > $o,Xs: list_nat] :
% 9.00/5.07 ( ( ( find_nat @ P @ Xs )
% 9.00/5.07 = none_nat )
% 9.00/5.07 = ( ~ ? [X: nat] :
% 9.00/5.07 ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4598_find__None__iff,axiom,
% 9.00/5.07 ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.00/5.07 ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.00/5.07 = none_P5556105721700978146at_nat )
% 9.00/5.07 = ( ~ ? [X: product_prod_nat_nat] :
% 9.00/5.07 ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4599_find__None__iff,axiom,
% 9.00/5.07 ! [P: num > $o,Xs: list_num] :
% 9.00/5.07 ( ( ( find_num @ P @ Xs )
% 9.00/5.07 = none_num )
% 9.00/5.07 = ( ~ ? [X: num] :
% 9.00/5.07 ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 9.00/5.07 & ( P @ X ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4600_real__root__gt__zero,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ord_less_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4601_real__root__strict__decreasing,axiom,
% 9.00/5.07 ! [N: nat,N6: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_nat @ N @ N6 )
% 9.00/5.07 => ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ord_less_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4602_root__abs__power,axiom,
% 9.00/5.07 ! [N: nat,Y3: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y3 @ N ) ) )
% 9.00/5.07 = ( abs_abs_real @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4603_mult__less__cancel__right2,axiom,
% 9.00/5.07 ! [A2: real,C: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ one_one_real ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4604_mult__less__cancel__right2,axiom,
% 9.00/5.07 ! [A2: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4605_mult__less__cancel__right2,axiom,
% 9.00/5.07 ! [A2: int,C: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ one_one_int ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4606_mult__less__cancel__right1,axiom,
% 9.00/5.07 ! [C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4607_mult__less__cancel__right1,axiom,
% 9.00/5.07 ! [C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ one_one_rat @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4608_mult__less__cancel__right1,axiom,
% 9.00/5.07 ! [C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ C @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ one_one_int @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4609_mult__less__cancel__left2,axiom,
% 9.00/5.07 ! [C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( times_times_real @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ A2 @ one_one_real ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4610_mult__less__cancel__left2,axiom,
% 9.00/5.07 ! [C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ A2 @ one_one_rat ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ one_one_rat @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4611_mult__less__cancel__left2,axiom,
% 9.00/5.07 ! [C: int,A2: int] :
% 9.00/5.07 ( ( ord_less_int @ ( times_times_int @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ A2 @ one_one_int ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ one_one_int @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4612_mult__less__cancel__left1,axiom,
% 9.00/5.07 ! [C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4613_mult__less__cancel__left1,axiom,
% 9.00/5.07 ! [C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ one_one_rat @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ B3 @ one_one_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4614_mult__less__cancel__left1,axiom,
% 9.00/5.07 ! [C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_int @ C @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_int @ one_one_int @ B3 ) )
% 9.00/5.07 & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_int @ B3 @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4615_mult__le__cancel__right2,axiom,
% 9.00/5.07 ! [A2: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ one_one_real ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4616_mult__le__cancel__right2,axiom,
% 9.00/5.07 ! [A2: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4617_mult__le__cancel__right2,axiom,
% 9.00/5.07 ! [A2: int,C: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ A2 @ C ) @ C )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4618_mult__le__cancel__right1,axiom,
% 9.00/5.07 ! [C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ one_one_real @ B3 ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4619_mult__le__cancel__right1,axiom,
% 9.00/5.07 ! [C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4620_mult__le__cancel__right1,axiom,
% 9.00/5.07 ! [C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ C @ ( times_times_int @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ one_one_int @ B3 ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4621_mult__le__cancel__left2,axiom,
% 9.00/5.07 ! [C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( times_times_real @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ one_one_real ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ one_one_real @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4622_mult__le__cancel__left2,axiom,
% 9.00/5.07 ! [C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ one_one_rat ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ one_one_rat @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4623_mult__le__cancel__left2,axiom,
% 9.00/5.07 ! [C: int,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( times_times_int @ C @ A2 ) @ C )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ A2 @ one_one_int ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ one_one_int @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4624_mult__le__cancel__left1,axiom,
% 9.00/5.07 ! [C: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ one_one_real @ B3 ) )
% 9.00/5.07 & ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4625_mult__le__cancel__left1,axiom,
% 9.00/5.07 ! [C: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ one_one_rat @ B3 ) )
% 9.00/5.07 & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ one_one_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4626_mult__le__cancel__left1,axiom,
% 9.00/5.07 ! [C: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B3 ) )
% 9.00/5.07 = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.00/5.07 => ( ord_less_eq_int @ one_one_int @ B3 ) )
% 9.00/5.07 & ( ( ord_less_int @ C @ zero_zero_int )
% 9.00/5.07 => ( ord_less_eq_int @ B3 @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4627_field__le__mult__one__interval,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ! [Z4: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Z4 )
% 9.00/5.07 => ( ( ord_less_real @ Z4 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ Z4 @ X2 ) @ Y3 ) ) )
% 9.00/5.07 => ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4628_field__le__mult__one__interval,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ! [Z4: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Z4 )
% 9.00/5.07 => ( ( ord_less_rat @ Z4 @ one_one_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ Z4 @ X2 ) @ Y3 ) ) )
% 9.00/5.07 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4629_divide__left__mono__neg,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4630_divide__left__mono__neg,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4631_mult__imp__le__div__pos,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y3 ) @ X2 )
% 9.00/5.07 => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4632_mult__imp__le__div__pos,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y3 ) @ X2 )
% 9.00/5.07 => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4633_mult__imp__div__pos__le,axiom,
% 9.00/5.07 ! [Y3: real,X2: real,Z: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ ( times_times_real @ Z @ Y3 ) )
% 9.00/5.07 => ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4634_mult__imp__div__pos__le,axiom,
% 9.00/5.07 ! [Y3: rat,X2: rat,Z: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ X2 @ ( times_times_rat @ Z @ Y3 ) )
% 9.00/5.07 => ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4635_pos__le__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4636_pos__le__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4637_pos__divide__le__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4638_pos__divide__le__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4639_neg__le__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4640_neg__le__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4641_neg__divide__le__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4642_neg__divide__le__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4643_divide__left__mono,axiom,
% 9.00/5.07 ! [B3: real,A2: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ B3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_eq_real @ ( divide_divide_real @ C @ A2 ) @ ( divide_divide_real @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4644_divide__left__mono,axiom,
% 9.00/5.07 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A2 @ B3 ) )
% 9.00/5.07 => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A2 ) @ ( divide_divide_rat @ C @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4645_le__divide__eq,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4646_le__divide__eq,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4647_divide__le__eq,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4648_divide__le__eq,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ B3 ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4649_convex__bound__le,axiom,
% 9.00/5.07 ! [X2: real,A2: real,Y3: real,U: real,V: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.00/5.07 => ( ( ( plus_plus_real @ U @ V )
% 9.00/5.07 = one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4650_convex__bound__le,axiom,
% 9.00/5.07 ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.00/5.07 => ( ( ( plus_plus_rat @ U @ V )
% 9.00/5.07 = one_one_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4651_convex__bound__le,axiom,
% 9.00/5.07 ! [X2: int,A2: int,Y3: int,U: int,V: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.00/5.07 => ( ( ( plus_plus_int @ U @ V )
% 9.00/5.07 = one_one_int )
% 9.00/5.07 => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4652_frac__le__eq,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real,W2: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4653_frac__le__eq,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4654_frac__less__eq,axiom,
% 9.00/5.07 ! [Y3: rat,Z: rat,X2: rat,W2: rat] :
% 9.00/5.07 ( ( Y3 != zero_zero_rat )
% 9.00/5.07 => ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ ( divide_divide_rat @ X2 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z ) )
% 9.00/5.07 = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z ) ) @ zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4655_frac__less__eq,axiom,
% 9.00/5.07 ! [Y3: real,Z: real,X2: real,W2: real] :
% 9.00/5.07 ( ( Y3 != zero_zero_real )
% 9.00/5.07 => ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ ( divide_divide_real @ X2 @ Y3 ) @ ( divide_divide_real @ W2 @ Z ) )
% 9.00/5.07 = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z ) ) @ zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4656_power__Suc__less,axiom,
% 9.00/5.07 ! [A2: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_real @ A2 @ one_one_real )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ ( power_power_real @ A2 @ N ) ) @ ( power_power_real @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4657_power__Suc__less,axiom,
% 9.00/5.07 ! [A2: rat,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ( ord_less_rat @ A2 @ one_one_rat )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ ( power_power_rat @ A2 @ N ) ) @ ( power_power_rat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4658_power__Suc__less,axiom,
% 9.00/5.07 ! [A2: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ( ord_less_nat @ A2 @ one_one_nat )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ A2 @ N ) ) @ ( power_power_nat @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4659_power__Suc__less,axiom,
% 9.00/5.07 ! [A2: int,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( ord_less_int @ A2 @ one_one_int )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ A2 @ ( power_power_int @ A2 @ N ) ) @ ( power_power_int @ A2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4660_pos__minus__divide__less__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4661_pos__minus__divide__less__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4662_pos__less__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4663_pos__less__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4664_neg__minus__divide__less__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4665_neg__minus__divide__less__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4666_neg__less__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4667_neg__less__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4668_minus__divide__less__eq,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4669_minus__divide__less__eq,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4670_less__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4671_less__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4672_add__divide__eq__if__simps_I3_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4673_add__divide__eq__if__simps_I3_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4674_add__divide__eq__if__simps_I3_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4675_minus__divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4676_minus__divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4677_minus__divide__add__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4678_add__divide__eq__if__simps_I6_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A2 ) @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4679_add__divide__eq__if__simps_I6_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A2 ) @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4680_add__divide__eq__if__simps_I6_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A2 @ Z ) ) @ B3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A2 ) @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4681_add__divide__eq__if__simps_I5_J,axiom,
% 9.00/5.07 ! [Z: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ( Z = zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( divide_divide_real @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ A2 @ ( times_times_real @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4682_add__divide__eq__if__simps_I5_J,axiom,
% 9.00/5.07 ! [Z: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ( Z = zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( divide_divide_rat @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ A2 @ ( times_times_rat @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4683_add__divide__eq__if__simps_I5_J,axiom,
% 9.00/5.07 ! [Z: complex,A2: complex,B3: complex] :
% 9.00/5.07 ( ( ( Z = zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( uminus1482373934393186551omplex @ B3 ) ) )
% 9.00/5.07 & ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A2 @ Z ) @ B3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A2 @ ( times_times_complex @ B3 @ Z ) ) @ Z ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4684_minus__divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( Z != zero_zero_real )
% 9.00/5.07 => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4685_minus__divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( Z != zero_zero_rat )
% 9.00/5.07 => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X2 ) @ ( times_times_rat @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4686_minus__divide__diff__eq__iff,axiom,
% 9.00/5.07 ! [Z: complex,X2: complex,Y3: complex] :
% 9.00/5.07 ( ( Z != zero_zero_complex )
% 9.00/5.07 => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X2 @ Z ) ) @ Y3 )
% 9.00/5.07 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X2 ) @ ( times_times_complex @ Y3 @ Z ) ) @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4687_zmult__zless__mono2__lemma,axiom,
% 9.00/5.07 ! [I: int,J: int,K: nat] :
% 9.00/5.07 ( ( ord_less_int @ I @ J )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4688_ln__realpow,axiom,
% 9.00/5.07 ! [X2: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ln_ln_real @ ( power_power_real @ X2 @ N ) )
% 9.00/5.07 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4689_q__pos__lemma,axiom,
% 9.00/5.07 ! [B7: int,Q5: int,R4: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.00/5.07 => ( ( ord_less_int @ R4 @ B7 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.00/5.07 => ( ord_less_eq_int @ zero_zero_int @ Q5 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4690_zdiv__mono2__lemma,axiom,
% 9.00/5.07 ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 9.00/5.07 ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.00/5.07 => ( ( ord_less_int @ R4 @ B7 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.00/5.07 => ( ( ord_less_eq_int @ B7 @ B3 )
% 9.00/5.07 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4691_zdiv__mono2__neg__lemma,axiom,
% 9.00/5.07 ! [B3: int,Q3: int,R2: int,B7: int,Q5: int,R4: int] :
% 9.00/5.07 ( ( ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 )
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) )
% 9.00/5.07 => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B7 @ Q5 ) @ R4 ) @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_int @ R2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ B7 )
% 9.00/5.07 => ( ( ord_less_eq_int @ B7 @ B3 )
% 9.00/5.07 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4692_unique__quotient__lemma,axiom,
% 9.00/5.07 ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
% 9.00/5.07 => ( ( ord_less_int @ R4 @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ R2 @ B3 )
% 9.00/5.07 => ( ord_less_eq_int @ Q5 @ Q3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4693_unique__quotient__lemma__neg,axiom,
% 9.00/5.07 ! [B3: int,Q5: int,R4: int,Q3: int,R2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B3 @ Q5 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_int @ B3 @ R2 )
% 9.00/5.07 => ( ( ord_less_int @ B3 @ R4 )
% 9.00/5.07 => ( ord_less_eq_int @ Q3 @ Q5 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4694_incr__mult__lemma,axiom,
% 9.00/5.07 ! [D: int,P: int > $o,K: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.07 => ( ! [X5: int] :
% 9.00/5.07 ( ( P @ X5 )
% 9.00/5.07 => ( P @ ( plus_plus_int @ X5 @ D ) ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.07 => ! [X4: int] :
% 9.00/5.07 ( ( P @ X4 )
% 9.00/5.07 => ( P @ ( plus_plus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4695_decr__mult__lemma,axiom,
% 9.00/5.07 ! [D: int,P: int > $o,K: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.07 => ( ! [X5: int] :
% 9.00/5.07 ( ( P @ X5 )
% 9.00/5.07 => ( P @ ( minus_minus_int @ X5 @ D ) ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.07 => ! [X4: int] :
% 9.00/5.07 ( ( P @ X4 )
% 9.00/5.07 => ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4696_real__root__pos__pos,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4697_real__root__strict__increasing,axiom,
% 9.00/5.07 ! [N: nat,N6: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_nat @ N @ N6 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.07 => ( ord_less_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4698_real__root__decreasing,axiom,
% 9.00/5.07 ! [N: nat,N6: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.07 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( root @ N6 @ X2 ) @ ( root @ N @ X2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4699_real__root__pow__pos,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( power_power_real @ ( root @ N @ X2 ) @ N )
% 9.00/5.07 = X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4700_real__root__pos__unique,axiom,
% 9.00/5.07 ! [N: nat,Y3: real,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ( power_power_real @ Y3 @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 => ( ( root @ N @ X2 )
% 9.00/5.07 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4701_real__root__power__cancel,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( root @ N @ ( power_power_real @ X2 @ N ) )
% 9.00/5.07 = X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4702_convex__bound__lt,axiom,
% 9.00/5.07 ! [X2: real,A2: real,Y3: real,U: real,V: real] :
% 9.00/5.07 ( ( ord_less_real @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_real @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.00/5.07 => ( ( ( plus_plus_real @ U @ V )
% 9.00/5.07 = one_one_real )
% 9.00/5.07 => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X2 ) @ ( times_times_real @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4703_convex__bound__lt,axiom,
% 9.00/5.07 ! [X2: rat,A2: rat,Y3: rat,U: rat,V: rat] :
% 9.00/5.07 ( ( ord_less_rat @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_rat @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.00/5.07 => ( ( ( plus_plus_rat @ U @ V )
% 9.00/5.07 = one_one_rat )
% 9.00/5.07 => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X2 ) @ ( times_times_rat @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4704_convex__bound__lt,axiom,
% 9.00/5.07 ! [X2: int,A2: int,Y3: int,U: int,V: int] :
% 9.00/5.07 ( ( ord_less_int @ X2 @ A2 )
% 9.00/5.07 => ( ( ord_less_int @ Y3 @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.00/5.07 => ( ( ( plus_plus_int @ U @ V )
% 9.00/5.07 = one_one_int )
% 9.00/5.07 => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X2 ) @ ( times_times_int @ V @ Y3 ) ) @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4705_le__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: real,B3: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ A2 @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4706_le__minus__divide__eq,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ A2 @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4707_minus__divide__le__eq,axiom,
% 9.00/5.07 ! [B3: real,C: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4708_minus__divide__le__eq,axiom,
% 9.00/5.07 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) )
% 9.00/5.07 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4709_neg__le__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4710_neg__le__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4711_neg__minus__divide__le__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4712_neg__minus__divide__le__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4713_pos__le__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_eq_real @ A2 @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ C ) @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4714_pos__le__minus__divide__eq,axiom,
% 9.00/5.07 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_eq_rat @ A2 @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ C ) @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4715_pos__minus__divide__le__eq,axiom,
% 9.00/5.07 ! [C: real,B3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ ( uminus_uminus_real @ B3 ) @ ( times_times_real @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4716_pos__minus__divide__le__eq,axiom,
% 9.00/5.07 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B3 @ C ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B3 ) @ ( times_times_rat @ A2 @ C ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4717_scaling__mono,axiom,
% 9.00/5.07 ! [U: real,V: real,R2: real,S2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ U @ V )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ R2 @ S2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4718_scaling__mono,axiom,
% 9.00/5.07 ! [U: rat,V: rat,R2: rat,S2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ U @ V )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ R2 @ S2 )
% 9.00/5.07 => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S2 ) ) @ V ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4719_power__eq__if,axiom,
% 9.00/5.07 ( power_power_complex
% 9.00/5.07 = ( ^ [P5: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4720_power__eq__if,axiom,
% 9.00/5.07 ( power_power_real
% 9.00/5.07 = ( ^ [P5: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4721_power__eq__if,axiom,
% 9.00/5.07 ( power_power_rat
% 9.00/5.07 = ( ^ [P5: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4722_power__eq__if,axiom,
% 9.00/5.07 ( power_power_nat
% 9.00/5.07 = ( ^ [P5: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4723_power__eq__if,axiom,
% 9.00/5.07 ( power_power_int
% 9.00/5.07 = ( ^ [P5: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4724_power__minus__mult,axiom,
% 9.00/5.07 ! [N: nat,A2: complex] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_complex @ ( power_power_complex @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 9.00/5.07 = ( power_power_complex @ A2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4725_power__minus__mult,axiom,
% 9.00/5.07 ! [N: nat,A2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_real @ ( power_power_real @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 9.00/5.07 = ( power_power_real @ A2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4726_power__minus__mult,axiom,
% 9.00/5.07 ! [N: nat,A2: rat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_rat @ ( power_power_rat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 9.00/5.07 = ( power_power_rat @ A2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4727_power__minus__mult,axiom,
% 9.00/5.07 ! [N: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_nat @ ( power_power_nat @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 9.00/5.07 = ( power_power_nat @ A2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4728_power__minus__mult,axiom,
% 9.00/5.07 ! [N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_int @ ( power_power_int @ A2 @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A2 )
% 9.00/5.07 = ( power_power_int @ A2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4729_real__archimedian__rdiv__eq__0,axiom,
% 9.00/5.07 ! [X2: real,C: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.00/5.07 => ( ! [M4: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ M4 )
% 9.00/5.07 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X2 ) @ C ) )
% 9.00/5.07 => ( X2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4730_log__eq__div__ln__mult__log,axiom,
% 9.00/5.07 ! [A2: real,B3: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( A2 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( B3 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( log @ A2 @ X2 )
% 9.00/5.07 = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( ln_ln_real @ A2 ) ) @ ( log @ B3 @ X2 ) ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4731_log__mult,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( A2 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( log @ A2 @ ( times_times_real @ X2 @ Y3 ) )
% 9.00/5.07 = ( plus_plus_real @ ( log @ A2 @ X2 ) @ ( log @ A2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4732_split__zdiv,axiom,
% 9.00/5.07 ! [P: int > $o,N: int,K: int] :
% 9.00/5.07 ( ( P @ ( divide_divide_int @ N @ K ) )
% 9.00/5.07 = ( ( ( K = zero_zero_int )
% 9.00/5.07 => ( P @ zero_zero_int ) )
% 9.00/5.07 & ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.07 => ! [I4: int,J3: int] :
% 9.00/5.07 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.00/5.07 & ( ord_less_int @ J3 @ K )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.07 => ( P @ I4 ) ) )
% 9.00/5.07 & ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.07 => ! [I4: int,J3: int] :
% 9.00/5.07 ( ( ( ord_less_int @ K @ J3 )
% 9.00/5.07 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.07 => ( P @ I4 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4733_int__div__neg__eq,axiom,
% 9.00/5.07 ! [A2: int,B3: int,Q3: int,R2: int] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_int @ B3 @ R2 )
% 9.00/5.07 => ( ( divide_divide_int @ A2 @ B3 )
% 9.00/5.07 = Q3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4734_int__div__pos__eq,axiom,
% 9.00/5.07 ! [A2: int,B3: int,Q3: int,R2: int] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.00/5.07 => ( ( ord_less_int @ R2 @ B3 )
% 9.00/5.07 => ( ( divide_divide_int @ A2 @ B3 )
% 9.00/5.07 = Q3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4735_log__nat__power,axiom,
% 9.00/5.07 ! [X2: real,B3: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( log @ B3 @ ( power_power_real @ X2 @ N ) )
% 9.00/5.07 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B3 @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4736_real__root__increasing,axiom,
% 9.00/5.07 ! [N: nat,N6: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ N6 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( root @ N @ X2 ) @ ( root @ N6 @ X2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4737_ln__root,axiom,
% 9.00/5.07 ! [N: nat,B3: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( ln_ln_real @ ( root @ N @ B3 ) )
% 9.00/5.07 = ( divide_divide_real @ ( ln_ln_real @ B3 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4738_mult__le__cancel__iff2,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Z )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X2 ) @ ( times_times_real @ Z @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4739_mult__le__cancel__iff2,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X2 ) @ ( times_times_rat @ Z @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4740_mult__le__cancel__iff2,axiom,
% 9.00/5.07 ! [Z: int,X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.07 => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X2 ) @ ( times_times_int @ Z @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4741_mult__le__cancel__iff1,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Z )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4742_mult__le__cancel__iff1,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.00/5.07 => ( ( ord_less_eq_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4743_mult__le__cancel__iff1,axiom,
% 9.00/5.07 ! [Z: int,X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.07 => ( ( ord_less_eq_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4744_mult__less__iff1,axiom,
% 9.00/5.07 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Z )
% 9.00/5.07 => ( ( ord_less_real @ ( times_times_real @ X2 @ Z ) @ ( times_times_real @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4745_mult__less__iff1,axiom,
% 9.00/5.07 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Z )
% 9.00/5.07 => ( ( ord_less_rat @ ( times_times_rat @ X2 @ Z ) @ ( times_times_rat @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4746_mult__less__iff1,axiom,
% 9.00/5.07 ! [Z: int,X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.07 => ( ( ord_less_int @ ( times_times_int @ X2 @ Z ) @ ( times_times_int @ Y3 @ Z ) )
% 9.00/5.07 = ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4747_arctan__add,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.07 => ( ( plus_plus_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 9.00/5.07 = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4748_root__powr__inverse,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( root @ N @ X2 )
% 9.00/5.07 = ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4749_eucl__rel__int__iff,axiom,
% 9.00/5.07 ! [K: int,L: int,Q3: int,R2: int] :
% 9.00/5.07 ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.07 = ( ( K
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
% 9.00/5.07 & ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.00/5.07 & ( ord_less_int @ R2 @ L ) ) )
% 9.00/5.07 & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.07 => ( ( ( ord_less_int @ L @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_int @ L @ R2 )
% 9.00/5.07 & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 9.00/5.07 & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 9.00/5.07 => ( Q3 = zero_zero_int ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4750_log__minus__eq__powr,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( B3 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( minus_minus_real @ ( log @ B3 @ X2 ) @ Y3 )
% 9.00/5.07 = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4751_split__root,axiom,
% 9.00/5.07 ! [P: real > $o,N: nat,X2: real] :
% 9.00/5.07 ( ( P @ ( root @ N @ X2 ) )
% 9.00/5.07 = ( ( ( N = zero_zero_nat )
% 9.00/5.07 => ( P @ zero_zero_real ) )
% 9.00/5.07 & ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ! [Y: real] :
% 9.00/5.07 ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 9.00/5.07 = X2 )
% 9.00/5.07 => ( P @ Y ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4752_mult__is__0,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ M @ N )
% 9.00/5.07 = zero_zero_nat )
% 9.00/5.07 = ( ( M = zero_zero_nat )
% 9.00/5.07 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4753_mult__0__right,axiom,
% 9.00/5.07 ! [M: nat] :
% 9.00/5.07 ( ( times_times_nat @ M @ zero_zero_nat )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4754_mult__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ K @ M )
% 9.00/5.07 = ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ( M = N )
% 9.00/5.07 | ( K = zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4755_mult__cancel2,axiom,
% 9.00/5.07 ! [M: nat,K: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ M @ K )
% 9.00/5.07 = ( times_times_nat @ N @ K ) )
% 9.00/5.07 = ( ( M = N )
% 9.00/5.07 | ( K = zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4756_sgn__0,axiom,
% 9.00/5.07 ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4757_sgn__0,axiom,
% 9.00/5.07 ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.00/5.07 = zero_zero_complex ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4758_sgn__0,axiom,
% 9.00/5.07 ( ( sgn_sgn_real @ zero_zero_real )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4759_sgn__0,axiom,
% 9.00/5.07 ( ( sgn_sgn_rat @ zero_zero_rat )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4760_sgn__0,axiom,
% 9.00/5.07 ( ( sgn_sgn_int @ zero_zero_int )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4761_powr__0,axiom,
% 9.00/5.07 ! [Z: real] :
% 9.00/5.07 ( ( powr_real @ zero_zero_real @ Z )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4762_powr__eq__0__iff,axiom,
% 9.00/5.07 ! [W2: real,Z: real] :
% 9.00/5.07 ( ( ( powr_real @ W2 @ Z )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( W2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4763_nat__mult__eq__1__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ M @ N )
% 9.00/5.07 = one_one_nat )
% 9.00/5.07 = ( ( M = one_one_nat )
% 9.00/5.07 & ( N = one_one_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4764_nat__1__eq__mult__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( one_one_nat
% 9.00/5.07 = ( times_times_nat @ M @ N ) )
% 9.00/5.07 = ( ( M = one_one_nat )
% 9.00/5.07 & ( N = one_one_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4765_sgn__less,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A2 ) @ zero_z3403309356797280102nteger )
% 9.00/5.07 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4766_sgn__less,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( sgn_sgn_real @ A2 ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4767_sgn__less,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( sgn_sgn_rat @ A2 ) @ zero_zero_rat )
% 9.00/5.07 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4768_sgn__less,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_int @ ( sgn_sgn_int @ A2 ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4769_sgn__greater,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A2 ) )
% 9.00/5.07 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4770_sgn__greater,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A2 ) )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4771_sgn__greater,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A2 ) )
% 9.00/5.07 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4772_sgn__greater,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4773_one__eq__mult__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( suc @ zero_zero_nat )
% 9.00/5.07 = ( times_times_nat @ M @ N ) )
% 9.00/5.07 = ( ( M
% 9.00/5.07 = ( suc @ zero_zero_nat ) )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4774_mult__eq__1__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ M @ N )
% 9.00/5.07 = ( suc @ zero_zero_nat ) )
% 9.00/5.07 = ( ( M
% 9.00/5.07 = ( suc @ zero_zero_nat ) )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4775_mult__less__cancel2,axiom,
% 9.00/5.07 ! [M: nat,K: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.00/5.07 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 & ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4776_nat__0__less__mult__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
% 9.00/5.07 = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.07 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4777_powr__zero__eq__one,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ( X2 = zero_zero_real )
% 9.00/5.07 => ( ( powr_real @ X2 @ zero_zero_real )
% 9.00/5.07 = zero_zero_real ) )
% 9.00/5.07 & ( ( X2 != zero_zero_real )
% 9.00/5.07 => ( ( powr_real @ X2 @ zero_zero_real )
% 9.00/5.07 = one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4778_mult__Suc__right,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( times_times_nat @ M @ ( suc @ N ) )
% 9.00/5.07 = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4779_powr__gt__zero,axiom,
% 9.00/5.07 ! [X2: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X2 @ A2 ) )
% 9.00/5.07 = ( X2 != zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4780_powr__nonneg__iff,axiom,
% 9.00/5.07 ! [A2: real,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real )
% 9.00/5.07 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4781_powr__less__cancel__iff,axiom,
% 9.00/5.07 ! [X2: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 9.00/5.07 = ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4782_zero__less__arctan__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( arctan @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4783_arctan__less__zero__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( arctan @ X2 ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4784_zero__le__arctan__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4785_arctan__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4786_sgn__pos,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.07 => ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = one_one_Code_integer ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4787_sgn__pos,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = one_one_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4788_sgn__pos,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = one_one_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4789_sgn__pos,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4790_one__le__mult__iff,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
% 9.00/5.07 = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.00/5.07 & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4791_mult__le__cancel2,axiom,
% 9.00/5.07 ! [M: nat,K: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
% 9.00/5.07 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4792_abs__sgn__eq__1,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( A2 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 9.00/5.07 = one_one_Code_integer ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4793_abs__sgn__eq__1,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( A2 != zero_zero_real )
% 9.00/5.07 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 9.00/5.07 = one_one_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4794_abs__sgn__eq__1,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( A2 != zero_zero_rat )
% 9.00/5.07 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 9.00/5.07 = one_one_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4795_abs__sgn__eq__1,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( A2 != zero_zero_int )
% 9.00/5.07 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 9.00/5.07 = one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4796_div__mult__self__is__m,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( divide_divide_nat @ ( times_times_nat @ M @ N ) @ N )
% 9.00/5.07 = M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4797_div__mult__self1__is__m,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( divide_divide_nat @ ( times_times_nat @ N @ M ) @ N )
% 9.00/5.07 = M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4798_powr__eq__one__iff,axiom,
% 9.00/5.07 ! [A2: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ A2 )
% 9.00/5.07 => ( ( ( powr_real @ A2 @ X2 )
% 9.00/5.07 = one_one_real )
% 9.00/5.07 = ( X2 = zero_zero_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4799_powr__one__gt__zero__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ( powr_real @ X2 @ one_one_real )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4800_powr__one,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( powr_real @ X2 @ one_one_real )
% 9.00/5.07 = X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4801_powr__le__cancel__iff,axiom,
% 9.00/5.07 ! [X2: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4802_sgn__neg,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.07 => ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4803_sgn__neg,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.07 => ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4804_sgn__neg,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.07 => ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4805_sgn__neg,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4806_powr__log__cancel,axiom,
% 9.00/5.07 ! [A2: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( A2 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( powr_real @ A2 @ ( log @ A2 @ X2 ) )
% 9.00/5.07 = X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4807_log__powr__cancel,axiom,
% 9.00/5.07 ! [A2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( A2 != one_one_real )
% 9.00/5.07 => ( ( log @ A2 @ ( powr_real @ A2 @ Y3 ) )
% 9.00/5.07 = Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4808_sgn__0__0,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = zero_z3403309356797280102nteger )
% 9.00/5.07 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4809_sgn__0__0,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4810_sgn__0__0,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4811_sgn__0__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4812_sgn__eq__0__iff,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = zero_z3403309356797280102nteger )
% 9.00/5.07 = ( A2 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4813_sgn__eq__0__iff,axiom,
% 9.00/5.07 ! [A2: complex] :
% 9.00/5.07 ( ( ( sgn_sgn_complex @ A2 )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 = ( A2 = zero_zero_complex ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4814_sgn__eq__0__iff,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4815_sgn__eq__0__iff,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4816_sgn__eq__0__iff,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4817_Suc__mult__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 9.00/5.07 = ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.00/5.07 = ( M = N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4818_mult__0,axiom,
% 9.00/5.07 ! [N: nat] :
% 9.00/5.07 ( ( times_times_nat @ zero_zero_nat @ N )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4819_le__cube,axiom,
% 9.00/5.07 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4820_le__square,axiom,
% 9.00/5.07 ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4821_mult__le__mono,axiom,
% 9.00/5.07 ! [I: nat,J: nat,K: nat,L: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ( ord_less_eq_nat @ K @ L )
% 9.00/5.07 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4822_mult__le__mono1,axiom,
% 9.00/5.07 ! [I: nat,J: nat,K: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4823_mult__le__mono2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,K: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4824_arctan__monotone,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4825_arctan__less__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 9.00/5.07 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4826_arctan__monotone_H,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4827_arctan__le__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( arctan @ X2 ) @ ( arctan @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4828_add__mult__distrib2,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4829_add__mult__distrib,axiom,
% 9.00/5.07 ! [M: nat,N: nat,K: nat] :
% 9.00/5.07 ( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4830_diff__mult__distrib2,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.07 = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4831_diff__mult__distrib,axiom,
% 9.00/5.07 ! [M: nat,N: nat,K: nat] :
% 9.00/5.07 ( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
% 9.00/5.07 = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4832_nat__mult__1__right,axiom,
% 9.00/5.07 ! [N: nat] :
% 9.00/5.07 ( ( times_times_nat @ N @ one_one_nat )
% 9.00/5.07 = N ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4833_nat__mult__1,axiom,
% 9.00/5.07 ! [N: nat] :
% 9.00/5.07 ( ( times_times_nat @ one_one_nat @ N )
% 9.00/5.07 = N ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4834_powr__non__neg,axiom,
% 9.00/5.07 ! [A2: real,X2: real] :
% 9.00/5.07 ~ ( ord_less_real @ ( powr_real @ A2 @ X2 ) @ zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4835_powr__less__mono2__neg,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4836_powr__ge__pzero,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4837_powr__mono2,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4838_powr__less__mono,axiom,
% 9.00/5.07 ! [A2: real,B3: real,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4839_powr__less__cancel,axiom,
% 9.00/5.07 ! [X2: real,A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) )
% 9.00/5.07 => ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ord_less_real @ A2 @ B3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4840_powr__mono,axiom,
% 9.00/5.07 ! [A2: real,B3: real,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ X2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4841_sgn__not__eq__imp,axiom,
% 9.00/5.07 ! [B3: int,A2: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ B3 )
% 9.00/5.07 != ( sgn_sgn_int @ A2 ) )
% 9.00/5.07 => ( ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 => ( ( ( sgn_sgn_int @ B3 )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 => ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = ( uminus_uminus_int @ ( sgn_sgn_int @ B3 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4842_sgn__not__eq__imp,axiom,
% 9.00/5.07 ! [B3: real,A2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ B3 )
% 9.00/5.07 != ( sgn_sgn_real @ A2 ) )
% 9.00/5.07 => ( ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( ( ( sgn_sgn_real @ B3 )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = ( uminus_uminus_real @ ( sgn_sgn_real @ B3 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4843_sgn__not__eq__imp,axiom,
% 9.00/5.07 ! [B3: rat,A2: rat] :
% 9.00/5.07 ( ( ( sgn_sgn_rat @ B3 )
% 9.00/5.07 != ( sgn_sgn_rat @ A2 ) )
% 9.00/5.07 => ( ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( ( ( sgn_sgn_rat @ B3 )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B3 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4844_sgn__not__eq__imp,axiom,
% 9.00/5.07 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.07 ( ( ( sgn_sgn_Code_integer @ B3 )
% 9.00/5.07 != ( sgn_sgn_Code_integer @ A2 ) )
% 9.00/5.07 => ( ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( ( sgn_sgn_Code_integer @ B3 )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B3 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4845_Suc__mult__less__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4846_mult__less__mono1,axiom,
% 9.00/5.07 ! [I: nat,J: nat,K: nat] :
% 9.00/5.07 ( ( ord_less_nat @ I @ J )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4847_mult__less__mono2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,K: nat] :
% 9.00/5.07 ( ( ord_less_nat @ I @ J )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4848_Suc__mult__le__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4849_mult__Suc,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( times_times_nat @ ( suc @ M ) @ N )
% 9.00/5.07 = ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4850_less__mult__imp__div__less,axiom,
% 9.00/5.07 ! [M: nat,I: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N ) )
% 9.00/5.07 => ( ord_less_nat @ ( divide_divide_nat @ M @ N ) @ I ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4851_mult__eq__self__implies__10,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( M
% 9.00/5.07 = ( times_times_nat @ M @ N ) )
% 9.00/5.07 => ( ( N = one_one_nat )
% 9.00/5.07 | ( M = zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4852_div__times__less__eq__dividend,axiom,
% 9.00/5.07 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) @ M ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4853_times__div__less__eq__dividend,axiom,
% 9.00/5.07 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) @ M ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4854_powr__mono2_H,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ zero_zero_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ Y3 @ A2 ) @ ( powr_real @ X2 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4855_powr__less__mono2,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4856_gr__one__powr,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ord_less_real @ one_one_real @ ( powr_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4857_powr__inj,axiom,
% 9.00/5.07 ! [A2: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( A2 != one_one_real )
% 9.00/5.07 => ( ( ( powr_real @ A2 @ X2 )
% 9.00/5.07 = ( powr_real @ A2 @ Y3 ) )
% 9.00/5.07 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4858_ge__one__powr__ge__zero,axiom,
% 9.00/5.07 ! [X2: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X2 @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4859_powr__mono__both,axiom,
% 9.00/5.07 ! [A2: real,B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ B3 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4860_powr__le1,axiom,
% 9.00/5.07 ! [A2: real,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ X2 @ A2 ) @ one_one_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4861_powr__divide,axiom,
% 9.00/5.07 ! [X2: real,Y3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( powr_real @ ( divide_divide_real @ X2 @ Y3 ) @ A2 )
% 9.00/5.07 = ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4862_powr__mult,axiom,
% 9.00/5.07 ! [X2: real,Y3: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( powr_real @ ( times_times_real @ X2 @ Y3 ) @ A2 )
% 9.00/5.07 = ( times_times_real @ ( powr_real @ X2 @ A2 ) @ ( powr_real @ Y3 @ A2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4863_sgn__1__pos,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = one_one_Code_integer )
% 9.00/5.07 = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4864_sgn__1__pos,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = one_one_real )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4865_sgn__1__pos,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = one_one_rat )
% 9.00/5.07 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4866_sgn__1__pos,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = one_one_int )
% 9.00/5.07 = ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4867_sgn__root,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( sgn_sgn_real @ ( root @ N @ X2 ) )
% 9.00/5.07 = ( sgn_sgn_real @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4868_abs__sgn__eq,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( A2 = zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) )
% 9.00/5.07 & ( ( A2 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A2 ) )
% 9.00/5.07 = one_one_Code_integer ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4869_abs__sgn__eq,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( A2 = zero_zero_real )
% 9.00/5.07 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 9.00/5.07 = zero_zero_real ) )
% 9.00/5.07 & ( ( A2 != zero_zero_real )
% 9.00/5.07 => ( ( abs_abs_real @ ( sgn_sgn_real @ A2 ) )
% 9.00/5.07 = one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4870_abs__sgn__eq,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( A2 = zero_zero_rat )
% 9.00/5.07 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 9.00/5.07 = zero_zero_rat ) )
% 9.00/5.07 & ( ( A2 != zero_zero_rat )
% 9.00/5.07 => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A2 ) )
% 9.00/5.07 = one_one_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4871_abs__sgn__eq,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( A2 = zero_zero_int )
% 9.00/5.07 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 9.00/5.07 = zero_zero_int ) )
% 9.00/5.07 & ( ( A2 != zero_zero_int )
% 9.00/5.07 => ( ( abs_abs_int @ ( sgn_sgn_int @ A2 ) )
% 9.00/5.07 = one_one_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4872_one__less__mult,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.07 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.00/5.07 => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4873_n__less__m__mult__n,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.00/5.07 => ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4874_n__less__n__mult__m,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.00/5.07 => ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4875_div__less__iff__less__mult,axiom,
% 9.00/5.07 ! [Q3: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 9.00/5.07 => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N )
% 9.00/5.07 = ( ord_less_nat @ M @ ( times_times_nat @ N @ Q3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4876_sgn__real__def,axiom,
% 9.00/5.07 ( sgn_sgn_real
% 9.00/5.07 = ( ^ [A4: real] : ( if_real @ ( A4 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A4 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4877_sgn__1__neg,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ A2 )
% 9.00/5.07 = ( uminus_uminus_int @ one_one_int ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4878_sgn__1__neg,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ A2 )
% 9.00/5.07 = ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.07 = ( ord_less_real @ A2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4879_sgn__1__neg,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( sgn_sgn_rat @ A2 )
% 9.00/5.07 = ( uminus_uminus_rat @ one_one_rat ) )
% 9.00/5.07 = ( ord_less_rat @ A2 @ zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4880_sgn__1__neg,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( sgn_sgn_Code_integer @ A2 )
% 9.00/5.07 = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.00/5.07 = ( ord_le6747313008572928689nteger @ A2 @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4881_sgn__if,axiom,
% 9.00/5.07 ( sgn_sgn_int
% 9.00/5.07 = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4882_sgn__if,axiom,
% 9.00/5.07 ( sgn_sgn_real
% 9.00/5.07 = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4883_sgn__if,axiom,
% 9.00/5.07 ( sgn_sgn_rat
% 9.00/5.07 = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4884_sgn__if,axiom,
% 9.00/5.07 ( sgn_sgn_Code_integer
% 9.00/5.07 = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4885_powr__realpow,axiom,
% 9.00/5.07 ! [X2: real,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( powr_real @ X2 @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.07 = ( power_power_real @ X2 @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4886_powr__less__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
% 9.00/5.07 = ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4887_less__powr__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
% 9.00/5.07 = ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4888_log__less__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ ( log @ B3 @ X2 ) @ Y3 )
% 9.00/5.07 = ( ord_less_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4889_less__log__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ Y3 @ ( log @ B3 @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4890_div__nat__eqI,axiom,
% 9.00/5.07 ! [N: nat,Q3: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M )
% 9.00/5.07 => ( ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
% 9.00/5.07 => ( ( divide_divide_nat @ M @ N )
% 9.00/5.07 = Q3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4891_less__eq__div__iff__mult__less__eq,axiom,
% 9.00/5.07 ! [Q3: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ Q3 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N @ Q3 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4892_dividend__less__times__div,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4893_dividend__less__div__times,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ord_less_nat @ M @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M @ N ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4894_split__div,axiom,
% 9.00/5.07 ! [P: nat > $o,M: nat,N: nat] :
% 9.00/5.07 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.00/5.07 = ( ( ( N = zero_zero_nat )
% 9.00/5.07 => ( P @ zero_zero_nat ) )
% 9.00/5.07 & ( ( N != zero_zero_nat )
% 9.00/5.07 => ! [I4: nat,J3: nat] :
% 9.00/5.07 ( ( ord_less_nat @ J3 @ N )
% 9.00/5.07 => ( ( M
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.00/5.07 => ( P @ I4 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4895_mult__eq__if,axiom,
% 9.00/5.07 ( times_times_nat
% 9.00/5.07 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4896_powr__neg__one,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.07 = ( divide_divide_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4897_powr__mult__base,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( times_times_real @ X2 @ ( powr_real @ X2 @ Y3 ) )
% 9.00/5.07 = ( powr_real @ X2 @ ( plus_plus_real @ one_one_real @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4898_sgn__power__injE,axiom,
% 9.00/5.07 ! [A2: real,N: nat,X2: real,B3: real] :
% 9.00/5.07 ( ( ( times_times_real @ ( sgn_sgn_real @ A2 ) @ ( power_power_real @ ( abs_abs_real @ A2 ) @ N ) )
% 9.00/5.07 = X2 )
% 9.00/5.07 => ( ( X2
% 9.00/5.07 = ( times_times_real @ ( sgn_sgn_real @ B3 ) @ ( power_power_real @ ( abs_abs_real @ B3 ) @ N ) ) )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( A2 = B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4899_le__log__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4900_log__le__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4901_le__powr__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( log @ B3 @ X2 ) @ Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4902_powr__le__iff,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ ( powr_real @ B3 @ Y3 ) @ X2 )
% 9.00/5.07 = ( ord_less_eq_real @ Y3 @ ( log @ B3 @ X2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4903_split__div_H,axiom,
% 9.00/5.07 ! [P: nat > $o,M: nat,N: nat] :
% 9.00/5.07 ( ( P @ ( divide_divide_nat @ M @ N ) )
% 9.00/5.07 = ( ( ( N = zero_zero_nat )
% 9.00/5.07 & ( P @ zero_zero_nat ) )
% 9.00/5.07 | ? [Q6: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q6 ) @ M )
% 9.00/5.07 & ( ord_less_nat @ M @ ( times_times_nat @ N @ ( suc @ Q6 ) ) )
% 9.00/5.07 & ( P @ Q6 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4904_ln__powr__bound,axiom,
% 9.00/5.07 ! [X2: real,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( ln_ln_real @ X2 ) @ ( divide_divide_real @ ( powr_real @ X2 @ A2 ) @ A2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4905_ln__powr__bound2,axiom,
% 9.00/5.07 ! [X2: real,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X2 ) @ A2 ) @ ( times_times_real @ ( powr_real @ A2 @ A2 ) @ X2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4906_add__log__eq__powr,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( B3 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( plus_plus_real @ Y3 @ ( log @ B3 @ X2 ) )
% 9.00/5.07 = ( log @ B3 @ ( times_times_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4907_log__add__eq__powr,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( B3 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( plus_plus_real @ ( log @ B3 @ X2 ) @ Y3 )
% 9.00/5.07 = ( log @ B3 @ ( times_times_real @ X2 @ ( powr_real @ B3 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4908_sgn__power__root,axiom,
% 9.00/5.07 ! [N: nat,X2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X2 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X2 ) ) @ N ) )
% 9.00/5.07 = X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4909_root__sgn__power,axiom,
% 9.00/5.07 ! [N: nat,Y3: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) ) )
% 9.00/5.07 = Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4910_minus__log__eq__powr,axiom,
% 9.00/5.07 ! [B3: real,X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ( B3 != one_one_real )
% 9.00/5.07 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( minus_minus_real @ Y3 @ ( log @ B3 @ X2 ) )
% 9.00/5.07 = ( log @ B3 @ ( divide_divide_real @ ( powr_real @ B3 @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4911_powr__def,axiom,
% 9.00/5.07 ( powr_real
% 9.00/5.07 = ( ^ [X: real,A4: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A4 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4912_nat__mult__le__cancel__disj,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4913_zero__le__sgn__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4914_sgn__le__0__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( sgn_sgn_real @ X2 ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4915_nat__mult__div__cancel__disj,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ( K = zero_zero_nat )
% 9.00/5.07 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = zero_zero_nat ) )
% 9.00/5.07 & ( ( K != zero_zero_nat )
% 9.00/5.07 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( divide_divide_nat @ M @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4916_nat__mult__less__cancel__disj,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 & ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4917_mul__def,axiom,
% 9.00/5.07 ( vEBT_VEBT_mul
% 9.00/5.07 = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4918_sgn__zero,axiom,
% 9.00/5.07 ( ( sgn_sgn_complex @ zero_zero_complex )
% 9.00/5.07 = zero_zero_complex ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4919_sgn__zero,axiom,
% 9.00/5.07 ( ( sgn_sgn_real @ zero_zero_real )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4920_nat__less__add__iff2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4921_mul__shift,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ X2 @ Y3 )
% 9.00/5.07 = Z )
% 9.00/5.07 = ( ( vEBT_VEBT_mul @ ( some_nat @ X2 ) @ ( some_nat @ Y3 ) )
% 9.00/5.07 = ( some_nat @ Z ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4922_zsgn__def,axiom,
% 9.00/5.07 ( sgn_sgn_int
% 9.00/5.07 = ( ^ [I4: int] : ( if_int @ ( I4 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I4 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4923_eucl__rel__int__remainderI,axiom,
% 9.00/5.07 ! [R2: int,L: int,K: int,Q3: int] :
% 9.00/5.07 ( ( ( sgn_sgn_int @ R2 )
% 9.00/5.07 = ( sgn_sgn_int @ L ) )
% 9.00/5.07 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 9.00/5.07 => ( ( K
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
% 9.00/5.07 => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4924_nat__mult__eq__cancel__disj,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ( times_times_nat @ K @ M )
% 9.00/5.07 = ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ( K = zero_zero_nat )
% 9.00/5.07 | ( M = N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4925_sgn__zero__iff,axiom,
% 9.00/5.07 ! [X2: complex] :
% 9.00/5.07 ( ( ( sgn_sgn_complex @ X2 )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 = ( X2 = zero_zero_complex ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4926_sgn__zero__iff,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ( sgn_sgn_real @ X2 )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( X2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4927_eucl__rel__int_Osimps,axiom,
% 9.00/5.07 ( eucl_rel_int
% 9.00/5.07 = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
% 9.00/5.07 ( ? [K3: int] :
% 9.00/5.07 ( ( A12 = K3 )
% 9.00/5.07 & ( A23 = zero_zero_int )
% 9.00/5.07 & ( A32
% 9.00/5.07 = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 9.00/5.07 | ? [L2: int,K3: int,Q6: int] :
% 9.00/5.07 ( ( A12 = K3 )
% 9.00/5.07 & ( A23 = L2 )
% 9.00/5.07 & ( A32
% 9.00/5.07 = ( product_Pair_int_int @ Q6 @ zero_zero_int ) )
% 9.00/5.07 & ( L2 != zero_zero_int )
% 9.00/5.07 & ( K3
% 9.00/5.07 = ( times_times_int @ Q6 @ L2 ) ) )
% 9.00/5.07 | ? [R5: int,L2: int,K3: int,Q6: int] :
% 9.00/5.07 ( ( A12 = K3 )
% 9.00/5.07 & ( A23 = L2 )
% 9.00/5.07 & ( A32
% 9.00/5.07 = ( product_Pair_int_int @ Q6 @ R5 ) )
% 9.00/5.07 & ( ( sgn_sgn_int @ R5 )
% 9.00/5.07 = ( sgn_sgn_int @ L2 ) )
% 9.00/5.07 & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 9.00/5.07 & ( K3
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ Q6 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4928_eucl__rel__int_Ocases,axiom,
% 9.00/5.07 ! [A13: int,A24: int,A33: product_prod_int_int] :
% 9.00/5.07 ( ( eucl_rel_int @ A13 @ A24 @ A33 )
% 9.00/5.07 => ( ( ( A24 = zero_zero_int )
% 9.00/5.07 => ( A33
% 9.00/5.07 != ( product_Pair_int_int @ zero_zero_int @ A13 ) ) )
% 9.00/5.07 => ( ! [Q4: int] :
% 9.00/5.07 ( ( A33
% 9.00/5.07 = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
% 9.00/5.07 => ( ( A24 != zero_zero_int )
% 9.00/5.07 => ( A13
% 9.00/5.07 != ( times_times_int @ Q4 @ A24 ) ) ) )
% 9.00/5.07 => ~ ! [R3: int,Q4: int] :
% 9.00/5.07 ( ( A33
% 9.00/5.07 = ( product_Pair_int_int @ Q4 @ R3 ) )
% 9.00/5.07 => ( ( ( sgn_sgn_int @ R3 )
% 9.00/5.07 = ( sgn_sgn_int @ A24 ) )
% 9.00/5.07 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A24 ) )
% 9.00/5.07 => ( A13
% 9.00/5.07 != ( plus_plus_int @ ( times_times_int @ Q4 @ A24 ) @ R3 ) ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4929_nat__mult__less__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ord_less_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4930_nat__mult__eq__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( ( times_times_nat @ K @ M )
% 9.00/5.07 = ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( M = N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4931_nat__mult__le__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4932_nat__mult__div__cancel1,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.07 = ( divide_divide_nat @ M @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4933_nat__diff__add__eq2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4934_nat__diff__add__eq1,axiom,
% 9.00/5.07 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ J @ I )
% 9.00/5.07 => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4935_nat__le__add__iff2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4936_nat__le__add__iff1,axiom,
% 9.00/5.07 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ J @ I )
% 9.00/5.07 => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4937_nat__eq__add__iff2,axiom,
% 9.00/5.07 ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.07 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( M
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4938_nat__eq__add__iff1,axiom,
% 9.00/5.07 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ J @ I )
% 9.00/5.07 => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.00/5.07 = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 9.00/5.07 = N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4939_nat__less__add__iff1,axiom,
% 9.00/5.07 ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ J @ I )
% 9.00/5.07 => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4940_ceiling__log__eq__powr__iff,axiom,
% 9.00/5.07 ! [X2: real,B3: real,K: nat] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.07 => ( ( ( archim7802044766580827645g_real @ ( log @ B3 @ X2 ) )
% 9.00/5.07 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 9.00/5.07 = ( ( ord_less_real @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ K ) ) @ X2 )
% 9.00/5.07 & ( ord_less_eq_real @ X2 @ ( powr_real @ B3 @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4941_bezw__0,axiom,
% 9.00/5.07 ! [X2: nat] :
% 9.00/5.07 ( ( bezw @ X2 @ zero_zero_nat )
% 9.00/5.07 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4942_length__mul__elem,axiom,
% 9.00/5.07 ! [Xs: list_list_VEBT_VEBT,N: nat] :
% 9.00/5.07 ( ! [X5: list_VEBT_VEBT] :
% 9.00/5.07 ( ( member2936631157270082147T_VEBT @ X5 @ ( set_list_VEBT_VEBT2 @ Xs ) )
% 9.00/5.07 => ( ( size_s6755466524823107622T_VEBT @ X5 )
% 9.00/5.07 = N ) )
% 9.00/5.07 => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
% 9.00/5.07 = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4943_length__mul__elem,axiom,
% 9.00/5.07 ! [Xs: list_list_o,N: nat] :
% 9.00/5.07 ( ! [X5: list_o] :
% 9.00/5.07 ( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
% 9.00/5.07 => ( ( size_size_list_o @ X5 )
% 9.00/5.07 = N ) )
% 9.00/5.07 => ( ( size_size_list_o @ ( concat_o @ Xs ) )
% 9.00/5.07 = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4944_length__mul__elem,axiom,
% 9.00/5.07 ! [Xs: list_list_nat,N: nat] :
% 9.00/5.07 ( ! [X5: list_nat] :
% 9.00/5.07 ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
% 9.00/5.07 => ( ( size_size_list_nat @ X5 )
% 9.00/5.07 = N ) )
% 9.00/5.07 => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
% 9.00/5.07 = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4945_gbinomial__absorption_H,axiom,
% 9.00/5.07 ! [K: nat,A2: rat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_rat @ A2 @ K )
% 9.00/5.07 = ( times_times_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4946_gbinomial__absorption_H,axiom,
% 9.00/5.07 ! [K: nat,A2: complex] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_complex @ A2 @ K )
% 9.00/5.07 = ( times_times_complex @ ( divide1717551699836669952omplex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4947_gbinomial__absorption_H,axiom,
% 9.00/5.07 ! [K: nat,A2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_real @ A2 @ K )
% 9.00/5.07 = ( times_times_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4948_nth__Cons__pos,axiom,
% 9.00/5.07 ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4949_nth__Cons__pos,axiom,
% 9.00/5.07 ! [N: nat,X2: int,Xs: list_int] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4950_nth__Cons__pos,axiom,
% 9.00/5.07 ! [N: nat,X2: nat,Xs: list_nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.07 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4951_rotate1__length01,axiom,
% 9.00/5.07 ! [Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ one_one_nat )
% 9.00/5.07 => ( ( rotate1_VEBT_VEBT @ Xs )
% 9.00/5.07 = Xs ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4952_rotate1__length01,axiom,
% 9.00/5.07 ! [Xs: list_o] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ one_one_nat )
% 9.00/5.07 => ( ( rotate1_o @ Xs )
% 9.00/5.07 = Xs ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4953_rotate1__length01,axiom,
% 9.00/5.07 ! [Xs: list_nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ one_one_nat )
% 9.00/5.07 => ( ( rotate1_nat @ Xs )
% 9.00/5.07 = Xs ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4954_remove__def,axiom,
% 9.00/5.07 ( remove6466555014256735590at_nat
% 9.00/5.07 = ( ^ [X: product_prod_nat_nat,A6: set_Pr1261947904930325089at_nat] : ( minus_1356011639430497352at_nat @ A6 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4955_remove__def,axiom,
% 9.00/5.07 ( remove_real
% 9.00/5.07 = ( ^ [X: real,A6: set_real] : ( minus_minus_set_real @ A6 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4956_remove__def,axiom,
% 9.00/5.07 ( remove_o
% 9.00/5.07 = ( ^ [X: $o,A6: set_o] : ( minus_minus_set_o @ A6 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4957_remove__def,axiom,
% 9.00/5.07 ( remove_int
% 9.00/5.07 = ( ^ [X: int,A6: set_int] : ( minus_minus_set_int @ A6 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4958_remove__def,axiom,
% 9.00/5.07 ( remove_nat
% 9.00/5.07 = ( ^ [X: nat,A6: set_nat] : ( minus_minus_set_nat @ A6 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4959_member__remove,axiom,
% 9.00/5.07 ! [X2: real,Y3: real,A3: set_real] :
% 9.00/5.07 ( ( member_real @ X2 @ ( remove_real @ Y3 @ A3 ) )
% 9.00/5.07 = ( ( member_real @ X2 @ A3 )
% 9.00/5.07 & ( X2 != Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4960_member__remove,axiom,
% 9.00/5.07 ! [X2: $o,Y3: $o,A3: set_o] :
% 9.00/5.07 ( ( member_o @ X2 @ ( remove_o @ Y3 @ A3 ) )
% 9.00/5.07 = ( ( member_o @ X2 @ A3 )
% 9.00/5.07 & ( X2 != Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4961_member__remove,axiom,
% 9.00/5.07 ! [X2: set_nat,Y3: set_nat,A3: set_set_nat] :
% 9.00/5.07 ( ( member_set_nat @ X2 @ ( remove_set_nat @ Y3 @ A3 ) )
% 9.00/5.07 = ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.07 & ( X2 != Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4962_member__remove,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat,A3: set_nat] :
% 9.00/5.07 ( ( member_nat @ X2 @ ( remove_nat @ Y3 @ A3 ) )
% 9.00/5.07 = ( ( member_nat @ X2 @ A3 )
% 9.00/5.07 & ( X2 != Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4963_member__remove,axiom,
% 9.00/5.07 ! [X2: int,Y3: int,A3: set_int] :
% 9.00/5.07 ( ( member_int @ X2 @ ( remove_int @ Y3 @ A3 ) )
% 9.00/5.07 = ( ( member_int @ X2 @ A3 )
% 9.00/5.07 & ( X2 != Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4964_nth__Cons__0,axiom,
% 9.00/5.07 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat )
% 9.00/5.07 = X2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4965_nth__Cons__0,axiom,
% 9.00/5.07 ! [X2: int,Xs: list_int] :
% 9.00/5.07 ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat )
% 9.00/5.07 = X2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4966_nth__Cons__0,axiom,
% 9.00/5.07 ! [X2: nat,Xs: list_nat] :
% 9.00/5.07 ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
% 9.00/5.07 = X2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4967_gbinomial__0_I2_J,axiom,
% 9.00/5.07 ! [K: nat] :
% 9.00/5.07 ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4968_gbinomial__0_I2_J,axiom,
% 9.00/5.07 ! [K: nat] :
% 9.00/5.07 ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4969_gbinomial__0_I2_J,axiom,
% 9.00/5.07 ! [K: nat] :
% 9.00/5.07 ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4970_gbinomial__0_I2_J,axiom,
% 9.00/5.07 ! [K: nat] :
% 9.00/5.07 ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4971_ceiling__zero,axiom,
% 9.00/5.07 ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4972_ceiling__zero,axiom,
% 9.00/5.07 ( ( archim7802044766580827645g_real @ zero_zero_real )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4973_gbinomial__0_I1_J,axiom,
% 9.00/5.07 ! [A2: complex] :
% 9.00/5.07 ( ( gbinomial_complex @ A2 @ zero_zero_nat )
% 9.00/5.07 = one_one_complex ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4974_gbinomial__0_I1_J,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( gbinomial_real @ A2 @ zero_zero_nat )
% 9.00/5.07 = one_one_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4975_gbinomial__0_I1_J,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( gbinomial_rat @ A2 @ zero_zero_nat )
% 9.00/5.07 = one_one_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4976_gbinomial__0_I1_J,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( gbinomial_nat @ A2 @ zero_zero_nat )
% 9.00/5.07 = one_one_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4977_gbinomial__0_I1_J,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( gbinomial_int @ A2 @ zero_zero_nat )
% 9.00/5.07 = one_one_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4978_ceiling__le__zero,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4979_ceiling__le__zero,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4980_zero__less__ceiling,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4981_zero__less__ceiling,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4982_ceiling__less__one,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4983_ceiling__less__one,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4984_one__le__ceiling,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ zero_zero_rat @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4985_one__le__ceiling,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4986_ceiling__le__one,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4987_ceiling__le__one,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ one_one_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4988_one__less__ceiling,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ one_one_rat @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4989_one__less__ceiling,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ one_one_real @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4990_ceiling__less__zero,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4991_ceiling__less__zero,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4992_zero__le__ceiling,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4993_zero__le__ceiling,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4994_ceiling__mono,axiom,
% 9.00/5.07 ! [Y3: real,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.07 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4995_ceiling__mono,axiom,
% 9.00/5.07 ! [Y3: rat,X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.07 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4996_ceiling__less__cancel,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 9.00/5.07 => ( ord_less_rat @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4997_ceiling__less__cancel,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 9.00/5.07 => ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4998_gbinomial__of__nat__symmetric,axiom,
% 9.00/5.07 ! [K: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.07 => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K )
% 9.00/5.07 = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_4999_set__subset__Cons,axiom,
% 9.00/5.07 ! [Xs: list_VEBT_VEBT,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ ( set_VEBT_VEBT2 @ ( cons_VEBT_VEBT @ X2 @ Xs ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5000_set__subset__Cons,axiom,
% 9.00/5.07 ! [Xs: list_nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5001_set__subset__Cons,axiom,
% 9.00/5.07 ! [Xs: list_int,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ ( set_int2 @ ( cons_int @ X2 @ Xs ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5002_impossible__Cons,axiom,
% 9.00/5.07 ! [Xs: list_int,Ys2: list_int,X2: int] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) )
% 9.00/5.07 => ( Xs
% 9.00/5.07 != ( cons_int @ X2 @ Ys2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5003_impossible__Cons,axiom,
% 9.00/5.07 ! [Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) )
% 9.00/5.07 => ( Xs
% 9.00/5.07 != ( cons_VEBT_VEBT @ X2 @ Ys2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5004_impossible__Cons,axiom,
% 9.00/5.07 ! [Xs: list_o,Ys2: list_o,X2: $o] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys2 ) )
% 9.00/5.07 => ( Xs
% 9.00/5.07 != ( cons_o @ X2 @ Ys2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5005_impossible__Cons,axiom,
% 9.00/5.07 ! [Xs: list_nat,Ys2: list_nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys2 ) )
% 9.00/5.07 => ( Xs
% 9.00/5.07 != ( cons_nat @ X2 @ Ys2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5006_find_Osimps_I2_J,axiom,
% 9.00/5.07 ! [P: int > $o,X2: int,Xs: list_int] :
% 9.00/5.07 ( ( ( P @ X2 )
% 9.00/5.07 => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
% 9.00/5.07 = ( some_int @ X2 ) ) )
% 9.00/5.07 & ( ~ ( P @ X2 )
% 9.00/5.07 => ( ( find_int @ P @ ( cons_int @ X2 @ Xs ) )
% 9.00/5.07 = ( find_int @ P @ Xs ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5007_find_Osimps_I2_J,axiom,
% 9.00/5.07 ! [P: nat > $o,X2: nat,Xs: list_nat] :
% 9.00/5.07 ( ( ( P @ X2 )
% 9.00/5.07 => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
% 9.00/5.07 = ( some_nat @ X2 ) ) )
% 9.00/5.07 & ( ~ ( P @ X2 )
% 9.00/5.07 => ( ( find_nat @ P @ ( cons_nat @ X2 @ Xs ) )
% 9.00/5.07 = ( find_nat @ P @ Xs ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5008_find_Osimps_I2_J,axiom,
% 9.00/5.07 ! [P: product_prod_nat_nat > $o,X2: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.00/5.07 ( ( ( P @ X2 )
% 9.00/5.07 => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
% 9.00/5.07 = ( some_P7363390416028606310at_nat @ X2 ) ) )
% 9.00/5.07 & ( ~ ( P @ X2 )
% 9.00/5.07 => ( ( find_P8199882355184865565at_nat @ P @ ( cons_P6512896166579812791at_nat @ X2 @ Xs ) )
% 9.00/5.07 = ( find_P8199882355184865565at_nat @ P @ Xs ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5009_find_Osimps_I2_J,axiom,
% 9.00/5.07 ! [P: num > $o,X2: num,Xs: list_num] :
% 9.00/5.07 ( ( ( P @ X2 )
% 9.00/5.07 => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
% 9.00/5.07 = ( some_num @ X2 ) ) )
% 9.00/5.07 & ( ~ ( P @ X2 )
% 9.00/5.07 => ( ( find_num @ P @ ( cons_num @ X2 @ Xs ) )
% 9.00/5.07 = ( find_num @ P @ Xs ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5010_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.00/5.07 ! [K: nat,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A2 )
% 9.00/5.07 => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A2 @ K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5011_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.00/5.07 ! [K: nat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A2 )
% 9.00/5.07 => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A2 @ K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5012_ceiling__add__le,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5013_ceiling__add__le,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X2 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5014_Suc__le__length__iff,axiom,
% 9.00/5.07 ! [N: nat,Xs: list_int] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
% 9.00/5.07 = ( ? [X: int,Ys3: list_int] :
% 9.00/5.07 ( ( Xs
% 9.00/5.07 = ( cons_int @ X @ Ys3 ) )
% 9.00/5.07 & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5015_Suc__le__length__iff,axiom,
% 9.00/5.07 ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.07 = ( ? [X: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.00/5.07 ( ( Xs
% 9.00/5.07 = ( cons_VEBT_VEBT @ X @ Ys3 ) )
% 9.00/5.07 & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5016_Suc__le__length__iff,axiom,
% 9.00/5.07 ! [N: nat,Xs: list_o] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) )
% 9.00/5.07 = ( ? [X: $o,Ys3: list_o] :
% 9.00/5.07 ( ( Xs
% 9.00/5.07 = ( cons_o @ X @ Ys3 ) )
% 9.00/5.07 & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5017_Suc__le__length__iff,axiom,
% 9.00/5.07 ! [N: nat,Xs: list_nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) )
% 9.00/5.07 = ( ? [X: nat,Ys3: list_nat] :
% 9.00/5.07 ( ( Xs
% 9.00/5.07 = ( cons_nat @ X @ Ys3 ) )
% 9.00/5.07 & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5018_gbinomial__trinomial__revision,axiom,
% 9.00/5.07 ! [K: nat,M: nat,A2: complex] :
% 9.00/5.07 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.07 => ( ( times_times_complex @ ( gbinomial_complex @ A2 @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 9.00/5.07 = ( times_times_complex @ ( gbinomial_complex @ A2 @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5019_gbinomial__trinomial__revision,axiom,
% 9.00/5.07 ! [K: nat,M: nat,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.07 => ( ( times_times_rat @ ( gbinomial_rat @ A2 @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 9.00/5.07 = ( times_times_rat @ ( gbinomial_rat @ A2 @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5020_gbinomial__trinomial__revision,axiom,
% 9.00/5.07 ! [K: nat,M: nat,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.07 => ( ( times_times_real @ ( gbinomial_real @ A2 @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 9.00/5.07 = ( times_times_real @ ( gbinomial_real @ A2 @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5021_list_Osize_I4_J,axiom,
% 9.00/5.07 ! [X21: int,X222: list_int] :
% 9.00/5.07 ( ( size_size_list_int @ ( cons_int @ X21 @ X222 ) )
% 9.00/5.07 = ( plus_plus_nat @ ( size_size_list_int @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5022_list_Osize_I4_J,axiom,
% 9.00/5.07 ! [X21: vEBT_VEBT,X222: list_VEBT_VEBT] :
% 9.00/5.07 ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X222 ) )
% 9.00/5.07 = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5023_list_Osize_I4_J,axiom,
% 9.00/5.07 ! [X21: $o,X222: list_o] :
% 9.00/5.07 ( ( size_size_list_o @ ( cons_o @ X21 @ X222 ) )
% 9.00/5.07 = ( plus_plus_nat @ ( size_size_list_o @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5024_list_Osize_I4_J,axiom,
% 9.00/5.07 ! [X21: nat,X222: list_nat] :
% 9.00/5.07 ( ( size_size_list_nat @ ( cons_nat @ X21 @ X222 ) )
% 9.00/5.07 = ( plus_plus_nat @ ( size_size_list_nat @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5025_nth__Cons_H,axiom,
% 9.00/5.07 ! [N: nat,X2: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.00/5.07 ( ( ( N = zero_zero_nat )
% 9.00/5.07 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 ) )
% 9.00/5.07 & ( ( N != zero_zero_nat )
% 9.00/5.07 => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5026_nth__Cons_H,axiom,
% 9.00/5.07 ! [N: nat,X2: int,Xs: list_int] :
% 9.00/5.07 ( ( ( N = zero_zero_nat )
% 9.00/5.07 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 ) )
% 9.00/5.07 & ( ( N != zero_zero_nat )
% 9.00/5.07 => ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5027_nth__Cons_H,axiom,
% 9.00/5.07 ! [N: nat,X2: nat,Xs: list_nat] :
% 9.00/5.07 ( ( ( N = zero_zero_nat )
% 9.00/5.07 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 ) )
% 9.00/5.07 & ( ( N != zero_zero_nat )
% 9.00/5.07 => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5028_mult__ceiling__le,axiom,
% 9.00/5.07 ! [A2: real,B3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ B3 )
% 9.00/5.07 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5029_mult__ceiling__le,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ B3 )
% 9.00/5.07 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5030_gbinomial__reduce__nat,axiom,
% 9.00/5.07 ! [K: nat,A2: complex] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_complex @ A2 @ K )
% 9.00/5.07 = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A2 @ one_one_complex ) @ K ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5031_gbinomial__reduce__nat,axiom,
% 9.00/5.07 ! [K: nat,A2: real] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_real @ A2 @ K )
% 9.00/5.07 = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A2 @ one_one_real ) @ K ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5032_gbinomial__reduce__nat,axiom,
% 9.00/5.07 ! [K: nat,A2: rat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.07 => ( ( gbinomial_rat @ A2 @ K )
% 9.00/5.07 = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A2 @ one_one_rat ) @ K ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5033_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: real,Xs: list_real,N: nat] :
% 9.00/5.07 ( ~ ( member_real @ X2 @ ( set_real2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.00/5.07 => ( ( ( nth_real @ ( cons_real @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5034_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: set_nat,Xs: list_set_nat,N: nat] :
% 9.00/5.07 ( ~ ( member_set_nat @ X2 @ ( set_set_nat2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.07 => ( ( ( nth_set_nat @ ( cons_set_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5035_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: int,Xs: list_int,N: nat] :
% 9.00/5.07 ( ~ ( member_int @ X2 @ ( set_int2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.07 => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5036_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
% 9.00/5.07 ( ~ ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.07 => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5037_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: $o,Xs: list_o,N: nat] :
% 9.00/5.07 ( ~ ( member_o @ X2 @ ( set_o2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.07 => ( ( ( nth_o @ ( cons_o @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5038_nth__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: nat,Xs: list_nat,N: nat] :
% 9.00/5.07 ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.07 => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = X2 )
% 9.00/5.07 = ( N = zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5039_nth__non__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,N: nat] :
% 9.00/5.07 ( ( X2 != Y3 )
% 9.00/5.07 => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ N )
% 9.00/5.07 = Y3 )
% 9.00/5.07 = ( ( ( nth_VEBT_VEBT @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.07 = Y3 )
% 9.00/5.07 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5040_nth__non__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: int,Y3: int,Xs: list_int,N: nat] :
% 9.00/5.07 ( ( X2 != Y3 )
% 9.00/5.07 => ( ( ( nth_int @ ( cons_int @ X2 @ Xs ) @ N )
% 9.00/5.07 = Y3 )
% 9.00/5.07 = ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.07 = Y3 )
% 9.00/5.07 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5041_nth__non__equal__first__eq,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat,Xs: list_nat,N: nat] :
% 9.00/5.07 ( ( X2 != Y3 )
% 9.00/5.07 => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N )
% 9.00/5.07 = Y3 )
% 9.00/5.07 = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
% 9.00/5.07 = Y3 )
% 9.00/5.07 & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5042_upto__aux__rec,axiom,
% 9.00/5.07 ( upto_aux
% 9.00/5.07 = ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5043_ceiling__eq,axiom,
% 9.00/5.07 ! [N: int,X2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.00/5.07 => ( ( archim7802044766580827645g_real @ X2 )
% 9.00/5.07 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5044_ceiling__eq,axiom,
% 9.00/5.07 ! [N: int,X2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ X2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
% 9.00/5.07 => ( ( archim2889992004027027881ng_rat @ X2 )
% 9.00/5.07 = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5045_ceiling__divide__lower,axiom,
% 9.00/5.07 ! [Q3: rat,P6: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.00/5.07 => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P6 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5046_ceiling__divide__lower,axiom,
% 9.00/5.07 ! [Q3: real,P6: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.00/5.07 => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P6 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5047__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 9.00/5.07 ~ ! [TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Info: option4927543243414619207at_nat] :
% 9.00/5.07 ~ ( ( sa
% 9.00/5.07 = ( vEBT_Node @ Info @ deg @ TreeList3 @ Summary3 ) )
% 9.00/5.07 & ( deg
% 9.00/5.07 = ( plus_plus_nat @ na @ m ) )
% 9.00/5.07 & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.00/5.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.00/5.07 & ( vEBT_invar_vebt @ Summary3 @ m )
% 9.00/5.07 & ! [X4: vEBT_VEBT] :
% 9.00/5.07 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.00/5.07 => ( vEBT_invar_vebt @ X4 @ na ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5048_split__pos__lemma,axiom,
% 9.00/5.07 ! [K: int,P: int > int > $o,N: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.07 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.00/5.07 = ( ! [I4: int,J3: int] :
% 9.00/5.07 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.00/5.07 & ( ord_less_int @ J3 @ K )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.07 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5049_split__neg__lemma,axiom,
% 9.00/5.07 ! [K: int,P: int > int > $o,N: int] :
% 9.00/5.07 ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.07 => ( ( P @ ( divide_divide_int @ N @ K ) @ ( modulo_modulo_int @ N @ K ) )
% 9.00/5.07 = ( ! [I4: int,J3: int] :
% 9.00/5.07 ( ( ( ord_less_int @ K @ J3 )
% 9.00/5.07 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.00/5.07 & ( N
% 9.00/5.07 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.07 => ( P @ I4 @ J3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5050_verit__le__mono__div__int,axiom,
% 9.00/5.07 ! [A3: int,B2: int,N: int] :
% 9.00/5.07 ( ( ord_less_int @ A3 @ B2 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ N )
% 9.00/5.07 => ( ord_less_eq_int
% 9.00/5.07 @ ( plus_plus_int @ ( divide_divide_int @ A3 @ N )
% 9.00/5.07 @ ( if_int
% 9.00/5.07 @ ( ( modulo_modulo_int @ B2 @ N )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 @ one_one_int
% 9.00/5.07 @ zero_zero_int ) )
% 9.00/5.07 @ ( divide_divide_int @ B2 @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5051_semiring__norm_I71_J,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5052_semiring__norm_I78_J,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5053_semiring__norm_I68_J,axiom,
% 9.00/5.07 ! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5054_semiring__norm_I75_J,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ~ ( ord_less_num @ M @ one ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5055_case4_I10_J,axiom,
% 9.00/5.07 ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5056_case4_I4_J,axiom,
% 9.00/5.07 ( ( size_s6755466524823107622T_VEBT @ treeList2 )
% 9.00/5.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5057_pow__sum,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A2 @ B3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) )
% 9.00/5.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5058_a0,axiom,
% 9.00/5.07 ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5059_case4_I7_J,axiom,
% 9.00/5.07 ! [I3: nat] :
% 9.00/5.07 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.00/5.07 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ X8 ) )
% 9.00/5.07 = ( vEBT_V8194947554948674370ptions @ summary2 @ I3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5060_power__minus__is__div,axiom,
% 9.00/5.07 ! [B3: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.07 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A2 @ B3 ) )
% 9.00/5.07 = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5061_member__bound,axiom,
% 9.00/5.07 ! [Tree: vEBT_VEBT,X2: nat,N: nat] :
% 9.00/5.07 ( ( vEBT_vebt_member @ Tree @ X2 )
% 9.00/5.07 => ( ( vEBT_invar_vebt @ Tree @ N )
% 9.00/5.07 => ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5062_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5063_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5064_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5065_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5066_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5067_numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.07 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5068_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5069_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5070_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5071_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5072_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5073_numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N ) )
% 9.00/5.07 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5074_semiring__norm_I69_J,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5075_semiring__norm_I76_J,axiom,
% 9.00/5.07 ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5076_bits__mod__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ zero_zero_int @ A2 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5077_bits__mod__0,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5078_mod__self,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ A2 @ A2 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5079_mod__self,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ A2 @ A2 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5080_mod__by__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ A2 @ zero_zero_int )
% 9.00/5.07 = A2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5081_mod__by__0,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ A2 @ zero_zero_nat )
% 9.00/5.07 = A2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5082_mod__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ zero_zero_int @ A2 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5083_mod__0,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5084_misiz,axiom,
% 9.00/5.07 ! [T: vEBT_VEBT,N: nat,M: nat] :
% 9.00/5.07 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.07 => ( ( ( some_nat @ M )
% 9.00/5.07 = ( vEBT_vebt_mint @ T ) )
% 9.00/5.07 => ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5085_helpypredd,axiom,
% 9.00/5.07 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.07 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.07 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 9.00/5.07 = ( some_nat @ Y3 ) )
% 9.00/5.07 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5086_helpyd,axiom,
% 9.00/5.07 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.07 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.07 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 9.00/5.07 = ( some_nat @ Y3 ) )
% 9.00/5.07 => ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5087_delt__out__of__range,axiom,
% 9.00/5.07 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.07 ( ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.07 | ( ord_less_nat @ Ma @ X2 ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.07 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.07 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5088_del__single__cont,axiom,
% 9.00/5.07 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.07 ( ( ( X2 = Mi )
% 9.00/5.07 & ( X2 = Ma ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.07 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.07 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5089_set__n__deg__not__0,axiom,
% 9.00/5.07 ! [TreeList2: list_VEBT_VEBT,N: nat,M: nat] :
% 9.00/5.07 ( ! [X5: vEBT_VEBT] :
% 9.00/5.07 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.07 => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.00/5.07 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5090_mi__ma__2__deg,axiom,
% 9.00/5.07 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.00/5.07 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.07 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.00/5.07 & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5091_pred__max,axiom,
% 9.00/5.07 ! [Deg: nat,Ma: nat,X2: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.07 => ( ( ord_less_nat @ Ma @ X2 )
% 9.00/5.07 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.07 = ( some_nat @ Ma ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5092_succ__min,axiom,
% 9.00/5.07 ! [Deg: nat,X2: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.07 => ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.07 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.07 = ( some_nat @ Mi ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5093_bit__concat__def,axiom,
% 9.00/5.07 ( vEBT_VEBT_bit_concat
% 9.00/5.07 = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5094_neg__numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.00/5.07 = ( ord_less_eq_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5095_neg__numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.00/5.07 = ( ord_less_eq_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5096_neg__numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.00/5.07 = ( ord_less_eq_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5097_neg__numeral__le__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.07 = ( ord_less_eq_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5098_neg__numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.07 = ( ord_less_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5099_neg__numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.00/5.07 = ( ord_less_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5100_neg__numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
% 9.00/5.07 = ( ord_less_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5101_neg__numeral__less__iff,axiom,
% 9.00/5.07 ! [M: num,N: num] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
% 9.00/5.07 = ( ord_less_num @ N @ M ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5102_power__zero__numeral,axiom,
% 9.00/5.07 ! [K: num] :
% 9.00/5.07 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5103_power__zero__numeral,axiom,
% 9.00/5.07 ! [K: num] :
% 9.00/5.07 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5104_power__zero__numeral,axiom,
% 9.00/5.07 ! [K: num] :
% 9.00/5.07 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5105_power__zero__numeral,axiom,
% 9.00/5.07 ! [K: num] :
% 9.00/5.07 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5106_power__zero__numeral,axiom,
% 9.00/5.07 ! [K: num] :
% 9.00/5.07 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 9.00/5.07 = zero_zero_complex ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5107_mod__mult__self1__is__0,axiom,
% 9.00/5.07 ! [B3: int,A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ ( times_times_int @ B3 @ A2 ) @ B3 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5108_mod__mult__self1__is__0,axiom,
% 9.00/5.07 ! [B3: nat,A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ ( times_times_nat @ B3 @ A2 ) @ B3 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5109_mod__mult__self2__is__0,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ ( times_times_int @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5110_mod__mult__self2__is__0,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5111_bits__mod__by__1,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ A2 @ one_one_int )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5112_bits__mod__by__1,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ A2 @ one_one_nat )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5113_mod__by__1,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ A2 @ one_one_int )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5114_mod__by__1,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( modulo_modulo_nat @ A2 @ one_one_nat )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5115_bits__mod__div__trivial,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5116_bits__mod__div__trivial,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5117_mod__div__trivial,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( divide_divide_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5118_mod__div__trivial,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( divide_divide_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5119_of__int__eq__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ( ring_1_of_int_int @ Z )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5120_of__int__eq__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ( ring_1_of_int_real @ Z )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5121_of__int__eq__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ( ring_1_of_int_rat @ Z )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5122_of__int__0__eq__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( zero_zero_int
% 9.00/5.07 = ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5123_of__int__0__eq__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( zero_zero_real
% 9.00/5.07 = ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5124_of__int__0__eq__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( zero_zero_rat
% 9.00/5.07 = ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( Z = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5125_of__int__0,axiom,
% 9.00/5.07 ( ( ring_1_of_int_int @ zero_zero_int )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5126_of__int__0,axiom,
% 9.00/5.07 ( ( ring_1_of_int_real @ zero_zero_int )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5127_of__int__0,axiom,
% 9.00/5.07 ( ( ring_1_of_int_rat @ zero_zero_int )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5128_of__int__le__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5129_of__int__le__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5130_of__int__le__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5131_of__int__less__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5132_of__int__less__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5133_of__int__less__iff,axiom,
% 9.00/5.07 ! [W2: int,Z: int] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_int @ W2 @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5134_frac__of__int,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5135_frac__of__int,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5136_mintlistlength,axiom,
% 9.00/5.07 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.00/5.07 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.07 => ( ( Mi != Ma )
% 9.00/5.07 => ( ( ord_less_nat @ Mi @ Ma )
% 9.00/5.07 & ? [M4: nat] :
% 9.00/5.07 ( ( ( some_nat @ M4 )
% 9.00/5.07 = ( vEBT_vebt_mint @ Summary ) )
% 9.00/5.07 & ( ord_less_nat @ M4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5137_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ one_one_real )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5138_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5139_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5140_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5141_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5142_numeral__le__one__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_num @ N @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5143_divide__le__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5144_divide__le__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5145_le__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5146_le__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5147_eq__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.07 = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( numeral_numeral_rat @ W2 )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5148_eq__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.07 = ( ( ( ( numeral_numeral_real @ W2 )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( numeral_numeral_real @ W2 )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5149_eq__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: complex,B3: complex,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.00/5.07 = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.07 != zero_zero_complex )
% 9.00/5.07 => ( ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5150_divide__eq__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.00/5.07 & ( ( ( numeral_numeral_rat @ W2 )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5151_divide__eq__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( numeral_numeral_real @ W2 )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.00/5.07 & ( ( ( numeral_numeral_real @ W2 )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5152_divide__eq__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: complex,W2: num,A2: complex] :
% 9.00/5.07 ( ( ( divide1717551699836669952omplex @ B3 @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.07 != zero_zero_complex )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_complex @ A2 @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.00/5.07 & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5153_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5154_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5155_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5156_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5157_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5158_one__less__numeral__iff,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
% 9.00/5.07 = ( ord_less_num @ one @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5159_less__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5160_less__divide__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5161_divide__less__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( numeral_numeral_rat @ W2 ) ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5162_divide__less__eq__numeral1_I1_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( numeral_numeral_real @ W2 ) ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5163_mod__minus1__right,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( modulo_modulo_int @ A2 @ ( uminus_uminus_int @ one_one_int ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5164_mod__minus1__right,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( modulo364778990260209775nteger @ A2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5165_of__int__numeral__le__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5166_of__int__numeral__le__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5167_of__int__numeral__le__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5168_of__int__numeral__le__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5169_of__int__le__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5170_of__int__le__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5171_of__int__le__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5172_of__int__le__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5173_of__int__numeral__less__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5174_of__int__numeral__less__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5175_of__int__numeral__less__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5176_of__int__numeral__less__iff,axiom,
% 9.00/5.07 ! [N: num,Z: int] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ ( ring_18347121197199848620nteger @ Z ) )
% 9.00/5.07 = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5177_of__int__less__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N ) )
% 9.00/5.07 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5178_of__int__less__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N ) )
% 9.00/5.07 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5179_of__int__less__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.07 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5180_of__int__less__numeral__iff,axiom,
% 9.00/5.07 ! [Z: int,N: num] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ Z ) @ ( numera6620942414471956472nteger @ N ) )
% 9.00/5.07 = ( ord_less_int @ Z @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5181_ceiling__le__numeral,axiom,
% 9.00/5.07 ! [X2: real,V: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( numeral_numeral_real @ V ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5182_ceiling__le__numeral,axiom,
% 9.00/5.07 ! [X2: rat,V: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5183_numeral__less__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5184_numeral__less__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: real] :
% 9.00/5.07 ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5185_mod__neg__neg__trivial,axiom,
% 9.00/5.07 ! [K: int,L: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.00/5.07 => ( ( ord_less_int @ L @ K )
% 9.00/5.07 => ( ( modulo_modulo_int @ K @ L )
% 9.00/5.07 = K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5186_mod__pos__pos__trivial,axiom,
% 9.00/5.07 ! [K: int,L: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.07 => ( ( ord_less_int @ K @ L )
% 9.00/5.07 => ( ( modulo_modulo_int @ K @ L )
% 9.00/5.07 = K ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5187_not__neg__one__le__neg__numeral__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5188_not__neg__one__le__neg__numeral__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5189_not__neg__one__le__neg__numeral__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5190_not__neg__one__le__neg__numeral__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5191_divide__le__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5192_divide__le__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
% 9.00/5.07 = ( ord_less_eq_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5193_le__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( ord_less_eq_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5194_le__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( ord_less_eq_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5195_eq__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.00/5.07 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5196_eq__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.00/5.07 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5197_eq__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: complex,B3: complex,W2: num] :
% 9.00/5.07 ( ( A2
% 9.00/5.07 = ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.00/5.07 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 != zero_zero_complex )
% 9.00/5.07 => ( ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.00/5.07 = B3 ) )
% 9.00/5.07 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5198_divide__eq__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 != zero_zero_real )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 9.00/5.07 & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 => ( A2 = zero_zero_real ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5199_divide__eq__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 != zero_zero_rat )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 9.00/5.07 & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 => ( A2 = zero_zero_rat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5200_divide__eq__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: complex,W2: num,A2: complex] :
% 9.00/5.07 ( ( ( divide1717551699836669952omplex @ B3 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 != zero_zero_complex )
% 9.00/5.07 => ( B3
% 9.00/5.07 = ( times_times_complex @ A2 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 9.00/5.07 & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 => ( A2 = zero_zero_complex ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5201_neg__numeral__less__neg__one__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5202_neg__numeral__less__neg__one__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5203_neg__numeral__less__neg__one__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5204_neg__numeral__less__neg__one__iff,axiom,
% 9.00/5.07 ! [M: num] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.00/5.07 = ( M != one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5205_less__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: real,B3: real,W2: num] :
% 9.00/5.07 ( ( ord_less_real @ A2 @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.00/5.07 = ( ord_less_real @ B3 @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5206_less__divide__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [A2: rat,B3: rat,W2: num] :
% 9.00/5.07 ( ( ord_less_rat @ A2 @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.00/5.07 = ( ord_less_rat @ B3 @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5207_divide__less__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: real,W2: num,A2: real] :
% 9.00/5.07 ( ( ord_less_real @ ( divide_divide_real @ B3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A2 )
% 9.00/5.07 = ( ord_less_real @ ( times_times_real @ A2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5208_divide__less__eq__numeral1_I2_J,axiom,
% 9.00/5.07 ! [B3: rat,W2: num,A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A2 )
% 9.00/5.07 = ( ord_less_rat @ ( times_times_rat @ A2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B3 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5209_zero__eq__power2,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5210_zero__eq__power2,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5211_zero__eq__power2,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( ( power_power_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_nat )
% 9.00/5.07 = ( A2 = zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5212_zero__eq__power2,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5213_zero__eq__power2,axiom,
% 9.00/5.07 ! [A2: complex] :
% 9.00/5.07 ( ( ( power_power_complex @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_complex )
% 9.00/5.07 = ( A2 = zero_zero_complex ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5214_of__int__0__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5215_of__int__0__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5216_of__int__0__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5217_of__int__le__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5218_of__int__le__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5219_of__int__le__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5220_of__int__less__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
% 9.00/5.07 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5221_of__int__less__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
% 9.00/5.07 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5222_of__int__less__0__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
% 9.00/5.07 = ( ord_less_int @ Z @ zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5223_of__int__0__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5224_of__int__0__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5225_of__int__0__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5226_of__int__1__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5227_of__int__1__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5228_of__int__1__le__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5229_of__int__le__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5230_of__int__le__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5231_of__int__le__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 9.00/5.07 = ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5232_of__int__1__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.07 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5233_of__int__1__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.07 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5234_of__int__1__less__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
% 9.00/5.07 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5235_of__int__less__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
% 9.00/5.07 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5236_of__int__less__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
% 9.00/5.07 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5237_of__int__less__1__iff,axiom,
% 9.00/5.07 ! [Z: int] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
% 9.00/5.07 = ( ord_less_int @ Z @ one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5238_of__int__le__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5239_of__int__le__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5240_of__int__le__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5241_of__int__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5242_of__int__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5243_of__int__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5244_of__int__less__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) @ ( ring_1_of_int_real @ X2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5245_of__int__less__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) @ ( ring_1_of_int_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5246_of__int__less__of__int__power__cancel__iff,axiom,
% 9.00/5.07 ! [B3: int,W2: nat,X2: int] :
% 9.00/5.07 ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) @ ( ring_1_of_int_int @ X2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ B3 @ W2 ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5247_of__int__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ X2 ) @ ( power_power_real @ ( ring_1_of_int_real @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5248_of__int__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ X2 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5249_of__int__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: int,B3: int,W2: nat] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ X2 ) @ ( power_power_int @ ( ring_1_of_int_int @ B3 ) @ W2 ) )
% 9.00/5.07 = ( ord_less_int @ X2 @ ( power_power_int @ B3 @ W2 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5250_one__div__two__eq__zero,axiom,
% 9.00/5.07 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5251_one__div__two__eq__zero,axiom,
% 9.00/5.07 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5252_one__div__two__eq__zero,axiom,
% 9.00/5.07 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5253_bits__1__div__2,axiom,
% 9.00/5.07 ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5254_bits__1__div__2,axiom,
% 9.00/5.07 ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5255_bits__1__div__2,axiom,
% 9.00/5.07 ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5256_power2__eq__iff__nonneg,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5257_power2__eq__iff__nonneg,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5258_power2__eq__iff__nonneg,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.00/5.07 => ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5259_power2__eq__iff__nonneg,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5260_power2__less__eq__zero__iff,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 9.00/5.07 = ( A2 = zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5261_power2__less__eq__zero__iff,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 9.00/5.07 = ( A2 = zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5262_power2__less__eq__zero__iff,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.00/5.07 = ( A2 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5263_zero__less__power2,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( A2 != zero_zero_real ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5264_zero__less__power2,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( A2 != zero_zero_rat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5265_zero__less__power2,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( A2 != zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5266_sum__power2__eq__zero__iff,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_zero_rat )
% 9.00/5.07 = ( ( X2 = zero_zero_rat )
% 9.00/5.07 & ( Y3 = zero_zero_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5267_sum__power2__eq__zero__iff,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( ( X2 = zero_zero_int )
% 9.00/5.07 & ( Y3 = zero_zero_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5268_sum__power2__eq__zero__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_zero_real )
% 9.00/5.07 = ( ( X2 = zero_zero_real )
% 9.00/5.07 & ( Y3 = zero_zero_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5269_not__mod__2__eq__0__eq__1,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 = one_one_Code_integer ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5270_not__mod__2__eq__0__eq__1,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 = one_one_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5271_not__mod__2__eq__0__eq__1,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 != zero_zero_nat )
% 9.00/5.07 = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = one_one_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5272_not__mod__2__eq__1__eq__0,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 != one_one_Code_integer )
% 9.00/5.07 = ( ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5273_not__mod__2__eq__1__eq__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 != one_one_int )
% 9.00/5.07 = ( ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5274_not__mod__2__eq__1__eq__0,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 != one_one_nat )
% 9.00/5.07 = ( ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5275_ceiling__less__numeral,axiom,
% 9.00/5.07 ! [X2: real,V: num] :
% 9.00/5.07 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5276_ceiling__less__numeral,axiom,
% 9.00/5.07 ! [X2: rat,V: num] :
% 9.00/5.07 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5277_numeral__le__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5278_numeral__le__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5279_ceiling__le__neg__numeral,axiom,
% 9.00/5.07 ! [X2: real,V: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5280_ceiling__le__neg__numeral,axiom,
% 9.00/5.07 ! [X2: rat,V: num] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5281_numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5282_numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5283_numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5284_numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5285_of__int__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5286_of__int__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5287_of__int__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5288_of__int__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5289_neg__numeral__less__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: real] :
% 9.00/5.07 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5290_neg__numeral__less__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: rat] :
% 9.00/5.07 ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5291_numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5292_numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5293_numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5294_numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5295_of__int__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5296_of__int__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( numeral_numeral_real @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5297_of__int__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5298_of__int__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ X2 ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5299_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5300_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5301_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5302_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5303_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5304_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5305_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 9.00/5.07 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5306_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
% 9.00/5.07 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5307_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 9.00/5.07 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5308_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 9.00/5.07 = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5309_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) @ ( semiri4939895301339042750nteger @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5310_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) @ ( semiri5074537144036343181t_real @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5311_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) @ ( semiri681578069525770553at_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5312_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ ( semiri1316708129612266289at_nat @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5313_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.00/5.07 ! [I: num,N: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) @ ( semiri1314217659103216013at_int @ X2 ) )
% 9.00/5.07 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5314_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( semiri4939895301339042750nteger @ X2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5315_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5316_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5317_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5318_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [X2: nat,I: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5319_ceiling__less__neg__numeral,axiom,
% 9.00/5.07 ! [X2: real,V: num] :
% 9.00/5.07 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.07 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5320_ceiling__less__neg__numeral,axiom,
% 9.00/5.07 ! [X2: rat,V: num] :
% 9.00/5.07 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.07 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5321_neg__numeral__le__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: real] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.07 = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5322_neg__numeral__le__ceiling,axiom,
% 9.00/5.07 ! [V: num,X2: rat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.07 = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5323_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5324_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5325_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5326_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5327_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5328_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5329_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5330_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5331_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_int @ ( ring_1_of_int_int @ A2 ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5332_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_real @ ( ring_1_of_int_real @ A2 ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5333_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_less_rat @ ( ring_1_of_int_rat @ A2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5334_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.00/5.07 ! [A2: int,X2: num,N: nat] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) )
% 9.00/5.07 = ( ord_less_int @ A2 @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5335_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ ( ring_1_of_int_int @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5336_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X2 ) ) @ N ) @ ( ring_1_of_int_real @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5337_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X2 ) ) @ N ) @ ( ring_1_of_int_rat @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5338_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.00/5.07 ! [X2: num,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X2 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A2 ) )
% 9.00/5.07 = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X2 ) ) @ N ) @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5339_sprop1,axiom,
% 9.00/5.07 ( ( sa
% 9.00/5.07 = ( vEBT_Node @ info @ deg @ treeList @ summary ) )
% 9.00/5.07 & ( deg
% 9.00/5.07 = ( plus_plus_nat @ na @ m ) )
% 9.00/5.07 & ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.00/5.07 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.00/5.07 & ( vEBT_invar_vebt @ summary @ m )
% 9.00/5.07 & ! [X4: vEBT_VEBT] :
% 9.00/5.07 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.00/5.07 => ( vEBT_invar_vebt @ X4 @ na ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5340_divmod__digit__0_I2_J,axiom,
% 9.00/5.07 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.07 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) )
% 9.00/5.07 = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5341_divmod__digit__0_I2_J,axiom,
% 9.00/5.07 ! [B3: int,A2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) )
% 9.00/5.07 = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5342_divmod__digit__0_I2_J,axiom,
% 9.00/5.07 ! [B3: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.07 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) )
% 9.00/5.07 = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5343_pos2,axiom,
% 9.00/5.07 ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5344_zero__power2,axiom,
% 9.00/5.07 ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5345_zero__power2,axiom,
% 9.00/5.07 ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5346_zero__power2,axiom,
% 9.00/5.07 ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5347_zero__power2,axiom,
% 9.00/5.07 ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5348_zero__power2,axiom,
% 9.00/5.07 ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = zero_zero_complex ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5349_numeral__2__eq__2,axiom,
% 9.00/5.07 ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.00/5.07 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5350_less__exp,axiom,
% 9.00/5.07 ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5351_self__le__ge2__pow,axiom,
% 9.00/5.07 ! [K: nat,M: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.00/5.07 => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5352_power2__nat__le__eq__le,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5353_power2__nat__le__imp__le,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
% 9.00/5.07 => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5354_cong__exp__iff__simps_I2_J,axiom,
% 9.00/5.07 ! [N: num,Q3: num] :
% 9.00/5.07 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.00/5.07 = zero_z3403309356797280102nteger )
% 9.00/5.07 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5355_cong__exp__iff__simps_I2_J,axiom,
% 9.00/5.07 ! [N: num,Q3: num] :
% 9.00/5.07 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.00/5.07 = zero_zero_int )
% 9.00/5.07 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 9.00/5.07 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5356_cong__exp__iff__simps_I2_J,axiom,
% 9.00/5.07 ! [N: num,Q3: num] :
% 9.00/5.07 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.00/5.07 = zero_zero_nat )
% 9.00/5.07 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.00/5.07 = zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5357_cong__exp__iff__simps_I1_J,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5358_cong__exp__iff__simps_I1_J,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) )
% 9.00/5.07 = zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5359_cong__exp__iff__simps_I1_J,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) )
% 9.00/5.07 = zero_zero_nat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5360_card__2__iff_H,axiom,
% 9.00/5.07 ! [S: set_complex] :
% 9.00/5.07 ( ( ( finite_card_complex @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: complex] :
% 9.00/5.07 ( ( member_complex @ X @ S )
% 9.00/5.07 & ? [Y: complex] :
% 9.00/5.07 ( ( member_complex @ Y @ S )
% 9.00/5.07 & ( X != Y )
% 9.00/5.07 & ! [Z2: complex] :
% 9.00/5.07 ( ( member_complex @ Z2 @ S )
% 9.00/5.07 => ( ( Z2 = X )
% 9.00/5.07 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5361_card__2__iff_H,axiom,
% 9.00/5.07 ! [S: set_list_nat] :
% 9.00/5.07 ( ( ( finite_card_list_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: list_nat] :
% 9.00/5.07 ( ( member_list_nat @ X @ S )
% 9.00/5.07 & ? [Y: list_nat] :
% 9.00/5.07 ( ( member_list_nat @ Y @ S )
% 9.00/5.07 & ( X != Y )
% 9.00/5.07 & ! [Z2: list_nat] :
% 9.00/5.07 ( ( member_list_nat @ Z2 @ S )
% 9.00/5.07 => ( ( Z2 = X )
% 9.00/5.07 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5362_card__2__iff_H,axiom,
% 9.00/5.07 ! [S: set_set_nat] :
% 9.00/5.07 ( ( ( finite_card_set_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: set_nat] :
% 9.00/5.07 ( ( member_set_nat @ X @ S )
% 9.00/5.07 & ? [Y: set_nat] :
% 9.00/5.07 ( ( member_set_nat @ Y @ S )
% 9.00/5.07 & ( X != Y )
% 9.00/5.07 & ! [Z2: set_nat] :
% 9.00/5.07 ( ( member_set_nat @ Z2 @ S )
% 9.00/5.07 => ( ( Z2 = X )
% 9.00/5.07 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5363_card__2__iff_H,axiom,
% 9.00/5.07 ! [S: set_nat] :
% 9.00/5.07 ( ( ( finite_card_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: nat] :
% 9.00/5.07 ( ( member_nat @ X @ S )
% 9.00/5.07 & ? [Y: nat] :
% 9.00/5.07 ( ( member_nat @ Y @ S )
% 9.00/5.07 & ( X != Y )
% 9.00/5.07 & ! [Z2: nat] :
% 9.00/5.07 ( ( member_nat @ Z2 @ S )
% 9.00/5.07 => ( ( Z2 = X )
% 9.00/5.07 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5364_card__2__iff_H,axiom,
% 9.00/5.07 ! [S: set_int] :
% 9.00/5.07 ( ( ( finite_card_int @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: int] :
% 9.00/5.07 ( ( member_int @ X @ S )
% 9.00/5.07 & ? [Y: int] :
% 9.00/5.07 ( ( member_int @ Y @ S )
% 9.00/5.07 & ( X != Y )
% 9.00/5.07 & ! [Z2: int] :
% 9.00/5.07 ( ( member_int @ Z2 @ S )
% 9.00/5.07 => ( ( Z2 = X )
% 9.00/5.07 | ( Z2 = Y ) ) ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5365_le__num__One__iff,axiom,
% 9.00/5.07 ! [X2: num] :
% 9.00/5.07 ( ( ord_less_eq_num @ X2 @ one )
% 9.00/5.07 = ( X2 = one ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5366_bits__stable__imp__add__self,axiom,
% 9.00/5.07 ! [A2: code_integer] :
% 9.00/5.07 ( ( ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 => ( ( plus_p5714425477246183910nteger @ A2 @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5367_bits__stable__imp__add__self,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ( ( ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 => ( ( plus_plus_int @ A2 @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5368_bits__stable__imp__add__self,axiom,
% 9.00/5.07 ! [A2: nat] :
% 9.00/5.07 ( ( ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = A2 )
% 9.00/5.07 => ( ( plus_plus_nat @ A2 @ ( modulo_modulo_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5369_divmod__digit__0_I1_J,axiom,
% 9.00/5.07 ! [B3: code_integer,A2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.07 => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5370_divmod__digit__0_I1_J,axiom,
% 9.00/5.07 ! [B3: int,A2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.07 => ( ( ord_less_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5371_divmod__digit__0_I1_J,axiom,
% 9.00/5.07 ! [B3: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.07 => ( ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5372_sum__squares__bound,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5373_sum__squares__bound,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X2 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5374_mult__exp__mod__exp__eq,axiom,
% 9.00/5.07 ! [M: nat,N: nat,A2: code_integer] :
% 9.00/5.07 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.07 => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.07 = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5375_mult__exp__mod__exp__eq,axiom,
% 9.00/5.07 ! [M: nat,N: nat,A2: int] :
% 9.00/5.07 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.07 => ( ( modulo_modulo_int @ ( times_times_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.07 = ( times_times_int @ ( modulo_modulo_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5376_mult__exp__mod__exp__eq,axiom,
% 9.00/5.07 ! [M: nat,N: nat,A2: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.07 => ( ( modulo_modulo_nat @ ( times_times_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.07 = ( times_times_nat @ ( modulo_modulo_nat @ A2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5377_half__gt__zero,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ A2 )
% 9.00/5.07 => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5378_half__gt__zero,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.07 => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5379_half__gt__zero__iff,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( ord_less_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5380_half__gt__zero__iff,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.07 = ( ord_less_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5381_field__less__half__sum,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_rat @ X2 @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5382_field__less__half__sum,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.07 => ( ord_less_real @ X2 @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5383_power2__le__imp__le,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ord_less_eq_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5384_power2__le__imp__le,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ord_less_eq_rat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5385_power2__le__imp__le,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.00/5.07 => ( ord_less_eq_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5386_power2__le__imp__le,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( ord_less_eq_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5387_power2__eq__imp__eq,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5388_power2__eq__imp__eq,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5389_power2__eq__imp__eq,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat] :
% 9.00/5.07 ( ( ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.00/5.07 => ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5390_power2__eq__imp__eq,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( X2 = Y3 ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5391_zero__le__power2,axiom,
% 9.00/5.07 ! [A2: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5392_zero__le__power2,axiom,
% 9.00/5.07 ! [A2: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5393_zero__le__power2,axiom,
% 9.00/5.07 ! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5394_power2__less__0,axiom,
% 9.00/5.07 ! [A2: real] :
% 9.00/5.07 ~ ( ord_less_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5395_power2__less__0,axiom,
% 9.00/5.07 ! [A2: rat] :
% 9.00/5.07 ~ ( ord_less_rat @ ( power_power_rat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5396_power2__less__0,axiom,
% 9.00/5.07 ! [A2: int] :
% 9.00/5.07 ~ ( ord_less_int @ ( power_power_int @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5397_of__nat__less__two__power,axiom,
% 9.00/5.07 ! [N: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5398_of__nat__less__two__power,axiom,
% 9.00/5.07 ! [N: nat] : ( ord_le6747313008572928689nteger @ ( semiri4939895301339042750nteger @ N ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5399_of__nat__less__two__power,axiom,
% 9.00/5.07 ! [N: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5400_of__nat__less__two__power,axiom,
% 9.00/5.07 ! [N: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5401_exp__add__not__zero__imp__right,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_zero_nat )
% 9.00/5.07 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5402_exp__add__not__zero__imp__right,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5403_exp__add__not__zero__imp__right,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5404_exp__add__not__zero__imp__left,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_zero_nat )
% 9.00/5.07 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.00/5.07 != zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5405_exp__add__not__zero__imp__left,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.00/5.07 != zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5406_exp__add__not__zero__imp__left,axiom,
% 9.00/5.07 ! [M: nat,N: nat] :
% 9.00/5.07 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 9.00/5.07 != zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5407_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_zero_nat )
% 9.00/5.07 => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.00/5.07 != zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5408_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_zero_int )
% 9.00/5.07 => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.00/5.07 != zero_zero_int ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5409_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.00/5.07 ! [N: nat,M: nat] :
% 9.00/5.07 ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
% 9.00/5.07 != zero_z3403309356797280102nteger )
% 9.00/5.07 => ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M ) )
% 9.00/5.07 != zero_z3403309356797280102nteger ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5410_abs__le__square__iff,axiom,
% 9.00/5.07 ! [X2: code_integer,Y3: code_integer] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 9.00/5.07 = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5411_abs__le__square__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5412_abs__le__square__iff,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ ( abs_abs_rat @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5413_abs__le__square__iff,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ ( abs_abs_int @ Y3 ) )
% 9.00/5.07 = ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5414_less__2__cases,axiom,
% 9.00/5.07 ! [N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 => ( ( N = zero_zero_nat )
% 9.00/5.07 | ( N
% 9.00/5.07 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5415_less__2__cases__iff,axiom,
% 9.00/5.07 ! [N: nat] :
% 9.00/5.07 ( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ( N = zero_zero_nat )
% 9.00/5.07 | ( N
% 9.00/5.07 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5416_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_Pr1261947904930325089at_nat] :
% 9.00/5.07 ( ( ( finite711546835091564841at_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5417_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_complex] :
% 9.00/5.07 ( ( ( finite_card_complex @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: complex,Y: complex] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_complex @ X @ ( insert_complex @ Y @ bot_bot_set_complex ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5418_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_list_nat] :
% 9.00/5.07 ( ( ( finite_card_list_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: list_nat,Y: list_nat] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ bot_bot_set_list_nat ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5419_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_set_nat] :
% 9.00/5.07 ( ( ( finite_card_set_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: set_nat,Y: set_nat] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ bot_bot_set_set_nat ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5420_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_real] :
% 9.00/5.07 ( ( ( finite_card_real @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: real,Y: real] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_real @ X @ ( insert_real @ Y @ bot_bot_set_real ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5421_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_o] :
% 9.00/5.07 ( ( ( finite_card_o @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: $o,Y: $o] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_o @ X @ ( insert_o @ Y @ bot_bot_set_o ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5422_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_nat] :
% 9.00/5.07 ( ( ( finite_card_nat @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: nat,Y: nat] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_nat @ X @ ( insert_nat @ Y @ bot_bot_set_nat ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5423_card__2__iff,axiom,
% 9.00/5.07 ! [S: set_int] :
% 9.00/5.07 ( ( ( finite_card_int @ S )
% 9.00/5.07 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.07 = ( ? [X: int,Y: int] :
% 9.00/5.07 ( ( S
% 9.00/5.07 = ( insert_int @ X @ ( insert_int @ Y @ bot_bot_set_int ) ) )
% 9.00/5.07 & ( X != Y ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5424_nat__induct2,axiom,
% 9.00/5.07 ! [P: nat > $o,N: nat] :
% 9.00/5.07 ( ( P @ zero_zero_nat )
% 9.00/5.07 => ( ( P @ one_one_nat )
% 9.00/5.07 => ( ! [N3: nat] :
% 9.00/5.07 ( ( P @ N3 )
% 9.00/5.07 => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.07 => ( P @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5425_diff__le__diff__pow,axiom,
% 9.00/5.07 ! [K: nat,M: nat,N: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.00/5.07 => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5426_realpow__square__minus__le,axiom,
% 9.00/5.07 ! [U: real,X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5427_numeral__1__eq__Suc__0,axiom,
% 9.00/5.07 ( ( numeral_numeral_nat @ one )
% 9.00/5.07 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5428_ex__le__of__int,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5429_ex__le__of__int,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5430_ex__less__of__int,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_real @ X2 @ ( ring_1_of_int_real @ Z4 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5431_ex__less__of__int,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ Z4 ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5432_ex__of__int__less,axiom,
% 9.00/5.07 ! [X2: real] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z4 ) @ X2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5433_ex__of__int__less,axiom,
% 9.00/5.07 ! [X2: rat] :
% 9.00/5.07 ? [Z4: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5434_mod__double__modulus,axiom,
% 9.00/5.07 ! [M: code_integer,X2: code_integer] :
% 9.00/5.07 ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 9.00/5.07 => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X2 )
% 9.00/5.07 => ( ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( modulo364778990260209775nteger @ X2 @ M ) )
% 9.00/5.07 | ( ( modulo364778990260209775nteger @ X2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X2 @ M ) @ M ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5435_mod__double__modulus,axiom,
% 9.00/5.07 ! [M: nat,X2: nat] :
% 9.00/5.07 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ X2 )
% 9.00/5.07 => ( ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( modulo_modulo_nat @ X2 @ M ) )
% 9.00/5.07 | ( ( modulo_modulo_nat @ X2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( plus_plus_nat @ ( modulo_modulo_nat @ X2 @ M ) @ M ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5436_mod__double__modulus,axiom,
% 9.00/5.07 ! [M: int,X2: int] :
% 9.00/5.07 ( ( ord_less_int @ zero_zero_int @ M )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.07 => ( ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( modulo_modulo_int @ X2 @ M ) )
% 9.00/5.07 | ( ( modulo_modulo_int @ X2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.07 = ( plus_plus_int @ ( modulo_modulo_int @ X2 @ M ) @ M ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5437_divmod__digit__1_I2_J,axiom,
% 9.00/5.07 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.07 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.07 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.07 => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 = ( modulo364778990260209775nteger @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5438_divmod__digit__1_I2_J,axiom,
% 9.00/5.07 ! [A2: nat,B3: nat] :
% 9.00/5.07 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.07 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 = ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5439_divmod__digit__1_I2_J,axiom,
% 9.00/5.07 ! [A2: int,B3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.07 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.07 => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.07 => ( ( minus_minus_int @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ B3 )
% 9.00/5.07 = ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5440_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_zero_rat
% 9.00/5.07 != ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5441_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_zero_nat
% 9.00/5.07 != ( numeral_numeral_nat @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5442_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_zero_real
% 9.00/5.07 != ( numeral_numeral_real @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5443_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_zero_int
% 9.00/5.07 != ( numeral_numeral_int @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5444_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_z5237406670263579293d_enat
% 9.00/5.07 != ( numera1916890842035813515d_enat @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5445_zero__neq__numeral,axiom,
% 9.00/5.07 ! [N: num] :
% 9.00/5.07 ( zero_z3403309356797280102nteger
% 9.00/5.07 != ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5446_power2__less__imp__less,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.07 => ( ord_less_real @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5447_power2__less__imp__less,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.07 => ( ord_less_rat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5448_power2__less__imp__less,axiom,
% 9.00/5.07 ! [X2: nat,Y3: nat] :
% 9.00/5.07 ( ( ord_less_nat @ ( power_power_nat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.00/5.07 => ( ord_less_nat @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5449_power2__less__imp__less,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.07 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.07 => ( ord_less_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5450_sum__power2__ge__zero,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5451_sum__power2__ge__zero,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5452_sum__power2__ge__zero,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5453_sum__power2__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: real,Y3: real] :
% 9.00/5.07 ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 9.00/5.07 = ( ( X2 = zero_zero_real )
% 9.00/5.07 & ( Y3 = zero_zero_real ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5454_sum__power2__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: rat,Y3: rat] :
% 9.00/5.07 ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 9.00/5.07 = ( ( X2 = zero_zero_rat )
% 9.00/5.07 & ( Y3 = zero_zero_rat ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5455_sum__power2__le__zero__iff,axiom,
% 9.00/5.07 ! [X2: int,Y3: int] :
% 9.00/5.07 ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 9.00/5.07 = ( ( X2 = zero_zero_int )
% 9.00/5.07 & ( Y3 = zero_zero_int ) ) ) ).
% 9.00/5.07
% 9.00/5.07 thf(fact_5456_sum__power2__gt__zero__iff,axiom,
% 9.00/5.08 ! [X2: real,Y3: real] :
% 9.00/5.08 ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 = ( ( X2 != zero_zero_real )
% 9.00/5.08 | ( Y3 != zero_zero_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5457_sum__power2__gt__zero__iff,axiom,
% 9.00/5.08 ! [X2: rat,Y3: rat] :
% 9.00/5.08 ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 = ( ( X2 != zero_zero_rat )
% 9.00/5.08 | ( Y3 != zero_zero_rat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5458_sum__power2__gt__zero__iff,axiom,
% 9.00/5.08 ! [X2: int,Y3: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 = ( ( X2 != zero_zero_int )
% 9.00/5.08 | ( Y3 != zero_zero_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5459_not__sum__power2__lt__zero,axiom,
% 9.00/5.08 ! [X2: real,Y3: real] :
% 9.00/5.08 ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5460_not__sum__power2__lt__zero,axiom,
% 9.00/5.08 ! [X2: rat,Y3: rat] :
% 9.00/5.08 ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5461_not__sum__power2__lt__zero,axiom,
% 9.00/5.08 ! [X2: int,Y3: int] :
% 9.00/5.08 ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5462_num_Osize_I4_J,axiom,
% 9.00/5.08 ( ( size_size_num @ one )
% 9.00/5.08 = zero_zero_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5463_square__le__1,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5464_square__le__1,axiom,
% 9.00/5.08 ! [X2: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X2 )
% 9.00/5.08 => ( ( ord_le3102999989581377725nteger @ X2 @ one_one_Code_integer )
% 9.00/5.08 => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5465_square__le__1,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_rat @ X2 @ one_one_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5466_square__le__1,axiom,
% 9.00/5.08 ! [X2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_int @ X2 @ one_one_int )
% 9.00/5.08 => ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5467_power2__le__iff__abs__le,axiom,
% 9.00/5.08 ! [Y3: code_integer,X2: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 9.00/5.08 => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5468_power2__le__iff__abs__le,axiom,
% 9.00/5.08 ! [Y3: real,X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.08 => ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5469_power2__le__iff__abs__le,axiom,
% 9.00/5.08 ! [Y3: rat,X2: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.08 => ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5470_power2__le__iff__abs__le,axiom,
% 9.00/5.08 ! [Y3: int,X2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.08 => ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5471_zero__le__even__power_H,axiom,
% 9.00/5.08 ! [A2: real,N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5472_zero__le__even__power_H,axiom,
% 9.00/5.08 ! [A2: rat,N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5473_zero__le__even__power_H,axiom,
% 9.00/5.08 ! [A2: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5474_abs__square__le__1,axiom,
% 9.00/5.08 ! [X2: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.00/5.08 = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5475_abs__square__le__1,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.00/5.08 = ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5476_abs__square__le__1,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.00/5.08 = ( ord_less_eq_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5477_abs__square__le__1,axiom,
% 9.00/5.08 ! [X2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.00/5.08 = ( ord_less_eq_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5478_abs__square__less__1,axiom,
% 9.00/5.08 ! [X2: code_integer] :
% 9.00/5.08 ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.00/5.08 = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X2 ) @ one_one_Code_integer ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5479_abs__square__less__1,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.00/5.08 = ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5480_abs__square__less__1,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.00/5.08 = ( ord_less_rat @ ( abs_abs_rat @ X2 ) @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5481_abs__square__less__1,axiom,
% 9.00/5.08 ! [X2: int] :
% 9.00/5.08 ( ( ord_less_int @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.00/5.08 = ( ord_less_int @ ( abs_abs_int @ X2 ) @ one_one_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5482_nat__bit__induct,axiom,
% 9.00/5.08 ! [P: nat > $o,N: nat] :
% 9.00/5.08 ( ( P @ zero_zero_nat )
% 9.00/5.08 => ( ! [N3: nat] :
% 9.00/5.08 ( ( P @ N3 )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.08 => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.00/5.08 => ( ! [N3: nat] :
% 9.00/5.08 ( ( P @ N3 )
% 9.00/5.08 => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.00/5.08 => ( P @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5483_div__2__gt__zero,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.08 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5484_Suc__n__div__2__gt__zero,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5485_arith__geo__mean,axiom,
% 9.00/5.08 ! [U: real,X2: real,Y3: real] :
% 9.00/5.08 ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 = ( times_times_real @ X2 @ Y3 ) )
% 9.00/5.08 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.08 => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5486_arith__geo__mean,axiom,
% 9.00/5.08 ! [U: rat,X2: rat,Y3: rat] :
% 9.00/5.08 ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 = ( times_times_rat @ X2 @ Y3 ) )
% 9.00/5.08 => ( ( ord_less_eq_rat @ zero_zero_rat @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.00/5.08 => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5487_divmod__digit__1_I1_J,axiom,
% 9.00/5.08 ! [A2: code_integer,B3: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A2 )
% 9.00/5.08 => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B3 )
% 9.00/5.08 => ( ( ord_le3102999989581377725nteger @ B3 @ ( modulo364778990260209775nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.08 => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_Code_integer )
% 9.00/5.08 = ( divide6298287555418463151nteger @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5488_divmod__digit__1_I1_J,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.08 => ( ( ord_less_eq_nat @ B3 @ ( modulo_modulo_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.08 => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_nat )
% 9.00/5.08 = ( divide_divide_nat @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5489_divmod__digit__1_I1_J,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.08 => ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ( ord_less_eq_int @ B3 @ ( modulo_modulo_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) )
% 9.00/5.08 => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) ) @ one_one_int )
% 9.00/5.08 = ( divide_divide_int @ A2 @ B3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5490_odd__0__le__power__imp__0__le,axiom,
% 9.00/5.08 ! [A2: real,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.08 => ( ord_less_eq_real @ zero_zero_real @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5491_odd__0__le__power__imp__0__le,axiom,
% 9.00/5.08 ! [A2: rat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.08 => ( ord_less_eq_rat @ zero_zero_rat @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5492_odd__0__le__power__imp__0__le,axiom,
% 9.00/5.08 ! [A2: int,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.08 => ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5493_odd__power__less__zero,axiom,
% 9.00/5.08 ! [A2: real,N: nat] :
% 9.00/5.08 ( ( ord_less_real @ A2 @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ ( power_power_real @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5494_odd__power__less__zero,axiom,
% 9.00/5.08 ! [A2: rat,N: nat] :
% 9.00/5.08 ( ( ord_less_rat @ A2 @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ ( power_power_rat @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5495_odd__power__less__zero,axiom,
% 9.00/5.08 ! [A2: int,N: nat] :
% 9.00/5.08 ( ( ord_less_int @ A2 @ zero_zero_int )
% 9.00/5.08 => ( ord_less_int @ ( power_power_int @ A2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ zero_zero_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5496_ex__power__ivl2,axiom,
% 9.00/5.08 ! [B3: nat,K: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.00/5.08 => ? [N3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
% 9.00/5.08 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5497_ex__power__ivl1,axiom,
% 9.00/5.08 ! [B3: nat,K: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.08 => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 9.00/5.08 => ? [N3: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N3 ) @ K )
% 9.00/5.08 & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5498_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.08 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5499_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.08 => ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5500_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.00/5.08 ! [B3: int,A2: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5501_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.00/5.08 ! [B3: nat,A2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.08 => ( ord_less_nat @ ( modulo_modulo_nat @ A2 @ B3 ) @ B3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5502_mod__eq__self__iff__div__eq__0,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ( modulo_modulo_int @ A2 @ B3 )
% 9.00/5.08 = A2 )
% 9.00/5.08 = ( ( divide_divide_int @ A2 @ B3 )
% 9.00/5.08 = zero_zero_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5503_mod__eq__self__iff__div__eq__0,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ A2 @ B3 )
% 9.00/5.08 = A2 )
% 9.00/5.08 = ( ( divide_divide_nat @ A2 @ B3 )
% 9.00/5.08 = zero_zero_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5504_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5505_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5506_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5507_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5508_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5509_zero__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5510_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5511_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5512_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5513_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5514_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5515_not__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5516_exp__bound,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5517_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5518_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5519_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5520_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5521_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5522_not__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5523_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5524_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5525_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5526_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5527_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5528_zero__less__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5529_le__of__int__ceiling,axiom,
% 9.00/5.08 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5530_le__of__int__ceiling,axiom,
% 9.00/5.08 ! [X2: rat] : ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5531_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5532_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5533_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5534_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5535_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5536_one__le__numeral,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5537_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5538_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5539_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5540_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5541_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5542_not__numeral__less__one,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5543_neg__numeral__le__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5544_neg__numeral__le__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5545_neg__numeral__le__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5546_neg__numeral__le__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5547_not__numeral__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5548_not__numeral__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5549_not__numeral__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5550_not__numeral__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5551_zero__neq__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ( zero_zero_int
% 9.00/5.08 != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5552_zero__neq__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ( zero_zero_real
% 9.00/5.08 != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5553_zero__neq__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ( zero_zero_rat
% 9.00/5.08 != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5554_zero__neq__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ( zero_z3403309356797280102nteger
% 9.00/5.08 != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5555_zero__neq__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ( zero_zero_complex
% 9.00/5.08 != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5556_zmod__le__nonneg__dividend,axiom,
% 9.00/5.08 ! [M: int,K: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.00/5.08 => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5557_not__numeral__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5558_not__numeral__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5559_not__numeral__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5560_not__numeral__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5561_neg__numeral__less__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5562_neg__numeral__less__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5563_neg__numeral__less__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5564_neg__numeral__less__numeral,axiom,
% 9.00/5.08 ! [M: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5565_Euclidean__Division_Opos__mod__bound,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.08 => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5566_neg__mod__bound,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_int @ L @ zero_zero_int )
% 9.00/5.08 => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5567_ln__one__plus__pos__lower__bound,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5568_invar__vebt_Ointros_I2_J,axiom,
% 9.00/5.08 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.00/5.08 ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.00/5.08 => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.00/5.08 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( M = N )
% 9.00/5.08 => ( ( Deg
% 9.00/5.08 = ( plus_plus_nat @ N @ M ) )
% 9.00/5.08 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.00/5.08 => ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5569_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5570_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.00/5.08 ! [B3: nat,A2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ B3 )
% 9.00/5.08 => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5571_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.00/5.08 ! [B3: int,A2: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5572_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ zero_zero_nat @ A2 )
% 9.00/5.08 => ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.08 => ( ( modulo_modulo_nat @ A2 @ B3 )
% 9.00/5.08 = A2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5573_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.08 => ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.08 => ( ( modulo_modulo_int @ A2 @ B3 )
% 9.00/5.08 = A2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5574_invar__vebt_Ointros_I3_J,axiom,
% 9.00/5.08 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.00/5.08 ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.00/5.08 => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.00/5.08 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( M
% 9.00/5.08 = ( suc @ N ) )
% 9.00/5.08 => ( ( Deg
% 9.00/5.08 = ( plus_plus_nat @ N @ M ) )
% 9.00/5.08 => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.00/5.08 => ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5575_neg__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5576_neg__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5577_neg__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5578_neg__numeral__le__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5579_not__zero__le__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5580_not__zero__le__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5581_not__zero__le__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5582_not__zero__le__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5583_not__zero__less__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5584_not__zero__less__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5585_not__zero__less__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5586_not__zero__less__neg__numeral,axiom,
% 9.00/5.08 ! [N: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5587_neg__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5588_neg__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) @ zero_zero_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5589_neg__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) @ zero_zero_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5590_neg__numeral__less__zero,axiom,
% 9.00/5.08 ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5591_ceiling__le,axiom,
% 9.00/5.08 ! [X2: real,A2: int] :
% 9.00/5.08 ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) )
% 9.00/5.08 => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5592_ceiling__le,axiom,
% 9.00/5.08 ! [X2: rat,A2: int] :
% 9.00/5.08 ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) )
% 9.00/5.08 => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ A2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5593_ceiling__le__iff,axiom,
% 9.00/5.08 ! [X2: real,Z: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5594_ceiling__le__iff,axiom,
% 9.00/5.08 ! [X2: rat,Z: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5595_eq__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ( numeral_numeral_rat @ W2 )
% 9.00/5.08 = ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_rat )
% 9.00/5.08 => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_rat )
% 9.00/5.08 => ( ( numeral_numeral_rat @ W2 )
% 9.00/5.08 = zero_zero_rat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5596_eq__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ( numeral_numeral_real @ W2 )
% 9.00/5.08 = ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_real )
% 9.00/5.08 => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_real )
% 9.00/5.08 => ( ( numeral_numeral_real @ W2 )
% 9.00/5.08 = zero_zero_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5597_eq__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: complex,C: complex] :
% 9.00/5.08 ( ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.08 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_complex )
% 9.00/5.08 => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_complex )
% 9.00/5.08 => ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.08 = zero_zero_complex ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5598_divide__eq__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ( divide_divide_rat @ B3 @ C )
% 9.00/5.08 = ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = ( ( ( C != zero_zero_rat )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_rat )
% 9.00/5.08 => ( ( numeral_numeral_rat @ W2 )
% 9.00/5.08 = zero_zero_rat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5599_divide__eq__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ( divide_divide_real @ B3 @ C )
% 9.00/5.08 = ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = ( ( ( C != zero_zero_real )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_real )
% 9.00/5.08 => ( ( numeral_numeral_real @ W2 )
% 9.00/5.08 = zero_zero_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5600_divide__eq__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: complex,C: complex,W2: num] :
% 9.00/5.08 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 9.00/5.08 = ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.08 = ( ( ( C != zero_zero_complex )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_complex )
% 9.00/5.08 => ( ( numera6690914467698888265omplex @ W2 )
% 9.00/5.08 = zero_zero_complex ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5601_less__ceiling__iff,axiom,
% 9.00/5.08 ! [Z: int,X2: rat] :
% 9.00/5.08 ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.08 = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5602_less__ceiling__iff,axiom,
% 9.00/5.08 ! [Z: int,X2: real] :
% 9.00/5.08 ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.08 = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5603_not__one__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5604_not__one__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5605_not__one__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5606_not__one__le__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5607_not__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5608_not__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5609_not__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5610_not__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5611_neg__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5612_neg__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5613_neg__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5614_neg__numeral__le__neg__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5615_neg__one__le__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5616_neg__one__le__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5617_neg__one__le__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5618_neg__one__le__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5619_neg__numeral__le__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5620_neg__numeral__le__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5621_neg__numeral__le__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5622_neg__numeral__le__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5623_neg__numeral__less__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5624_neg__numeral__less__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5625_neg__numeral__less__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5626_neg__numeral__less__one,axiom,
% 9.00/5.08 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5627_neg__one__less__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5628_neg__one__less__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5629_neg__one__less__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5630_neg__one__less__numeral,axiom,
% 9.00/5.08 ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5631_not__numeral__less__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5632_not__numeral__less__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5633_not__numeral__less__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5634_not__numeral__less__neg__one,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5635_not__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5636_not__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5637_not__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5638_not__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5639_not__neg__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5640_not__neg__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5641_not__neg__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5642_not__neg__one__less__neg__numeral,axiom,
% 9.00/5.08 ! [M: num] :
% 9.00/5.08 ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5643_neg__mod__conj,axiom,
% 9.00/5.08 ! [B3: int,A2: int] :
% 9.00/5.08 ( ( ord_less_int @ B3 @ zero_zero_int )
% 9.00/5.08 => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A2 @ B3 ) @ zero_zero_int )
% 9.00/5.08 & ( ord_less_int @ B3 @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5644_pos__mod__conj,axiom,
% 9.00/5.08 ! [B3: int,A2: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A2 @ B3 ) )
% 9.00/5.08 & ( ord_less_int @ ( modulo_modulo_int @ A2 @ B3 ) @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5645_zmod__trivial__iff,axiom,
% 9.00/5.08 ! [I: int,K: int] :
% 9.00/5.08 ( ( ( modulo_modulo_int @ I @ K )
% 9.00/5.08 = I )
% 9.00/5.08 = ( ( K = zero_zero_int )
% 9.00/5.08 | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.00/5.08 & ( ord_less_int @ I @ K ) )
% 9.00/5.08 | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.00/5.08 & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5646_neg__mod__sign,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_int @ L @ zero_zero_int )
% 9.00/5.08 => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5647_Euclidean__Division_Opos__mod__sign,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.08 => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5648_real__of__int__div4,axiom,
% 9.00/5.08 ! [N: int,X2: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5649_zdiv__mono__strict,axiom,
% 9.00/5.08 ! [A3: int,B2: int,N: int] :
% 9.00/5.08 ( ( ord_less_int @ A3 @ B2 )
% 9.00/5.08 => ( ( ord_less_int @ zero_zero_int @ N )
% 9.00/5.08 => ( ( ( modulo_modulo_int @ A3 @ N )
% 9.00/5.08 = zero_zero_int )
% 9.00/5.08 => ( ( ( modulo_modulo_int @ B2 @ N )
% 9.00/5.08 = zero_zero_int )
% 9.00/5.08 => ( ord_less_int @ ( divide_divide_int @ A3 @ N ) @ ( divide_divide_int @ B2 @ N ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5650_abs__mod__less,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( L != zero_zero_int )
% 9.00/5.08 => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5651_num_Osize_I5_J,axiom,
% 9.00/5.08 ! [X22: num] :
% 9.00/5.08 ( ( size_size_num @ ( bit0 @ X22 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5652_of__int__nonneg,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5653_of__int__nonneg,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5654_of__int__nonneg,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5655_of__int__leD,axiom,
% 9.00/5.08 ! [N: int,X2: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5656_of__int__leD,axiom,
% 9.00/5.08 ! [N: int,X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_eq_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5657_of__int__leD,axiom,
% 9.00/5.08 ! [N: int,X2: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_eq_rat @ one_one_rat @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5658_of__int__leD,axiom,
% 9.00/5.08 ! [N: int,X2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_eq_int @ one_one_int @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5659_of__int__pos,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5660_of__int__pos,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5661_of__int__pos,axiom,
% 9.00/5.08 ! [Z: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.08 => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5662_of__int__lessD,axiom,
% 9.00/5.08 ! [N: int,X2: code_integer] :
% 9.00/5.08 ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5663_of__int__lessD,axiom,
% 9.00/5.08 ! [N: int,X2: real] :
% 9.00/5.08 ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_real @ one_one_real @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5664_of__int__lessD,axiom,
% 9.00/5.08 ! [N: int,X2: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_rat @ one_one_rat @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5665_of__int__lessD,axiom,
% 9.00/5.08 ! [N: int,X2: int] :
% 9.00/5.08 ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X2 )
% 9.00/5.08 => ( ( N = zero_zero_int )
% 9.00/5.08 | ( ord_less_int @ one_one_int @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5666_floor__exists1,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ? [X5: int] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X2 )
% 9.00/5.08 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 9.00/5.08 & ! [Y5: int] :
% 9.00/5.08 ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y5 ) @ X2 )
% 9.00/5.08 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 9.00/5.08 => ( Y5 = X5 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5667_floor__exists1,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ? [X5: int] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X2 )
% 9.00/5.08 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
% 9.00/5.08 & ! [Y5: int] :
% 9.00/5.08 ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y5 ) @ X2 )
% 9.00/5.08 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y5 @ one_one_int ) ) ) )
% 9.00/5.08 => ( Y5 = X5 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5668_floor__exists,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ? [Z4: int] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z4 ) @ X2 )
% 9.00/5.08 & ( ord_less_real @ X2 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5669_floor__exists,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ? [Z4: int] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z4 ) @ X2 )
% 9.00/5.08 & ( ord_less_rat @ X2 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z4 @ one_one_int ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5670_of__int__ceiling__le__add__one,axiom,
% 9.00/5.08 ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5671_of__int__ceiling__le__add__one,axiom,
% 9.00/5.08 ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5672_of__int__ceiling__diff__one__le,axiom,
% 9.00/5.08 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5673_of__int__ceiling__diff__one__le,axiom,
% 9.00/5.08 ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5674_of__nat__less__of__int__iff,axiom,
% 9.00/5.08 ! [N: nat,X2: int] :
% 9.00/5.08 ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N ) @ ( ring_1_of_int_rat @ X2 ) )
% 9.00/5.08 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5675_of__nat__less__of__int__iff,axiom,
% 9.00/5.08 ! [N: nat,X2: int] :
% 9.00/5.08 ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X2 ) )
% 9.00/5.08 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5676_of__nat__less__of__int__iff,axiom,
% 9.00/5.08 ! [N: nat,X2: int] :
% 9.00/5.08 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( ring_1_of_int_real @ X2 ) )
% 9.00/5.08 = ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5677_ceiling__log__nat__eq__if,axiom,
% 9.00/5.08 ! [B3: nat,N: nat,K: nat] :
% 9.00/5.08 ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 9.00/5.08 => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.08 => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.00/5.08 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5678_less__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5679_less__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5680_divide__less__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5681_divide__less__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5682_eq__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_real )
% 9.00/5.08 => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_real )
% 9.00/5.08 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = zero_zero_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5683_eq__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_rat )
% 9.00/5.08 => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_rat )
% 9.00/5.08 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = zero_zero_rat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5684_eq__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: complex,C: complex] :
% 9.00/5.08 ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.08 = ( divide1717551699836669952omplex @ B3 @ C ) )
% 9.00/5.08 = ( ( ( C != zero_zero_complex )
% 9.00/5.08 => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 9.00/5.08 = B3 ) )
% 9.00/5.08 & ( ( C = zero_zero_complex )
% 9.00/5.08 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.08 = zero_zero_complex ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5685_divide__eq__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ( divide_divide_real @ B3 @ C )
% 9.00/5.08 = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.08 = ( ( ( C != zero_zero_real )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_real )
% 9.00/5.08 => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = zero_zero_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5686_divide__eq__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ( divide_divide_rat @ B3 @ C )
% 9.00/5.08 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.08 = ( ( ( C != zero_zero_rat )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_rat )
% 9.00/5.08 => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = zero_zero_rat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5687_divide__eq__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: complex,C: complex,W2: num] :
% 9.00/5.08 ( ( ( divide1717551699836669952omplex @ B3 @ C )
% 9.00/5.08 = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.00/5.08 = ( ( ( C != zero_zero_complex )
% 9.00/5.08 => ( B3
% 9.00/5.08 = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ( C = zero_zero_complex )
% 9.00/5.08 => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.00/5.08 = zero_zero_complex ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5688_int__le__real__less,axiom,
% 9.00/5.08 ( ord_less_eq_int
% 9.00/5.08 = ( ^ [N2: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5689_int__less__real__le,axiom,
% 9.00/5.08 ( ord_less_int
% 9.00/5.08 = ( ^ [N2: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5690_mod__pos__neg__trivial,axiom,
% 9.00/5.08 ! [K: int,L: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.08 => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.00/5.08 => ( ( modulo_modulo_int @ K @ L )
% 9.00/5.08 = ( plus_plus_int @ K @ L ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5691_mod__pos__geq,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ L )
% 9.00/5.08 => ( ( ord_less_eq_int @ L @ K )
% 9.00/5.08 => ( ( modulo_modulo_int @ K @ L )
% 9.00/5.08 = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5692_ceiling__log__nat__eq__powr__iff,axiom,
% 9.00/5.08 ! [B3: nat,K: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.08 => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.00/5.08 = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
% 9.00/5.08 = ( ( ord_less_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 9.00/5.08 & ( ord_less_eq_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5693_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.00/5.08 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.00/5.08 => ( ( modulo_modulo_nat @ A2 @ ( times_times_nat @ B3 @ C ) )
% 9.00/5.08 = ( plus_plus_nat @ ( times_times_nat @ B3 @ ( modulo_modulo_nat @ ( divide_divide_nat @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5694_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.00/5.08 ! [C: int,A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.08 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5695_ceiling__correct,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) @ one_one_real ) @ X2 )
% 9.00/5.08 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5696_ceiling__correct,axiom,
% 9.00/5.08 ! [X2: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) @ one_one_rat ) @ X2 )
% 9.00/5.08 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5697_ceiling__unique,axiom,
% 9.00/5.08 ! [Z: int,X2: real] :
% 9.00/5.08 ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ Z ) )
% 9.00/5.08 => ( ( archim7802044766580827645g_real @ X2 )
% 9.00/5.08 = Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5698_ceiling__unique,axiom,
% 9.00/5.08 ! [Z: int,X2: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ Z ) )
% 9.00/5.08 => ( ( archim2889992004027027881ng_rat @ X2 )
% 9.00/5.08 = Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5699_ceiling__eq__iff,axiom,
% 9.00/5.08 ! [X2: real,A2: int] :
% 9.00/5.08 ( ( ( archim7802044766580827645g_real @ X2 )
% 9.00/5.08 = A2 )
% 9.00/5.08 = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A2 ) @ one_one_real ) @ X2 )
% 9.00/5.08 & ( ord_less_eq_real @ X2 @ ( ring_1_of_int_real @ A2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5700_ceiling__eq__iff,axiom,
% 9.00/5.08 ! [X2: rat,A2: int] :
% 9.00/5.08 ( ( ( archim2889992004027027881ng_rat @ X2 )
% 9.00/5.08 = A2 )
% 9.00/5.08 = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A2 ) @ one_one_rat ) @ X2 )
% 9.00/5.08 & ( ord_less_eq_rat @ X2 @ ( ring_1_of_int_rat @ A2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5701_ceiling__split,axiom,
% 9.00/5.08 ! [P: int > $o,T: real] :
% 9.00/5.08 ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 9.00/5.08 = ( ! [I4: int] :
% 9.00/5.08 ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
% 9.00/5.08 & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
% 9.00/5.08 => ( P @ I4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5702_ceiling__split,axiom,
% 9.00/5.08 ! [P: int > $o,T: rat] :
% 9.00/5.08 ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 9.00/5.08 = ( ! [I4: int] :
% 9.00/5.08 ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
% 9.00/5.08 & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
% 9.00/5.08 => ( P @ I4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5703_ceiling__less__iff,axiom,
% 9.00/5.08 ! [X2: real,Z: int] :
% 9.00/5.08 ( ( ord_less_int @ ( archim7802044766580827645g_real @ X2 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_real @ X2 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5704_ceiling__less__iff,axiom,
% 9.00/5.08 ! [X2: rat,Z: int] :
% 9.00/5.08 ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X2 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_rat @ X2 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5705_le__ceiling__iff,axiom,
% 9.00/5.08 ! [Z: int,X2: rat] :
% 9.00/5.08 ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X2 ) )
% 9.00/5.08 = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5706_le__ceiling__iff,axiom,
% 9.00/5.08 ! [Z: int,X2: real] :
% 9.00/5.08 ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X2 ) )
% 9.00/5.08 = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5707_divide__le__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5708_divide__le__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5709_le__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5710_le__divide__eq__numeral_I1_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5711_divide__less__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ord_less_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5712_divide__less__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ord_less_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5713_less__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5714_less__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5715_real__of__int__div2,axiom,
% 9.00/5.08 ! [N: int,X2: 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 @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5716_split__zmod,axiom,
% 9.00/5.08 ! [P: int > $o,N: int,K: int] :
% 9.00/5.08 ( ( P @ ( modulo_modulo_int @ N @ K ) )
% 9.00/5.08 = ( ( ( K = zero_zero_int )
% 9.00/5.08 => ( P @ N ) )
% 9.00/5.08 & ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.08 => ! [I4: int,J3: int] :
% 9.00/5.08 ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.00/5.08 & ( ord_less_int @ J3 @ K )
% 9.00/5.08 & ( N
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.08 => ( P @ J3 ) ) )
% 9.00/5.08 & ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.08 => ! [I4: int,J3: int] :
% 9.00/5.08 ( ( ( ord_less_int @ K @ J3 )
% 9.00/5.08 & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.00/5.08 & ( N
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ K @ I4 ) @ J3 ) ) )
% 9.00/5.08 => ( P @ J3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5717_int__mod__neg__eq,axiom,
% 9.00/5.08 ! [A2: int,B3: int,Q3: int,R2: int] :
% 9.00/5.08 ( ( A2
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.00/5.08 => ( ( ord_less_int @ B3 @ R2 )
% 9.00/5.08 => ( ( modulo_modulo_int @ A2 @ B3 )
% 9.00/5.08 = R2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5718_int__mod__pos__eq,axiom,
% 9.00/5.08 ! [A2: int,B3: int,Q3: int,R2: int] :
% 9.00/5.08 ( ( A2
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ B3 @ Q3 ) @ R2 ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.00/5.08 => ( ( ord_less_int @ R2 @ B3 )
% 9.00/5.08 => ( ( modulo_modulo_int @ A2 @ B3 )
% 9.00/5.08 = R2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5719_real__of__int__div3,axiom,
% 9.00/5.08 ! [N: int,X2: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N ) @ ( ring_1_of_int_real @ X2 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N @ X2 ) ) ) @ one_one_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5720_minus__mod__int__eq,axiom,
% 9.00/5.08 ! [L: int,K: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.00/5.08 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
% 9.00/5.08 = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5721_zmod__minus1,axiom,
% 9.00/5.08 ! [B3: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B3 )
% 9.00/5.08 = ( minus_minus_int @ B3 @ one_one_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5722_zmod__zmult2__eq,axiom,
% 9.00/5.08 ! [C: int,A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.00/5.08 => ( ( modulo_modulo_int @ A2 @ ( times_times_int @ B3 @ C ) )
% 9.00/5.08 = ( plus_plus_int @ ( times_times_int @ B3 @ ( modulo_modulo_int @ ( divide_divide_int @ A2 @ B3 ) @ C ) ) @ ( modulo_modulo_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5723_ceiling__divide__upper,axiom,
% 9.00/5.08 ! [Q3: real,P6: real] :
% 9.00/5.08 ( ( ord_less_real @ zero_zero_real @ Q3 )
% 9.00/5.08 => ( ord_less_eq_real @ P6 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5724_ceiling__divide__upper,axiom,
% 9.00/5.08 ! [Q3: rat,P6: rat] :
% 9.00/5.08 ( ( ord_less_rat @ zero_zero_rat @ Q3 )
% 9.00/5.08 => ( ord_less_eq_rat @ P6 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P6 @ Q3 ) ) ) @ Q3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5725_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.08 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5726_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.08 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.08 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5727_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ A2 )
% 9.00/5.08 => ( ( member_real @ A2 @ ring_1_Ints_real )
% 9.00/5.08 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A2 ) @ ( archim7802044766580827645g_real @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5728_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.08 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.08 => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5729_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.08 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5730_mult__ceiling__le__Ints,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ A2 )
% 9.00/5.08 => ( ( member_rat @ A2 @ ring_1_Ints_rat )
% 9.00/5.08 => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A2 @ B3 ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A2 ) @ ( archim2889992004027027881ng_rat @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5731_divide__le__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: real,C: real,W2: num] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( divide_divide_real @ B3 @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5732_divide__le__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [B3: rat,C: rat,W2: num] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( divide_divide_rat @ B3 @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5733_le__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: real,C: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.08 => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ B3 @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5734_le__divide__eq__numeral_I2_J,axiom,
% 9.00/5.08 ! [W2: num,B3: rat,C: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B3 @ C ) )
% 9.00/5.08 = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B3 ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.00/5.08 => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ B3 @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.00/5.08 & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.00/5.08 => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5735_divmod__step__eq,axiom,
% 9.00/5.08 ! [L: num,R2: nat,Q3: nat] :
% 9.00/5.08 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.00/5.08 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
% 9.00/5.08 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.00/5.08 => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5736_divmod__step__eq,axiom,
% 9.00/5.08 ! [L: num,R2: int,Q3: int] :
% 9.00/5.08 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.00/5.08 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.08 = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
% 9.00/5.08 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.00/5.08 => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.08 = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5737_divmod__step__eq,axiom,
% 9.00/5.08 ! [L: num,R2: code_integer,Q3: code_integer] :
% 9.00/5.08 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.00/5.08 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 9.00/5.08 = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 9.00/5.08 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.00/5.08 => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
% 9.00/5.08 = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5738_post__member__pre__member,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X2 ) @ Y3 )
% 9.00/5.08 => ( ( vEBT_vebt_member @ T @ Y3 )
% 9.00/5.08 | ( X2 = Y3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5739_valid__insert__both__member__options__pres,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat,Y3: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( ord_less_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( vEBT_V8194947554948674370ptions @ T @ X2 )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y3 ) @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5740_valid__insert__both__member__options__add,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X2 ) @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5741_insert__simp__mima,axiom,
% 9.00/5.08 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( ( X2 = Mi )
% 9.00/5.08 | ( X2 = Ma ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.08 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5742_valid__pres__insert,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X2 ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5743_inrange,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ord_less_eq_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5744_mod__less,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ M @ N )
% 9.00/5.08 => ( ( modulo_modulo_nat @ M @ N )
% 9.00/5.08 = M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5745_mod__by__Suc__0,axiom,
% 9.00/5.08 ! [M: nat] :
% 9.00/5.08 ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.00/5.08 = zero_zero_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5746_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 9.00/5.08 ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 9.00/5.08 = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5747_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 9.00/5.08 ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 9.00/5.08 = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5748_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: num,H2: num,L3: num,H3: num] :
% 9.00/5.08 ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 9.00/5.08 = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5749_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 9.00/5.08 ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 9.00/5.08 = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5750_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: int,H2: int,L3: int,H3: int] :
% 9.00/5.08 ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 9.00/5.08 = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5751_Icc__eq__Icc,axiom,
% 9.00/5.08 ! [L: real,H2: real,L3: real,H3: real] :
% 9.00/5.08 ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 9.00/5.08 = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 9.00/5.08 = ( ( ( L = L3 )
% 9.00/5.08 & ( H2 = H3 ) )
% 9.00/5.08 | ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.00/5.08 & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5752_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: $o,L: $o,U: $o] :
% 9.00/5.08 ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_o @ L @ I )
% 9.00/5.08 & ( ord_less_eq_o @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5753_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: set_nat,L: set_nat,U: set_nat] :
% 9.00/5.08 ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_set_nat @ L @ I )
% 9.00/5.08 & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5754_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: set_int,L: set_int,U: set_int] :
% 9.00/5.08 ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_set_int @ L @ I )
% 9.00/5.08 & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5755_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: rat,L: rat,U: rat] :
% 9.00/5.08 ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_rat @ L @ I )
% 9.00/5.08 & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5756_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: num,L: num,U: num] :
% 9.00/5.08 ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_num @ L @ I )
% 9.00/5.08 & ( ord_less_eq_num @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5757_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: nat,L: nat,U: nat] :
% 9.00/5.08 ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_nat @ L @ I )
% 9.00/5.08 & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5758_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: int,L: int,U: int] :
% 9.00/5.08 ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_int @ L @ I )
% 9.00/5.08 & ( ord_less_eq_int @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5759_atLeastAtMost__iff,axiom,
% 9.00/5.08 ! [I: real,L: real,U: real] :
% 9.00/5.08 ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 9.00/5.08 = ( ( ord_less_eq_real @ L @ I )
% 9.00/5.08 & ( ord_less_eq_real @ I @ U ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5760_finite__atLeastAtMost,axiom,
% 9.00/5.08 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5761_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: $o,B3: $o] :
% 9.00/5.08 ( ( bot_bot_set_o
% 9.00/5.08 = ( set_or8904488021354931149Most_o @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5762_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: set_int,B3: set_int] :
% 9.00/5.08 ( ( bot_bot_set_set_int
% 9.00/5.08 = ( set_or370866239135849197et_int @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5763_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( bot_bot_set_rat
% 9.00/5.08 = ( set_or633870826150836451st_rat @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5764_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: num,B3: num] :
% 9.00/5.08 ( ( bot_bot_set_num
% 9.00/5.08 = ( set_or7049704709247886629st_num @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5765_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( bot_bot_set_nat
% 9.00/5.08 = ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5766_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( bot_bot_set_int
% 9.00/5.08 = ( set_or1266510415728281911st_int @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5767_atLeastatMost__empty__iff2,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( bot_bot_set_real
% 9.00/5.08 = ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5768_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: $o,B3: $o] :
% 9.00/5.08 ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ~ ( ord_less_eq_o @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5769_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: set_int,B3: set_int] :
% 9.00/5.08 ( ( ( set_or370866239135849197et_int @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_set_int )
% 9.00/5.08 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5770_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ( set_or633870826150836451st_rat @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_rat )
% 9.00/5.08 = ( ~ ( ord_less_eq_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5771_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: num,B3: num] :
% 9.00/5.08 ( ( ( set_or7049704709247886629st_num @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_num )
% 9.00/5.08 = ( ~ ( ord_less_eq_num @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5772_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ~ ( ord_less_eq_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5773_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ~ ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5774_atLeastatMost__empty__iff,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ~ ( ord_less_eq_real @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5775_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
% 9.00/5.08 ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_set_int @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_set_int @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5776_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.08 ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_rat @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5777_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: num,B3: num,C: num,D: num] :
% 9.00/5.08 ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_num @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_num @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5778_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_nat @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5779_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_int @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5780_atLeastatMost__subset__iff,axiom,
% 9.00/5.08 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.08 ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.00/5.08 = ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_real @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_real @ B3 @ D ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5781_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: $o,A2: $o] :
% 9.00/5.08 ( ( ord_less_o @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_o ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5782_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: rat,A2: rat] :
% 9.00/5.08 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or633870826150836451st_rat @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_rat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5783_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: num,A2: num] :
% 9.00/5.08 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or7049704709247886629st_num @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_num ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5784_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: nat,A2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5785_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: int,A2: int] :
% 9.00/5.08 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5786_atLeastatMost__empty,axiom,
% 9.00/5.08 ! [B3: real,A2: real] :
% 9.00/5.08 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.08 => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 9.00/5.08 = bot_bot_set_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5787_infinite__Icc__iff,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) )
% 9.00/5.08 = ( ord_less_rat @ A2 @ B3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5788_infinite__Icc__iff,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) )
% 9.00/5.08 = ( ord_less_real @ A2 @ B3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5789_atLeastAtMost__singleton,axiom,
% 9.00/5.08 ! [A2: $o] :
% 9.00/5.08 ( ( set_or8904488021354931149Most_o @ A2 @ A2 )
% 9.00/5.08 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5790_atLeastAtMost__singleton,axiom,
% 9.00/5.08 ! [A2: nat] :
% 9.00/5.08 ( ( set_or1269000886237332187st_nat @ A2 @ A2 )
% 9.00/5.08 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5791_atLeastAtMost__singleton,axiom,
% 9.00/5.08 ! [A2: int] :
% 9.00/5.08 ( ( set_or1266510415728281911st_int @ A2 @ A2 )
% 9.00/5.08 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5792_atLeastAtMost__singleton,axiom,
% 9.00/5.08 ! [A2: real] :
% 9.00/5.08 ( ( set_or1222579329274155063t_real @ A2 @ A2 )
% 9.00/5.08 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5793_atLeastAtMost__singleton__iff,axiom,
% 9.00/5.08 ! [A2: $o,B3: $o,C: $o] :
% 9.00/5.08 ( ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 9.00/5.08 = ( insert_o @ C @ bot_bot_set_o ) )
% 9.00/5.08 = ( ( A2 = B3 )
% 9.00/5.08 & ( B3 = C ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5794_atLeastAtMost__singleton__iff,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.08 ( ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 9.00/5.08 = ( insert_nat @ C @ bot_bot_set_nat ) )
% 9.00/5.08 = ( ( A2 = B3 )
% 9.00/5.08 & ( B3 = C ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5795_atLeastAtMost__singleton__iff,axiom,
% 9.00/5.08 ! [A2: int,B3: int,C: int] :
% 9.00/5.08 ( ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 9.00/5.08 = ( insert_int @ C @ bot_bot_set_int ) )
% 9.00/5.08 = ( ( A2 = B3 )
% 9.00/5.08 & ( B3 = C ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5796_atLeastAtMost__singleton__iff,axiom,
% 9.00/5.08 ! [A2: real,B3: real,C: real] :
% 9.00/5.08 ( ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 9.00/5.08 = ( insert_real @ C @ bot_bot_set_real ) )
% 9.00/5.08 = ( ( A2 = B3 )
% 9.00/5.08 & ( B3 = C ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5797_real__of__nat__less__numeral__iff,axiom,
% 9.00/5.08 ! [N: nat,W2: num] :
% 9.00/5.08 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( numeral_numeral_real @ W2 ) )
% 9.00/5.08 = ( ord_less_nat @ N @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5798_numeral__less__real__of__nat__iff,axiom,
% 9.00/5.08 ! [W2: num,N: nat] :
% 9.00/5.08 ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N ) )
% 9.00/5.08 = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5799_numeral__le__real__of__nat__iff,axiom,
% 9.00/5.08 ! [N: num,M: nat] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M ) )
% 9.00/5.08 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5800_card__atLeastAtMost,axiom,
% 9.00/5.08 ! [L: nat,U: nat] :
% 9.00/5.08 ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.00/5.08 = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5801_not__mod2__eq__Suc__0__eq__0,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 != ( suc @ zero_zero_nat ) )
% 9.00/5.08 = ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 = zero_zero_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5802_add__self__mod__2,axiom,
% 9.00/5.08 ! [M: nat] :
% 9.00/5.08 ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 = zero_zero_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5803_powr__numeral,axiom,
% 9.00/5.08 ! [X2: real,N: num] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( powr_real @ X2 @ ( numeral_numeral_real @ N ) )
% 9.00/5.08 = ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5804_mod2__gr__0,axiom,
% 9.00/5.08 ! [M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.08 = one_one_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5805_half__nonnegative__int__iff,axiom,
% 9.00/5.08 ! [K: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.00/5.08 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5806_half__negative__int__iff,axiom,
% 9.00/5.08 ! [K: int] :
% 9.00/5.08 ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.00/5.08 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5807_mod__less__eq__dividend,axiom,
% 9.00/5.08 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ M ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5808_infinite__Icc,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat] :
% 9.00/5.08 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.08 => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5809_infinite__Icc,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.08 => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5810_atLeastAtMost__singleton_H,axiom,
% 9.00/5.08 ! [A2: $o,B3: $o] :
% 9.00/5.08 ( ( A2 = B3 )
% 9.00/5.08 => ( ( set_or8904488021354931149Most_o @ A2 @ B3 )
% 9.00/5.08 = ( insert_o @ A2 @ bot_bot_set_o ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5811_atLeastAtMost__singleton_H,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat] :
% 9.00/5.08 ( ( A2 = B3 )
% 9.00/5.08 => ( ( set_or1269000886237332187st_nat @ A2 @ B3 )
% 9.00/5.08 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5812_atLeastAtMost__singleton_H,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( A2 = B3 )
% 9.00/5.08 => ( ( set_or1266510415728281911st_int @ A2 @ B3 )
% 9.00/5.08 = ( insert_int @ A2 @ bot_bot_set_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5813_atLeastAtMost__singleton_H,axiom,
% 9.00/5.08 ! [A2: real,B3: real] :
% 9.00/5.08 ( ( A2 = B3 )
% 9.00/5.08 => ( ( set_or1222579329274155063t_real @ A2 @ B3 )
% 9.00/5.08 = ( insert_real @ A2 @ bot_bot_set_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5814_mod__Suc,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.00/5.08 = N )
% 9.00/5.08 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.00/5.08 = zero_zero_nat ) )
% 9.00/5.08 & ( ( ( suc @ ( modulo_modulo_nat @ M @ N ) )
% 9.00/5.08 != N )
% 9.00/5.08 => ( ( modulo_modulo_nat @ ( suc @ M ) @ N )
% 9.00/5.08 = ( suc @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5815_all__nat__less,axiom,
% 9.00/5.08 ! [N: nat,P: nat > $o] :
% 9.00/5.08 ( ( ! [M2: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M2 @ N )
% 9.00/5.08 => ( P @ M2 ) ) )
% 9.00/5.08 = ( ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.08 => ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5816_ex__nat__less,axiom,
% 9.00/5.08 ! [N: nat,P: nat > $o] :
% 9.00/5.08 ( ( ? [M2: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M2 @ N )
% 9.00/5.08 & ( P @ M2 ) ) )
% 9.00/5.08 = ( ? [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.08 & ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5817_mod__induct,axiom,
% 9.00/5.08 ! [P: nat > $o,N: nat,P6: nat,M: nat] :
% 9.00/5.08 ( ( P @ N )
% 9.00/5.08 => ( ( ord_less_nat @ N @ P6 )
% 9.00/5.08 => ( ( ord_less_nat @ M @ P6 )
% 9.00/5.08 => ( ! [N3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ N3 @ P6 )
% 9.00/5.08 => ( ( P @ N3 )
% 9.00/5.08 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P6 ) ) ) )
% 9.00/5.08 => ( P @ M ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5818_mod__less__divisor,axiom,
% 9.00/5.08 ! [N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5819_gcd__nat__induct,axiom,
% 9.00/5.08 ! [P: nat > nat > $o,M: nat,N: nat] :
% 9.00/5.08 ( ! [M4: nat] : ( P @ M4 @ zero_zero_nat )
% 9.00/5.08 => ( ! [M4: nat,N3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.08 => ( ( P @ N3 @ ( modulo_modulo_nat @ M4 @ N3 ) )
% 9.00/5.08 => ( P @ M4 @ N3 ) ) )
% 9.00/5.08 => ( P @ M @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5820_mod__Suc__le__divisor,axiom,
% 9.00/5.08 ! [M: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N ) ) @ N ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5821_mod__eq__0D,axiom,
% 9.00/5.08 ! [M: nat,D: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ M @ D )
% 9.00/5.08 = zero_zero_nat )
% 9.00/5.08 => ? [Q4: nat] :
% 9.00/5.08 ( M
% 9.00/5.08 = ( times_times_nat @ D @ Q4 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5822_mod__geq,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ~ ( ord_less_nat @ M @ N )
% 9.00/5.08 => ( ( modulo_modulo_nat @ M @ N )
% 9.00/5.08 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5823_mod__if,axiom,
% 9.00/5.08 ( modulo_modulo_nat
% 9.00/5.08 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5824_le__mod__geq,axiom,
% 9.00/5.08 ! [N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.08 => ( ( modulo_modulo_nat @ M @ N )
% 9.00/5.08 = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N ) @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5825_vebt__insert_Osimps_I2_J,axiom,
% 9.00/5.08 ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 9.00/5.08 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X2 )
% 9.00/5.08 = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5826_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: set_int,B3: set_int,C: set_int,D: set_int] :
% 9.00/5.08 ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A2 @ B3 ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_set_int @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_set_int @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_set_int @ C @ A2 )
% 9.00/5.08 | ( ord_less_set_int @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5827_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: rat,B3: rat,C: rat,D: rat] :
% 9.00/5.08 ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A2 @ B3 ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_rat @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_rat @ C @ A2 )
% 9.00/5.08 | ( ord_less_rat @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5828_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: num,B3: num,C: num,D: num] :
% 9.00/5.08 ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A2 @ B3 ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_num @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_num @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_num @ C @ A2 )
% 9.00/5.08 | ( ord_less_num @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_num @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5829_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: nat,B3: nat,C: nat,D: nat] :
% 9.00/5.08 ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_nat @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_nat @ C @ A2 )
% 9.00/5.08 | ( ord_less_nat @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5830_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: int,B3: int,C: int,D: int] :
% 9.00/5.08 ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A2 @ B3 ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_int @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_int @ C @ A2 )
% 9.00/5.08 | ( ord_less_int @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5831_atLeastatMost__psubset__iff,axiom,
% 9.00/5.08 ! [A2: real,B3: real,C: real,D: real] :
% 9.00/5.08 ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.00/5.08 = ( ( ~ ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.08 | ( ( ord_less_eq_real @ C @ A2 )
% 9.00/5.08 & ( ord_less_eq_real @ B3 @ D )
% 9.00/5.08 & ( ( ord_less_real @ C @ A2 )
% 9.00/5.08 | ( ord_less_real @ B3 @ D ) ) ) )
% 9.00/5.08 & ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5832_mod__le__divisor,axiom,
% 9.00/5.08 ! [N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N ) @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5833_div__less__mono,axiom,
% 9.00/5.08 ! [A3: nat,B2: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ( ( modulo_modulo_nat @ A3 @ N )
% 9.00/5.08 = zero_zero_nat )
% 9.00/5.08 => ( ( ( modulo_modulo_nat @ B2 @ N )
% 9.00/5.08 = zero_zero_nat )
% 9.00/5.08 => ( ord_less_nat @ ( divide_divide_nat @ A3 @ N ) @ ( divide_divide_nat @ B2 @ N ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5834_mod__eq__nat1E,axiom,
% 9.00/5.08 ! [M: nat,Q3: nat,N: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.00/5.08 = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.08 => ~ ! [S3: nat] :
% 9.00/5.08 ( M
% 9.00/5.08 != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5835_mod__eq__nat2E,axiom,
% 9.00/5.08 ! [M: nat,Q3: nat,N: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.00/5.08 = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.08 => ~ ! [S3: nat] :
% 9.00/5.08 ( N
% 9.00/5.08 != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5836_nat__mod__eq__lemma,axiom,
% 9.00/5.08 ! [X2: nat,N: nat,Y3: nat] :
% 9.00/5.08 ( ( ( modulo_modulo_nat @ X2 @ N )
% 9.00/5.08 = ( modulo_modulo_nat @ Y3 @ N ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.08 => ? [Q4: nat] :
% 9.00/5.08 ( X2
% 9.00/5.08 = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N @ Q4 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5837_atLeast0__atMost__Suc,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.00/5.08 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5838_Icc__eq__insert__lb__nat,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.08 => ( ( set_or1269000886237332187st_nat @ M @ N )
% 9.00/5.08 = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5839_atLeastAtMostSuc__conv,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.08 => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) )
% 9.00/5.08 = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5840_atLeastAtMost__insertL,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.08 => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.00/5.08 = ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5841_subset__eq__atLeast0__atMost__finite,axiom,
% 9.00/5.08 ! [N6: set_nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.08 => ( finite_finite_nat @ N6 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5842_vebt__insert_Osimps_I3_J,axiom,
% 9.00/5.08 ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 9.00/5.08 ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X2 )
% 9.00/5.08 = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5843_vebt__insert_Osimps_I1_J,axiom,
% 9.00/5.08 ! [X2: nat,A2: $o,B3: $o] :
% 9.00/5.08 ( ( ( X2 = zero_zero_nat )
% 9.00/5.08 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.08 = ( vEBT_Leaf @ $true @ B3 ) ) )
% 9.00/5.08 & ( ( X2 != zero_zero_nat )
% 9.00/5.08 => ( ( ( X2 = one_one_nat )
% 9.00/5.08 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.08 = ( vEBT_Leaf @ A2 @ $true ) ) )
% 9.00/5.08 & ( ( X2 != one_one_nat )
% 9.00/5.08 => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.08 = ( vEBT_Leaf @ A2 @ B3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5844_split__mod,axiom,
% 9.00/5.08 ! [P: nat > $o,M: nat,N: nat] :
% 9.00/5.08 ( ( P @ ( modulo_modulo_nat @ M @ N ) )
% 9.00/5.08 = ( ( ( N = zero_zero_nat )
% 9.00/5.08 => ( P @ M ) )
% 9.00/5.08 & ( ( N != zero_zero_nat )
% 9.00/5.08 => ! [I4: nat,J3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ J3 @ N )
% 9.00/5.08 => ( ( M
% 9.00/5.08 = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
% 9.00/5.08 => ( P @ J3 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5845_Suc__times__mod__eq,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.00/5.08 => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N ) ) @ M )
% 9.00/5.08 = one_one_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5846_two__realpow__ge__one,axiom,
% 9.00/5.08 ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5847_nth__rotate1,axiom,
% 9.00/5.08 ! [N: nat,Xs: list_int] :
% 9.00/5.08 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.08 => ( ( nth_int @ ( rotate1_int @ Xs ) @ N )
% 9.00/5.08 = ( nth_int @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5848_nth__rotate1,axiom,
% 9.00/5.08 ! [N: nat,Xs: list_VEBT_VEBT] :
% 9.00/5.08 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.08 => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs ) @ N )
% 9.00/5.08 = ( nth_VEBT_VEBT @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5849_nth__rotate1,axiom,
% 9.00/5.08 ! [N: nat,Xs: list_o] :
% 9.00/5.08 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.08 => ( ( nth_o @ ( rotate1_o @ Xs ) @ N )
% 9.00/5.08 = ( nth_o @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5850_nth__rotate1,axiom,
% 9.00/5.08 ! [N: nat,Xs: list_nat] :
% 9.00/5.08 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.08 => ( ( nth_nat @ ( rotate1_nat @ Xs ) @ N )
% 9.00/5.08 = ( nth_nat @ Xs @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5851_ln__2__less__1,axiom,
% 9.00/5.08 ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5852_not__exp__less__eq__0__int,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ zero_zero_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5853_verit__le__mono__div,axiom,
% 9.00/5.08 ! [A3: nat,B2: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ord_less_eq_nat
% 9.00/5.08 @ ( plus_plus_nat @ ( divide_divide_nat @ A3 @ N )
% 9.00/5.08 @ ( if_nat
% 9.00/5.08 @ ( ( modulo_modulo_nat @ B2 @ N )
% 9.00/5.08 = zero_zero_nat )
% 9.00/5.08 @ one_one_nat
% 9.00/5.08 @ zero_zero_nat ) )
% 9.00/5.08 @ ( divide_divide_nat @ B2 @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5854_exp__half__le2,axiom,
% 9.00/5.08 ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5855_L2__set__mult__ineq__lemma,axiom,
% 9.00/5.08 ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A2 @ C ) ) @ ( times_times_real @ B3 @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5856_less__log2__of__power,axiom,
% 9.00/5.08 ! [N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.00/5.08 => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5857_le__log2__of__power,axiom,
% 9.00/5.08 ! [N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M )
% 9.00/5.08 => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5858_powr__neg__numeral,axiom,
% 9.00/5.08 ! [X2: real,N: num] :
% 9.00/5.08 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( powr_real @ X2 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
% 9.00/5.08 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5859_pos__zdiv__mult__2,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 9.00/5.08 = ( divide_divide_int @ B3 @ A2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5860_neg__zdiv__mult__2,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.08 => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 9.00/5.08 = ( divide_divide_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5861_pos__zmod__mult__2,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ A2 )
% 9.00/5.08 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 9.00/5.08 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B3 @ A2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5862_log2__of__power__less,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.08 => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5863_real__exp__bound__lemma,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.08 => ( ord_less_eq_real @ ( exp_real @ X2 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5864_neg__zmod__mult__2,axiom,
% 9.00/5.08 ! [A2: int,B3: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ A2 @ zero_zero_int )
% 9.00/5.08 => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) )
% 9.00/5.08 = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B3 @ one_one_int ) @ A2 ) ) @ one_one_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5865_pos__eucl__rel__int__mult__2,axiom,
% 9.00/5.08 ! [B3: int,A2: int,Q3: int,R2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.08 => ( ( eucl_rel_int @ A2 @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.08 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5866_log2__of__power__le,axiom,
% 9.00/5.08 ! [M: nat,N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.08 => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5867_exp__lower__Taylor__quadratic,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X2 ) @ ( divide_divide_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5868_neg__eucl__rel__int__mult__2,axiom,
% 9.00/5.08 ! [B3: int,A2: int,Q3: int,R2: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ B3 @ zero_zero_int )
% 9.00/5.08 => ( ( eucl_rel_int @ ( plus_plus_int @ A2 @ one_one_int ) @ B3 @ ( product_Pair_int_int @ Q3 @ R2 ) )
% 9.00/5.08 => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B3 ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5869_arctan__double,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.08 => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X2 ) )
% 9.00/5.08 = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5870_ln__one__minus__pos__lower__bound,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.08 => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5871_abs__ln__one__plus__x__minus__x__bound,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.08 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5872_ceiling__log2__div2,axiom,
% 9.00/5.08 ! [N: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.08 => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.00/5.08 = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5873_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
% 9.00/5.08 ! [X2: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.08 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X2 ) ) @ X2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5874_VEBT__internal_Oinsert_H_Oelims,axiom,
% 9.00/5.08 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.08 ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
% 9.00/5.08 = Y3 )
% 9.00/5.08 => ( ! [A: $o,B: $o] :
% 9.00/5.08 ( ( X2
% 9.00/5.08 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.08 => ( Y3
% 9.00/5.08 != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) )
% 9.00/5.08 => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.08 ( ( X2
% 9.00/5.08 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.08 => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 9.00/5.08 => ( Y3
% 9.00/5.08 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
% 9.00/5.08 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 9.00/5.08 => ( Y3
% 9.00/5.08 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5875_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
% 9.00/5.08 ! [Deg: nat,X2: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
% 9.00/5.08 => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) )
% 9.00/5.08 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X2 )
% 9.00/5.08 => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5876_insert__correct,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5877_insert__corr,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5878_abs__sqrt__wlog,axiom,
% 9.00/5.08 ! [P: code_integer > code_integer > $o,X2: code_integer] :
% 9.00/5.08 ( ! [X5: code_integer] :
% 9.00/5.08 ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
% 9.00/5.08 => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( P @ ( abs_abs_Code_integer @ X2 ) @ ( power_8256067586552552935nteger @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5879_abs__sqrt__wlog,axiom,
% 9.00/5.08 ! [P: real > real > $o,X2: real] :
% 9.00/5.08 ( ! [X5: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.00/5.08 => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( P @ ( abs_abs_real @ X2 ) @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5880_abs__sqrt__wlog,axiom,
% 9.00/5.08 ! [P: rat > rat > $o,X2: rat] :
% 9.00/5.08 ( ! [X5: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
% 9.00/5.08 => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( P @ ( abs_abs_rat @ X2 ) @ ( power_power_rat @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5881_abs__sqrt__wlog,axiom,
% 9.00/5.08 ! [P: int > int > $o,X2: int] :
% 9.00/5.08 ( ! [X5: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ zero_zero_int @ X5 )
% 9.00/5.08 => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( P @ ( abs_abs_int @ X2 ) @ ( power_power_int @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5882_pred__list__to__short,axiom,
% 9.00/5.08 ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.08 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 = none_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5883_succ__list__to__short,axiom,
% 9.00/5.08 ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.08 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 = none_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5884_UnCI,axiom,
% 9.00/5.08 ! [C: real,B2: set_real,A3: set_real] :
% 9.00/5.08 ( ( ~ ( member_real @ C @ B2 )
% 9.00/5.08 => ( member_real @ C @ A3 ) )
% 9.00/5.08 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5885_UnCI,axiom,
% 9.00/5.08 ! [C: $o,B2: set_o,A3: set_o] :
% 9.00/5.08 ( ( ~ ( member_o @ C @ B2 )
% 9.00/5.08 => ( member_o @ C @ A3 ) )
% 9.00/5.08 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5886_UnCI,axiom,
% 9.00/5.08 ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.08 ( ( ~ ( member_set_nat @ C @ B2 )
% 9.00/5.08 => ( member_set_nat @ C @ A3 ) )
% 9.00/5.08 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5887_UnCI,axiom,
% 9.00/5.08 ! [C: int,B2: set_int,A3: set_int] :
% 9.00/5.08 ( ( ~ ( member_int @ C @ B2 )
% 9.00/5.08 => ( member_int @ C @ A3 ) )
% 9.00/5.08 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5888_UnCI,axiom,
% 9.00/5.08 ! [C: nat,B2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( ~ ( member_nat @ C @ B2 )
% 9.00/5.08 => ( member_nat @ C @ A3 ) )
% 9.00/5.08 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5889_UnCI,axiom,
% 9.00/5.08 ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ~ ( member8206827879206165904at_nat @ C @ B2 )
% 9.00/5.08 => ( member8206827879206165904at_nat @ C @ A3 ) )
% 9.00/5.08 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5890_UnCI,axiom,
% 9.00/5.08 ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ~ ( member8757157785044589968at_nat @ C @ B2 )
% 9.00/5.08 => ( member8757157785044589968at_nat @ C @ A3 ) )
% 9.00/5.08 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5891_Un__iff,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_real @ C @ A3 )
% 9.00/5.08 | ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5892_Un__iff,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_o @ C @ A3 )
% 9.00/5.08 | ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5893_Un__iff,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_set_nat @ C @ A3 )
% 9.00/5.08 | ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5894_Un__iff,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_int @ C @ A3 )
% 9.00/5.08 | ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5895_Un__iff,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_nat @ C @ A3 )
% 9.00/5.08 | ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5896_Un__iff,axiom,
% 9.00/5.08 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member8206827879206165904at_nat @ C @ A3 )
% 9.00/5.08 | ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5897_Un__iff,axiom,
% 9.00/5.08 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member8757157785044589968at_nat @ C @ A3 )
% 9.00/5.08 | ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5898_finite__atLeastAtMost__int,axiom,
% 9.00/5.08 ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5899_high__def,axiom,
% 9.00/5.08 ( vEBT_VEBT_high
% 9.00/5.08 = ( ^ [X: nat,N2: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5900_high__bound__aux,axiom,
% 9.00/5.08 ! [Ma: nat,N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.00/5.08 => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5901_high__inv,axiom,
% 9.00/5.08 ! [X2: nat,N: nat,Y3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 9.00/5.08 = Y3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5902_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo5327735625951526323at_nat )
% 9.00/5.08 = ( ( A3 = bot_bo5327735625951526323at_nat )
% 9.00/5.08 & ( B2 = bot_bo5327735625951526323at_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5903_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo228742789529271731at_nat )
% 9.00/5.08 = ( ( A3 = bot_bo228742789529271731at_nat )
% 9.00/5.08 & ( B2 = bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5904_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( sup_sup_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ( A3 = bot_bot_set_real )
% 9.00/5.08 & ( B2 = bot_bot_set_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5905_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( sup_sup_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ( A3 = bot_bot_set_o )
% 9.00/5.08 & ( B2 = bot_bot_set_o ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5906_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( sup_sup_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ( A3 = bot_bot_set_nat )
% 9.00/5.08 & ( B2 = bot_bot_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5907_Un__empty,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( sup_sup_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ( A3 = bot_bot_set_int )
% 9.00/5.08 & ( B2 = bot_bot_set_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5908_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_int,G2: set_int] :
% 9.00/5.08 ( ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite_finite_int @ F2 )
% 9.00/5.08 & ( finite_finite_int @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5909_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_complex,G2: set_complex] :
% 9.00/5.08 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.08 & ( finite3207457112153483333omplex @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5910_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.08 & ( finite6177210948735845034at_nat @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5911_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 9.00/5.08 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.08 & ( finite4001608067531595151d_enat @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5912_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_nat,G2: set_nat] :
% 9.00/5.08 ( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite_finite_nat @ F2 )
% 9.00/5.08 & ( finite_finite_nat @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5913_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite4392333629123659920at_nat @ F2 )
% 9.00/5.08 & ( finite4392333629123659920at_nat @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5914_finite__Un,axiom,
% 9.00/5.08 ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) )
% 9.00/5.08 = ( ( finite4343798906461161616at_nat @ F2 )
% 9.00/5.08 & ( finite4343798906461161616at_nat @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5915_Un__subset__iff,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 9.00/5.08 & ( ord_less_eq_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5916_Un__subset__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 9.00/5.08 & ( ord_le3000389064537975527at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5917_Un__subset__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 9.00/5.08 & ( ord_le1268244103169919719at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5918_Un__subset__iff,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ( ord_less_eq_set_int @ A3 @ C2 )
% 9.00/5.08 & ( ord_less_eq_set_int @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5919_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5920_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: real,B2: set_real,C2: set_real] :
% 9.00/5.08 ( ( sup_sup_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_real @ A2 @ ( sup_sup_set_real @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5921_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: $o,B2: set_o,C2: set_o] :
% 9.00/5.08 ( ( sup_sup_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_o @ A2 @ ( sup_sup_set_o @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5922_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: int,B2: set_int,C2: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_int @ A2 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5923_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5924_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5925_Un__insert__left,axiom,
% 9.00/5.08 ! [A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5926_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5927_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_real,A2: real,B2: set_real] :
% 9.00/5.08 ( ( sup_sup_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_real @ A2 @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5928_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_o,A2: $o,B2: set_o] :
% 9.00/5.08 ( ( sup_sup_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_o @ A2 @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5929_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_int,A2: int,B2: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_int @ A2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5930_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_nat,A2: nat,B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_nat @ A2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5931_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,A2: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ ( insert5050368324300391991at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert5050368324300391991at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5932_Un__insert__right,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,A2: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ ( insert9069300056098147895at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert9069300056098147895at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5933_Un__Diff__cancel,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5934_Un__Diff__cancel,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5935_Un__Diff__cancel,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5936_Un__Diff__cancel2,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) @ A3 )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ B2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5937_Un__Diff__cancel2,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) @ A3 )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ B2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5938_Un__Diff__cancel2,axiom,
% 9.00/5.08 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B2 @ A3 ) @ A3 )
% 9.00/5.08 = ( sup_sup_set_nat @ B2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5939_Compl__Diff__eq,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( uminus4384627049435823934at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5940_Compl__Diff__eq,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5941_Compl__Diff__eq,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5942_case4_I11_J,axiom,
% 9.00/5.08 ( ( mi != ma )
% 9.00/5.08 => ! [I3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.00/5.08 => ( ( ( ( vEBT_VEBT_high @ ma @ na )
% 9.00/5.08 = I3 )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
% 9.00/5.08 & ! [X4: nat] :
% 9.00/5.08 ( ( ( ( vEBT_VEBT_high @ X4 @ na )
% 9.00/5.08 = I3 )
% 9.00/5.08 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ na ) ) )
% 9.00/5.08 => ( ( ord_less_nat @ mi @ X4 )
% 9.00/5.08 & ( ord_less_eq_nat @ X4 @ ma ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5943_UnE,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member_real @ C @ A3 )
% 9.00/5.08 => ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5944_UnE,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member_o @ C @ A3 )
% 9.00/5.08 => ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5945_UnE,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member_set_nat @ C @ A3 )
% 9.00/5.08 => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5946_UnE,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member_int @ C @ A3 )
% 9.00/5.08 => ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5947_UnE,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member_nat @ C @ A3 )
% 9.00/5.08 => ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5948_UnE,axiom,
% 9.00/5.08 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member8206827879206165904at_nat @ C @ A3 )
% 9.00/5.08 => ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5949_UnE,axiom,
% 9.00/5.08 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ~ ( member8757157785044589968at_nat @ C @ A3 )
% 9.00/5.08 => ( member8757157785044589968at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5950_UnI1,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ A3 )
% 9.00/5.08 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5951_UnI1,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ A3 )
% 9.00/5.08 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5952_UnI1,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ A3 )
% 9.00/5.08 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5953_UnI1,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ A3 )
% 9.00/5.08 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5954_UnI1,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ A3 )
% 9.00/5.08 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5955_UnI1,axiom,
% 9.00/5.08 ! [C: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ C @ A3 )
% 9.00/5.08 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5956_UnI1,axiom,
% 9.00/5.08 ! [C: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ C @ A3 )
% 9.00/5.08 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5957_UnI2,axiom,
% 9.00/5.08 ! [C: real,B2: set_real,A3: set_real] :
% 9.00/5.08 ( ( member_real @ C @ B2 )
% 9.00/5.08 => ( member_real @ C @ ( sup_sup_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5958_UnI2,axiom,
% 9.00/5.08 ! [C: $o,B2: set_o,A3: set_o] :
% 9.00/5.08 ( ( member_o @ C @ B2 )
% 9.00/5.08 => ( member_o @ C @ ( sup_sup_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5959_UnI2,axiom,
% 9.00/5.08 ! [C: set_nat,B2: set_set_nat,A3: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ B2 )
% 9.00/5.08 => ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5960_UnI2,axiom,
% 9.00/5.08 ! [C: int,B2: set_int,A3: set_int] :
% 9.00/5.08 ( ( member_int @ C @ B2 )
% 9.00/5.08 => ( member_int @ C @ ( sup_sup_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5961_UnI2,axiom,
% 9.00/5.08 ! [C: nat,B2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ B2 )
% 9.00/5.08 => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5962_UnI2,axiom,
% 9.00/5.08 ! [C: produc859450856879609959at_nat,B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ C @ B2 )
% 9.00/5.08 => ( member8206827879206165904at_nat @ C @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5963_UnI2,axiom,
% 9.00/5.08 ! [C: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ C @ B2 )
% 9.00/5.08 => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5964_bex__Un,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,P: nat > $o] :
% 9.00/5.08 ( ( ? [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 & ( P @ X ) ) )
% 9.00/5.08 = ( ? [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ A3 )
% 9.00/5.08 & ( P @ X ) )
% 9.00/5.08 | ? [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ B2 )
% 9.00/5.08 & ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5965_bex__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 9.00/5.08 ( ( ? [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 & ( P @ X ) ) )
% 9.00/5.08 = ( ? [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ A3 )
% 9.00/5.08 & ( P @ X ) )
% 9.00/5.08 | ? [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ B2 )
% 9.00/5.08 & ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5966_bex__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
% 9.00/5.08 ( ( ? [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 & ( P @ X ) ) )
% 9.00/5.08 = ( ? [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ A3 )
% 9.00/5.08 & ( P @ X ) )
% 9.00/5.08 | ? [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.00/5.08 & ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5967_ball__Un,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,P: nat > $o] :
% 9.00/5.08 ( ( ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( P @ X ) ) )
% 9.00/5.08 = ( ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ A3 )
% 9.00/5.08 => ( P @ X ) )
% 9.00/5.08 & ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ B2 )
% 9.00/5.08 => ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5968_ball__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,P: produc859450856879609959at_nat > $o] :
% 9.00/5.08 ( ( ! [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( P @ X ) ) )
% 9.00/5.08 = ( ! [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ A3 )
% 9.00/5.08 => ( P @ X ) )
% 9.00/5.08 & ! [X: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( member8206827879206165904at_nat @ X @ B2 )
% 9.00/5.08 => ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5969_ball__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
% 9.00/5.08 ( ( ! [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( P @ X ) ) )
% 9.00/5.08 = ( ! [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ A3 )
% 9.00/5.08 => ( P @ X ) )
% 9.00/5.08 & ! [X: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( member8757157785044589968at_nat @ X @ B2 )
% 9.00/5.08 => ( P @ X ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5970_Un__assoc,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5971_Un__assoc,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5972_Un__assoc,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5973_Un__absorb,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ A3 )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5974_Un__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ A3 )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5975_Un__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ A3 )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5976_Un__commute,axiom,
% 9.00/5.08 ( sup_sup_set_nat
% 9.00/5.08 = ( ^ [A6: set_nat,B6: set_nat] : ( sup_sup_set_nat @ B6 @ A6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5977_Un__commute,axiom,
% 9.00/5.08 ( sup_su718114333110466843at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] : ( sup_su718114333110466843at_nat @ B6 @ A6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5978_Un__commute,axiom,
% 9.00/5.08 ( sup_su5525570899277871387at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B6 @ A6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5979_Un__left__absorb,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5980_Un__left__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5981_Un__left__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5982_Un__left__commute,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5983_Un__left__commute,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5984_Un__left__commute,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5985_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5987_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_real] :
% 9.00/5.08 ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5988_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_o] :
% 9.00/5.08 ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5989_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5990_boolean__algebra_Odisj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5991_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ bot_bo5327735625951526323at_nat )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5992_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ bot_bo228742789529271731at_nat )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5993_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_real] :
% 9.00/5.08 ( ( sup_sup_set_real @ A3 @ bot_bot_set_real )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5994_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_o] :
% 9.00/5.08 ( ( sup_sup_set_o @ A3 @ bot_bot_set_o )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5995_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ bot_bot_set_nat )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5996_Un__empty__right,axiom,
% 9.00/5.08 ! [A3: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ A3 @ bot_bot_set_int )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5997_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5998_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_5999_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_real] :
% 9.00/5.08 ( ( sup_sup_set_real @ bot_bot_set_real @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6000_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_o] :
% 9.00/5.08 ( ( sup_sup_set_o @ bot_bot_set_o @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6001_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ bot_bot_set_nat @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6002_Un__empty__left,axiom,
% 9.00/5.08 ! [B2: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ bot_bot_set_int @ B2 )
% 9.00/5.08 = B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6003_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_int,T2: set_int] :
% 9.00/5.08 ( ( ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite_finite_int @ S )
% 9.00/5.08 | ~ ( finite_finite_int @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6004_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_complex,T2: set_complex] :
% 9.00/5.08 ( ( ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite3207457112153483333omplex @ S )
% 9.00/5.08 | ~ ( finite3207457112153483333omplex @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6005_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite6177210948735845034at_nat @ S )
% 9.00/5.08 | ~ ( finite6177210948735845034at_nat @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6006_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 9.00/5.08 ( ( ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.08 | ~ ( finite4001608067531595151d_enat @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6007_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite_finite_nat @ S )
% 9.00/5.08 | ~ ( finite_finite_nat @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6008_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite4392333629123659920at_nat @ S )
% 9.00/5.08 | ~ ( finite4392333629123659920at_nat @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6009_infinite__Un,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) )
% 9.00/5.08 = ( ~ ( finite4343798906461161616at_nat @ S )
% 9.00/5.08 | ~ ( finite4343798906461161616at_nat @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6010_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_int,T2: set_int] :
% 9.00/5.08 ( ~ ( finite_finite_int @ S )
% 9.00/5.08 => ~ ( finite_finite_int @ ( sup_sup_set_int @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6011_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_complex,T2: set_complex] :
% 9.00/5.08 ( ~ ( finite3207457112153483333omplex @ S )
% 9.00/5.08 => ~ ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6012_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ~ ( finite6177210948735845034at_nat @ S )
% 9.00/5.08 => ~ ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6013_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_Extended_enat,T2: set_Extended_enat] :
% 9.00/5.08 ( ~ ( finite4001608067531595151d_enat @ S )
% 9.00/5.08 => ~ ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6014_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ~ ( finite_finite_nat @ S )
% 9.00/5.08 => ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6015_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ~ ( finite4392333629123659920at_nat @ S )
% 9.00/5.08 => ~ ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6016_Un__infinite,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ~ ( finite4343798906461161616at_nat @ S )
% 9.00/5.08 => ~ ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6017_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_int,G2: set_int] :
% 9.00/5.08 ( ( finite_finite_int @ F2 )
% 9.00/5.08 => ( ( finite_finite_int @ G2 )
% 9.00/5.08 => ( finite_finite_int @ ( sup_sup_set_int @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6018_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_complex,G2: set_complex] :
% 9.00/5.08 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.08 => ( ( finite3207457112153483333omplex @ G2 )
% 9.00/5.08 => ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6019_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.08 => ( ( finite6177210948735845034at_nat @ G2 )
% 9.00/5.08 => ( finite6177210948735845034at_nat @ ( sup_su6327502436637775413at_nat @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6020_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 9.00/5.08 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.08 => ( ( finite4001608067531595151d_enat @ G2 )
% 9.00/5.08 => ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6021_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_nat,G2: set_nat] :
% 9.00/5.08 ( ( finite_finite_nat @ F2 )
% 9.00/5.08 => ( ( finite_finite_nat @ G2 )
% 9.00/5.08 => ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6022_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_Pr8693737435421807431at_nat,G2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( finite4392333629123659920at_nat @ F2 )
% 9.00/5.08 => ( ( finite4392333629123659920at_nat @ G2 )
% 9.00/5.08 => ( finite4392333629123659920at_nat @ ( sup_su718114333110466843at_nat @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6023_finite__UnI,axiom,
% 9.00/5.08 ! [F2: set_Pr4329608150637261639at_nat,G2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( finite4343798906461161616at_nat @ F2 )
% 9.00/5.08 => ( ( finite4343798906461161616at_nat @ G2 )
% 9.00/5.08 => ( finite4343798906461161616at_nat @ ( sup_su5525570899277871387at_nat @ F2 @ G2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6024_subset__Un__eq,axiom,
% 9.00/5.08 ( ord_less_eq_set_nat
% 9.00/5.08 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A6 @ B6 )
% 9.00/5.08 = B6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6025_subset__Un__eq,axiom,
% 9.00/5.08 ( ord_le3000389064537975527at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A6 @ B6 )
% 9.00/5.08 = B6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6026_subset__Un__eq,axiom,
% 9.00/5.08 ( ord_le1268244103169919719at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A6 @ B6 )
% 9.00/5.08 = B6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6027_subset__Un__eq,axiom,
% 9.00/5.08 ( ord_less_eq_set_int
% 9.00/5.08 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.08 ( ( sup_sup_set_int @ A6 @ B6 )
% 9.00/5.08 = B6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6028_subset__UnE,axiom,
% 9.00/5.08 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ! [A8: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A8 @ A3 )
% 9.00/5.08 => ! [B10: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ B10 @ B2 )
% 9.00/5.08 => ( C2
% 9.00/5.08 != ( sup_sup_set_nat @ A8 @ B10 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6029_subset__UnE,axiom,
% 9.00/5.08 ! [C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ C2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ! [A8: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ A8 @ A3 )
% 9.00/5.08 => ! [B10: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ B10 @ B2 )
% 9.00/5.08 => ( C2
% 9.00/5.08 != ( sup_su718114333110466843at_nat @ A8 @ B10 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6030_subset__UnE,axiom,
% 9.00/5.08 ! [C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ! [A8: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ A8 @ A3 )
% 9.00/5.08 => ! [B10: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ B10 @ B2 )
% 9.00/5.08 => ( C2
% 9.00/5.08 != ( sup_su5525570899277871387at_nat @ A8 @ B10 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6031_subset__UnE,axiom,
% 9.00/5.08 ! [C2: set_int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ C2 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ~ ! [A8: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A8 @ A3 )
% 9.00/5.08 => ! [B10: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ B10 @ B2 )
% 9.00/5.08 => ( C2
% 9.00/5.08 != ( sup_sup_set_int @ A8 @ B10 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6032_Un__absorb2,axiom,
% 9.00/5.08 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.08 => ( ( sup_sup_set_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6033_Un__absorb2,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ B2 @ A3 )
% 9.00/5.08 => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6034_Un__absorb2,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ B2 @ A3 )
% 9.00/5.08 => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6035_Un__absorb2,axiom,
% 9.00/5.08 ! [B2: set_int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.08 => ( ( sup_sup_set_int @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6036_Un__absorb1,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_sup_set_nat @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6037_Un__absorb1,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6038_Un__absorb1,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6039_Un__absorb1,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_sup_set_int @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6040_Un__upper2,axiom,
% 9.00/5.08 ! [B2: set_nat,A3: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6041_Un__upper2,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B2 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6042_Un__upper2,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6043_Un__upper2,axiom,
% 9.00/5.08 ! [B2: set_int,A3: set_int] : ( ord_less_eq_set_int @ B2 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6044_Un__upper1,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6045_Un__upper1,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6046_Un__upper1,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6047_Un__upper1,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6048_Un__least,axiom,
% 9.00/5.08 ! [A3: set_nat,C2: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_nat @ B2 @ C2 )
% 9.00/5.08 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6049_Un__least,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_le3000389064537975527at_nat @ B2 @ C2 )
% 9.00/5.08 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6050_Un__least,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_le1268244103169919719at_nat @ B2 @ C2 )
% 9.00/5.08 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6051_Un__least,axiom,
% 9.00/5.08 ! [A3: set_int,C2: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_int @ B2 @ C2 )
% 9.00/5.08 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6052_Un__mono,axiom,
% 9.00/5.08 ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_nat @ B2 @ D4 )
% 9.00/5.08 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6053_Un__mono,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,D4: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_le3000389064537975527at_nat @ B2 @ D4 )
% 9.00/5.08 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6054_Un__mono,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,D4: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_le1268244103169919719at_nat @ B2 @ D4 )
% 9.00/5.08 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6055_Un__mono,axiom,
% 9.00/5.08 ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_int @ B2 @ D4 )
% 9.00/5.08 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A3 @ B2 ) @ ( sup_sup_set_int @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6056_Un__Diff,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( minus_8321449233255521966at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) @ ( minus_8321449233255521966at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6057_Un__Diff,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) @ ( minus_3314409938677909166at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6058_Un__Diff,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C2 ) @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6059_ivl__disj__un__two__touch_I4_J,axiom,
% 9.00/5.08 ! [L: rat,M: rat,U: rat] :
% 9.00/5.08 ( ( ord_less_eq_rat @ L @ M )
% 9.00/5.08 => ( ( ord_less_eq_rat @ M @ U )
% 9.00/5.08 => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M ) @ ( set_or633870826150836451st_rat @ M @ U ) )
% 9.00/5.08 = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6060_ivl__disj__un__two__touch_I4_J,axiom,
% 9.00/5.08 ! [L: num,M: num,U: num] :
% 9.00/5.08 ( ( ord_less_eq_num @ L @ M )
% 9.00/5.08 => ( ( ord_less_eq_num @ M @ U )
% 9.00/5.08 => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M ) @ ( set_or7049704709247886629st_num @ M @ U ) )
% 9.00/5.08 = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6061_ivl__disj__un__two__touch_I4_J,axiom,
% 9.00/5.08 ! [L: nat,M: nat,U: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ L @ M )
% 9.00/5.08 => ( ( ord_less_eq_nat @ M @ U )
% 9.00/5.08 => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M ) @ ( set_or1269000886237332187st_nat @ M @ U ) )
% 9.00/5.08 = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6062_ivl__disj__un__two__touch_I4_J,axiom,
% 9.00/5.08 ! [L: int,M: int,U: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ L @ M )
% 9.00/5.08 => ( ( ord_less_eq_int @ M @ U )
% 9.00/5.08 => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M ) @ ( set_or1266510415728281911st_int @ M @ U ) )
% 9.00/5.08 = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6063_ivl__disj__un__two__touch_I4_J,axiom,
% 9.00/5.08 ! [L: real,M: real,U: real] :
% 9.00/5.08 ( ( ord_less_eq_real @ L @ M )
% 9.00/5.08 => ( ( ord_less_eq_real @ M @ U )
% 9.00/5.08 => ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M ) @ ( set_or1222579329274155063t_real @ M @ U ) )
% 9.00/5.08 = ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6064_insert__is__Un,axiom,
% 9.00/5.08 ( insert8211810215607154385at_nat
% 9.00/5.08 = ( ^ [A4: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6065_insert__is__Un,axiom,
% 9.00/5.08 ( insert5050368324300391991at_nat
% 9.00/5.08 = ( ^ [A4: produc859450856879609959at_nat] : ( sup_su718114333110466843at_nat @ ( insert5050368324300391991at_nat @ A4 @ bot_bo5327735625951526323at_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6066_insert__is__Un,axiom,
% 9.00/5.08 ( insert9069300056098147895at_nat
% 9.00/5.08 = ( ^ [A4: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6067_insert__is__Un,axiom,
% 9.00/5.08 ( insert_real
% 9.00/5.08 = ( ^ [A4: real] : ( sup_sup_set_real @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6068_insert__is__Un,axiom,
% 9.00/5.08 ( insert_o
% 9.00/5.08 = ( ^ [A4: $o] : ( sup_sup_set_o @ ( insert_o @ A4 @ bot_bot_set_o ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6069_insert__is__Un,axiom,
% 9.00/5.08 ( insert_nat
% 9.00/5.08 = ( ^ [A4: nat] : ( sup_sup_set_nat @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6070_insert__is__Un,axiom,
% 9.00/5.08 ( insert_int
% 9.00/5.08 = ( ^ [A4: int] : ( sup_sup_set_int @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6071_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat] :
% 9.00/5.08 ( ( ( sup_su6327502436637775413at_nat @ A3 @ B2 )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6072_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,X2: produc859450856879609959at_nat] :
% 9.00/5.08 ( ( ( sup_su718114333110466843at_nat @ A3 @ B2 )
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo5327735625951526323at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6073_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X2: produc3843707927480180839at_nat] :
% 9.00/5.08 ( ( ( sup_su5525570899277871387at_nat @ A3 @ B2 )
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo228742789529271731at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo228742789529271731at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6074_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real,X2: real] :
% 9.00/5.08 ( ( ( sup_sup_set_real @ A3 @ B2 )
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_real )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.08 & ( B2 = bot_bot_set_real ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6075_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o,X2: $o] :
% 9.00/5.08 ( ( ( sup_sup_set_o @ A3 @ B2 )
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_o )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.08 & ( B2 = bot_bot_set_o ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6076_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,X2: nat] :
% 9.00/5.08 ( ( ( sup_sup_set_nat @ A3 @ B2 )
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 & ( B2 = bot_bot_set_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6077_Un__singleton__iff,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int,X2: int] :
% 9.00/5.08 ( ( ( sup_sup_set_int @ A3 @ B2 )
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_int )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.08 & ( B2 = bot_bot_set_int ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6078_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6079_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: produc859450856879609959at_nat,A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo5327735625951526323at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert5050368324300391991at_nat @ X2 @ bot_bo5327735625951526323at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6080_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: produc3843707927480180839at_nat,A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bo228742789529271731at_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 9.00/5.08 & ( B2 = bot_bo228742789529271731at_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6081_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( insert_real @ X2 @ bot_bot_set_real )
% 9.00/5.08 = ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_real )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.08 & ( B2 = bot_bot_set_real ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6082_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( insert_o @ X2 @ bot_bot_set_o )
% 9.00/5.08 = ( sup_sup_set_o @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_o )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.08 & ( B2 = bot_bot_set_o ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_o @ X2 @ bot_bot_set_o ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6083_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( insert_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.08 = ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_nat )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 & ( B2 = bot_bot_set_nat ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6084_singleton__Un__iff,axiom,
% 9.00/5.08 ! [X2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( insert_int @ X2 @ bot_bot_set_int )
% 9.00/5.08 = ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ( A3 = bot_bot_set_int )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.08 & ( B2 = bot_bot_set_int ) )
% 9.00/5.08 | ( ( A3
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.08 & ( B2
% 9.00/5.08 = ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6085_Diff__subset__conv,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ord_le3000389064537975527at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6086_Diff__subset__conv,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ord_le1268244103169919719at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6087_Diff__subset__conv,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ord_less_eq_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6088_Diff__subset__conv,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( ord_less_eq_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6089_Diff__partition,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_su718114333110466843at_nat @ A3 @ ( minus_8321449233255521966at_nat @ B2 @ A3 ) )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6090_Diff__partition,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_su5525570899277871387at_nat @ A3 @ ( minus_3314409938677909166at_nat @ B2 @ A3 ) )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6091_Diff__partition,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6092_Diff__partition,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.08 => ( ( sup_sup_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6093_atLeastAtMostPlus1__int__conv,axiom,
% 9.00/5.08 ! [M: int,N: int] :
% 9.00/5.08 ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.00/5.08 => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N ) )
% 9.00/5.08 = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M @ N ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6094_simp__from__to,axiom,
% 9.00/5.08 ( set_or1266510415728281911st_int
% 9.00/5.08 = ( ^ [I4: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I4 ) @ bot_bot_set_int @ ( insert_int @ I4 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6095_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_complex,B2: set_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6096_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_list_nat,B2: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6097_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6098_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6099_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6100_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] : ( ord_less_eq_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6101_card__Un__le,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] : ( ord_less_eq_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6102_periodic__finite__ex,axiom,
% 9.00/5.08 ! [D: int,P: int > $o] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D )
% 9.00/5.08 => ( ! [X5: int,K2: int] :
% 9.00/5.08 ( ( P @ X5 )
% 9.00/5.08 = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.00/5.08 => ( ( ? [X8: int] : ( P @ X8 ) )
% 9.00/5.08 = ( ? [X: int] :
% 9.00/5.08 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 9.00/5.08 & ( P @ X ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6103_aset_I7_J,axiom,
% 9.00/5.08 ! [D4: int,A3: set_int,T: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_int @ T @ X4 )
% 9.00/5.08 => ( ord_less_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6104_aset_I5_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ T @ A3 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_int @ X4 @ T )
% 9.00/5.08 => ( ord_less_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6105_aset_I4_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ T @ A3 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( X4 != T )
% 9.00/5.08 => ( ( plus_plus_int @ X4 @ D4 )
% 9.00/5.08 != T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6106_aset_I3_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( X4 = T )
% 9.00/5.08 => ( ( plus_plus_int @ X4 @ D4 )
% 9.00/5.08 = T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6107_bset_I7_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ T @ B2 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_int @ T @ X4 )
% 9.00/5.08 => ( ord_less_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6108_bset_I5_J,axiom,
% 9.00/5.08 ! [D4: int,B2: set_int,T: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_int @ X4 @ T )
% 9.00/5.08 => ( ord_less_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6109_bset_I4_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ T @ B2 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( X4 != T )
% 9.00/5.08 => ( ( minus_minus_int @ X4 @ D4 )
% 9.00/5.08 != T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6110_bset_I3_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( X4 = T )
% 9.00/5.08 => ( ( minus_minus_int @ X4 @ D4 )
% 9.00/5.08 = T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6111_aset_I8_J,axiom,
% 9.00/5.08 ! [D4: int,A3: set_int,T: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ T @ X4 )
% 9.00/5.08 => ( ord_less_eq_int @ T @ ( plus_plus_int @ X4 @ D4 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6112_aset_I6_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A3 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ A3 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ X4 @ T )
% 9.00/5.08 => ( ord_less_eq_int @ ( plus_plus_int @ X4 @ D4 ) @ T ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6113_bset_I8_J,axiom,
% 9.00/5.08 ! [D4: int,T: int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ T @ X4 )
% 9.00/5.08 => ( ord_less_eq_int @ T @ ( minus_minus_int @ X4 @ D4 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6114_bset_I6_J,axiom,
% 9.00/5.08 ! [D4: int,B2: set_int,T: int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ! [X4: int] :
% 9.00/5.08 ( ! [Xa3: int] :
% 9.00/5.08 ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb2: int] :
% 9.00/5.08 ( ( member_int @ Xb2 @ B2 )
% 9.00/5.08 => ( X4
% 9.00/5.08 != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.00/5.08 => ( ( ord_less_eq_int @ X4 @ T )
% 9.00/5.08 => ( ord_less_eq_int @ ( minus_minus_int @ X4 @ D4 ) @ T ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6115_cppi,axiom,
% 9.00/5.08 ! [D4: int,P: int > $o,P4: int > $o,A3: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ? [Z5: int] :
% 9.00/5.08 ! [X5: int] :
% 9.00/5.08 ( ( ord_less_int @ Z5 @ X5 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 = ( P4 @ X5 ) ) )
% 9.00/5.08 => ( ! [X5: int] :
% 9.00/5.08 ( ! [Xa: int] :
% 9.00/5.08 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb3: int] :
% 9.00/5.08 ( ( member_int @ Xb3 @ A3 )
% 9.00/5.08 => ( X5
% 9.00/5.08 != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
% 9.00/5.08 => ( ! [X5: int,K2: int] :
% 9.00/5.08 ( ( P4 @ X5 )
% 9.00/5.08 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 9.00/5.08 => ( ( ? [X8: int] : ( P @ X8 ) )
% 9.00/5.08 = ( ? [X: int] :
% 9.00/5.08 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 & ( P4 @ X ) )
% 9.00/5.08 | ? [X: int] :
% 9.00/5.08 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 & ? [Y: int] :
% 9.00/5.08 ( ( member_int @ Y @ A3 )
% 9.00/5.08 & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6116_cpmi,axiom,
% 9.00/5.08 ! [D4: int,P: int > $o,P4: int > $o,B2: set_int] :
% 9.00/5.08 ( ( ord_less_int @ zero_zero_int @ D4 )
% 9.00/5.08 => ( ? [Z5: int] :
% 9.00/5.08 ! [X5: int] :
% 9.00/5.08 ( ( ord_less_int @ X5 @ Z5 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 = ( P4 @ X5 ) ) )
% 9.00/5.08 => ( ! [X5: int] :
% 9.00/5.08 ( ! [Xa: int] :
% 9.00/5.08 ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 => ! [Xb3: int] :
% 9.00/5.08 ( ( member_int @ Xb3 @ B2 )
% 9.00/5.08 => ( X5
% 9.00/5.08 != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
% 9.00/5.08 => ( ! [X5: int,K2: int] :
% 9.00/5.08 ( ( P4 @ X5 )
% 9.00/5.08 = ( P4 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
% 9.00/5.08 => ( ( ? [X8: int] : ( P @ X8 ) )
% 9.00/5.08 = ( ? [X: int] :
% 9.00/5.08 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 & ( P4 @ X ) )
% 9.00/5.08 | ? [X: int] :
% 9.00/5.08 ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
% 9.00/5.08 & ? [Y: int] :
% 9.00/5.08 ( ( member_int @ Y @ B2 )
% 9.00/5.08 & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6117_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 9.00/5.08 ! [X2: nat,N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.08 => ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6118_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
% 9.00/5.08 ! [A2: $o,B3: $o,X2: nat] :
% 9.00/5.08 ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 )
% 9.00/5.08 = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6119_insert_H__correct,axiom,
% 9.00/5.08 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.08 => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6120_nested__mint,axiom,
% 9.00/5.08 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.08 => ( ( N
% 9.00/5.08 = ( suc @ ( suc @ Va ) ) )
% 9.00/5.08 => ( ~ ( ord_less_nat @ Ma @ Mi )
% 9.00/5.08 => ( ( Ma != Mi )
% 9.00/5.08 => ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6121_both__member__options__from__chilf__to__complete__tree,axiom,
% 9.00/5.08 ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.00/5.08 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6122_member__inv,axiom,
% 9.00/5.08 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 9.00/5.08 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.08 & ( ( X2 = Mi )
% 9.00/5.08 | ( X2 = Ma )
% 9.00/5.08 | ( ( ord_less_nat @ X2 @ Ma )
% 9.00/5.08 & ( ord_less_nat @ Mi @ X2 )
% 9.00/5.08 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.08 & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6123_both__member__options__from__complete__tree__to__child,axiom,
% 9.00/5.08 ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 9.00/5.08 ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.00/5.08 => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.08 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 | ( X2 = Mi )
% 9.00/5.08 | ( X2 = Ma ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6124_both__member__options__ding,axiom,
% 9.00/5.08 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.08 => ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.00/5.08 => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6125_bit__split__inv,axiom,
% 9.00/5.08 ! [X2: nat,D: nat] :
% 9.00/5.08 ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X2 @ D ) @ ( vEBT_VEBT_low @ X2 @ D ) @ D )
% 9.00/5.08 = X2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6126_Int__iff,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_real @ C @ A3 )
% 9.00/5.08 & ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6127_Int__iff,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_o @ C @ A3 )
% 9.00/5.08 & ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6128_Int__iff,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_set_nat @ C @ A3 )
% 9.00/5.08 & ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6129_Int__iff,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_int @ C @ A3 )
% 9.00/5.08 & ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6130_Int__iff,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member_nat @ C @ A3 )
% 9.00/5.08 & ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6131_Int__iff,axiom,
% 9.00/5.08 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( member8440522571783428010at_nat @ C @ A3 )
% 9.00/5.08 & ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6132_IntI,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ A3 )
% 9.00/5.08 => ( ( member_real @ C @ B2 )
% 9.00/5.08 => ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6133_IntI,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ A3 )
% 9.00/5.08 => ( ( member_o @ C @ B2 )
% 9.00/5.08 => ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6134_IntI,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ A3 )
% 9.00/5.08 => ( ( member_set_nat @ C @ B2 )
% 9.00/5.08 => ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6135_IntI,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ A3 )
% 9.00/5.08 => ( ( member_int @ C @ B2 )
% 9.00/5.08 => ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6136_IntI,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ A3 )
% 9.00/5.08 => ( ( member_nat @ C @ B2 )
% 9.00/5.08 => ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6137_IntI,axiom,
% 9.00/5.08 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ C @ A3 )
% 9.00/5.08 => ( ( member8440522571783428010at_nat @ C @ B2 )
% 9.00/5.08 => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6138_low__def,axiom,
% 9.00/5.08 ( vEBT_VEBT_low
% 9.00/5.08 = ( ^ [X: nat,N2: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6139_low__inv,axiom,
% 9.00/5.08 ! [X2: nat,N: nat,Y3: nat] :
% 9.00/5.08 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.08 => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X2 ) @ N )
% 9.00/5.08 = X2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6140_boolean__algebra_Oconj__zero__left,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6141_boolean__algebra_Oconj__zero__left,axiom,
% 9.00/5.08 ! [X2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6142_boolean__algebra_Oconj__zero__left,axiom,
% 9.00/5.08 ! [X2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6143_boolean__algebra_Oconj__zero__left,axiom,
% 9.00/5.08 ! [X2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6144_boolean__algebra_Oconj__zero__left,axiom,
% 9.00/5.08 ! [X2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6145_boolean__algebra_Oconj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6146_boolean__algebra_Oconj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6147_boolean__algebra_Oconj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6148_boolean__algebra_Oconj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6149_boolean__algebra_Oconj__zero__right,axiom,
% 9.00/5.08 ! [X2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6150_finite__Int,axiom,
% 9.00/5.08 ! [F2: set_int,G2: set_int] :
% 9.00/5.08 ( ( ( finite_finite_int @ F2 )
% 9.00/5.08 | ( finite_finite_int @ G2 ) )
% 9.00/5.08 => ( finite_finite_int @ ( inf_inf_set_int @ F2 @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6151_finite__Int,axiom,
% 9.00/5.08 ! [F2: set_complex,G2: set_complex] :
% 9.00/5.08 ( ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.08 | ( finite3207457112153483333omplex @ G2 ) )
% 9.00/5.08 => ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ F2 @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6152_finite__Int,axiom,
% 9.00/5.08 ! [F2: set_Extended_enat,G2: set_Extended_enat] :
% 9.00/5.08 ( ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.08 | ( finite4001608067531595151d_enat @ G2 ) )
% 9.00/5.08 => ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ F2 @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6153_finite__Int,axiom,
% 9.00/5.08 ! [F2: set_nat,G2: set_nat] :
% 9.00/5.08 ( ( ( finite_finite_nat @ F2 )
% 9.00/5.08 | ( finite_finite_nat @ G2 ) )
% 9.00/5.08 => ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6154_finite__Int,axiom,
% 9.00/5.08 ! [F2: set_Pr1261947904930325089at_nat,G2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.08 | ( finite6177210948735845034at_nat @ G2 ) )
% 9.00/5.08 => ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ F2 @ G2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6155_Int__subset__iff,axiom,
% 9.00/5.08 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ord_less_eq_set_nat @ C2 @ A3 )
% 9.00/5.08 & ( ord_less_eq_set_nat @ C2 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6156_Int__subset__iff,axiom,
% 9.00/5.08 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
% 9.00/5.08 & ( ord_le3146513528884898305at_nat @ C2 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6157_Int__subset__iff,axiom,
% 9.00/5.08 ! [C2: set_int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( ( ord_less_eq_set_int @ C2 @ A3 )
% 9.00/5.08 & ( ord_less_eq_set_int @ C2 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6158_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6159_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6160_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6161_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6162_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6163_Int__insert__right__if1,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6164_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6165_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6166_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ~ ( member_set_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6167_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6168_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ~ ( member_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6169_Int__insert__right__if0,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6170_insert__inter__insert,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6171_insert__inter__insert,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6172_insert__inter__insert,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6173_insert__inter__insert,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6174_insert__inter__insert,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6175_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: real,C2: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6176_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: $o,C2: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6177_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6178_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: int,C2: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6179_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6180_Int__insert__left__if1,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6181_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: real,C2: set_real,B2: set_real] :
% 9.00/5.08 ( ~ ( member_real @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_real @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6182_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: $o,C2: set_o,B2: set_o] :
% 9.00/5.08 ( ~ ( member_o @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_o @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6183_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ~ ( member_set_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6184_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: int,C2: set_int,B2: set_int] :
% 9.00/5.08 ( ~ ( member_int @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_int @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6185_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 9.00/5.08 ( ~ ( member_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6186_Int__insert__left__if0,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6187_Un__Int__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6188_Un__Int__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6189_Un__Int__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6190_Un__Int__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6191_Un__Int__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6192_Un__Int__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6193_Un__Int__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6194_Un__Int__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6195_Un__Int__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ S @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6196_Un__Int__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6197_Un__Int__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ S @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6198_Un__Int__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ S @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6199_Un__Int__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ T2 @ ( sup_su6327502436637775413at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6200_Un__Int__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_nat,S: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ T2 @ ( sup_sup_set_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6201_Un__Int__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ T2 @ ( sup_su718114333110466843at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6202_Un__Int__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ T2 @ ( sup_su5525570899277871387at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6203_Int__Un__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6204_Int__Un__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6205_Int__Un__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6206_Int__Un__eq_I1_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ S )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6207_Int__Un__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6208_Int__Un__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6209_Int__Un__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6210_Int__Un__eq_I2_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S @ T2 ) @ T2 )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6211_Int__Un__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr1261947904930325089at_nat,T2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ S @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6212_Int__Un__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_nat,T2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6213_Int__Un__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr8693737435421807431at_nat,T2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ S @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6214_Int__Un__eq_I3_J,axiom,
% 9.00/5.08 ! [S: set_Pr4329608150637261639at_nat,T2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ S @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
% 9.00/5.08 = S ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6215_Int__Un__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ T2 @ ( inf_in2572325071724192079at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6216_Int__Un__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_nat,S: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ T2 @ ( inf_inf_set_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6217_Int__Un__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ T2 @ ( inf_in4302113700860409141at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6218_Int__Un__eq_I4_J,axiom,
% 9.00/5.08 ! [T2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ T2 @ ( inf_in7913087082777306421at_nat @ S @ T2 ) )
% 9.00/5.08 = T2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6219_summaxma,axiom,
% 9.00/5.08 ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.08 ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 9.00/5.08 => ( ( Mi != Ma )
% 9.00/5.08 => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
% 9.00/5.08 = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6220_inf__compl__bot__left1,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6221_inf__compl__bot__left1,axiom,
% 9.00/5.08 ! [X2: set_real,Y3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ ( inf_inf_set_real @ X2 @ Y3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6222_inf__compl__bot__left1,axiom,
% 9.00/5.08 ! [X2: set_o,Y3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ ( inf_inf_set_o @ X2 @ Y3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6223_inf__compl__bot__left1,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ ( inf_inf_set_nat @ X2 @ Y3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6224_inf__compl__bot__left1,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ ( inf_inf_set_int @ X2 @ Y3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6225_inf__compl__bot__left2,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ Y3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6226_inf__compl__bot__left2,axiom,
% 9.00/5.08 ! [X2: set_real,Y3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ Y3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6227_inf__compl__bot__left2,axiom,
% 9.00/5.08 ! [X2: set_o,Y3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ Y3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6228_inf__compl__bot__left2,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ Y3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6229_inf__compl__bot__left2,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ Y3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6230_inf__compl__bot__right,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ ( uminus6524753893492686040at_nat @ X2 ) ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6231_inf__compl__bot__right,axiom,
% 9.00/5.08 ! [X2: set_real,Y3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ X2 @ ( inf_inf_set_real @ Y3 @ ( uminus612125837232591019t_real @ X2 ) ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6232_inf__compl__bot__right,axiom,
% 9.00/5.08 ! [X2: set_o,Y3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ X2 @ ( inf_inf_set_o @ Y3 @ ( uminus_uminus_set_o @ X2 ) ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6233_inf__compl__bot__right,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ ( uminus5710092332889474511et_nat @ X2 ) ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6234_inf__compl__bot__right,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X2 ) ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6235_boolean__algebra_Oconj__cancel__left,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ X2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6236_boolean__algebra_Oconj__cancel__left,axiom,
% 9.00/5.08 ! [X2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ X2 )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6237_boolean__algebra_Oconj__cancel__left,axiom,
% 9.00/5.08 ! [X2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ X2 )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6238_boolean__algebra_Oconj__cancel__left,axiom,
% 9.00/5.08 ! [X2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ X2 )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6239_boolean__algebra_Oconj__cancel__left,axiom,
% 9.00/5.08 ! [X2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ X2 )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6240_boolean__algebra_Oconj__cancel__right,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ X2 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6241_boolean__algebra_Oconj__cancel__right,axiom,
% 9.00/5.08 ! [X2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ X2 @ ( uminus612125837232591019t_real @ X2 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6242_boolean__algebra_Oconj__cancel__right,axiom,
% 9.00/5.08 ! [X2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ X2 @ ( uminus_uminus_set_o @ X2 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6243_boolean__algebra_Oconj__cancel__right,axiom,
% 9.00/5.08 ! [X2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ X2 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6244_boolean__algebra_Oconj__cancel__right,axiom,
% 9.00/5.08 ! [X2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ X2 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6245_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B3: set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( bot_bot_set_set_nat
% 9.00/5.08 = ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member_set_nat @ B3 @ A3 )
% 9.00/5.08 & ( bot_bot_set_set_nat
% 9.00/5.08 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6246_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( bot_bo2099793752762293965at_nat
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member8440522571783428010at_nat @ B3 @ A3 )
% 9.00/5.08 & ( bot_bo2099793752762293965at_nat
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6247_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_real,B3: real,B2: set_real] :
% 9.00/5.08 ( ( bot_bot_set_real
% 9.00/5.08 = ( inf_inf_set_real @ A3 @ ( insert_real @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member_real @ B3 @ A3 )
% 9.00/5.08 & ( bot_bot_set_real
% 9.00/5.08 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6248_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_o,B3: $o,B2: set_o] :
% 9.00/5.08 ( ( bot_bot_set_o
% 9.00/5.08 = ( inf_inf_set_o @ A3 @ ( insert_o @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member_o @ B3 @ A3 )
% 9.00/5.08 & ( bot_bot_set_o
% 9.00/5.08 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6249_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_nat,B3: nat,B2: set_nat] :
% 9.00/5.08 ( ( bot_bot_set_nat
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ ( insert_nat @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member_nat @ B3 @ A3 )
% 9.00/5.08 & ( bot_bot_set_nat
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6250_disjoint__insert_I2_J,axiom,
% 9.00/5.08 ! [A3: set_int,B3: int,B2: set_int] :
% 9.00/5.08 ( ( bot_bot_set_int
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ ( insert_int @ B3 @ B2 ) ) )
% 9.00/5.08 = ( ~ ( member_int @ B3 @ A3 )
% 9.00/5.08 & ( bot_bot_set_int
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6251_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_set_nat,A2: set_nat,A3: set_set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_set_nat @ B2 @ ( insert_set_nat @ A2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_set_nat )
% 9.00/5.08 = ( ~ ( member_set_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_set_nat @ B2 @ A3 )
% 9.00/5.08 = bot_bot_set_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6252_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_Pr1261947904930325089at_nat,A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ B2 @ ( insert8211810215607154385at_nat @ A2 @ A3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_in2572325071724192079at_nat @ B2 @ A3 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6253_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_real,A2: real,A3: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ B2 @ ( insert_real @ A2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ~ ( member_real @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_real @ B2 @ A3 )
% 9.00/5.08 = bot_bot_set_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6254_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_o,A2: $o,A3: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ B2 @ ( insert_o @ A2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ~ ( member_o @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_o @ B2 @ A3 )
% 9.00/5.08 = bot_bot_set_o ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6255_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_nat,A2: nat,A3: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ B2 @ ( insert_nat @ A2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ~ ( member_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_nat @ B2 @ A3 )
% 9.00/5.08 = bot_bot_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6256_disjoint__insert_I1_J,axiom,
% 9.00/5.08 ! [B2: set_int,A2: int,A3: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ B2 @ ( insert_int @ A2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ~ ( member_int @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_int @ B2 @ A3 )
% 9.00/5.08 = bot_bot_set_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6257_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( bot_bot_set_set_nat
% 9.00/5.08 = ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member_set_nat @ A2 @ B2 )
% 9.00/5.08 & ( bot_bot_set_set_nat
% 9.00/5.08 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6258_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( bot_bo2099793752762293965at_nat
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.08 & ( bot_bo2099793752762293965at_nat
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6259_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( bot_bot_set_real
% 9.00/5.08 = ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member_real @ A2 @ B2 )
% 9.00/5.08 & ( bot_bot_set_real
% 9.00/5.08 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6260_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( bot_bot_set_o
% 9.00/5.08 = ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member_o @ A2 @ B2 )
% 9.00/5.08 & ( bot_bot_set_o
% 9.00/5.08 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6261_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( bot_bot_set_nat
% 9.00/5.08 = ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member_nat @ A2 @ B2 )
% 9.00/5.08 & ( bot_bot_set_nat
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6262_insert__disjoint_I2_J,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( bot_bot_set_int
% 9.00/5.08 = ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 ) )
% 9.00/5.08 = ( ~ ( member_int @ A2 @ B2 )
% 9.00/5.08 & ( bot_bot_set_int
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6263_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bot_set_set_nat )
% 9.00/5.08 = ( ~ ( member_set_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6264_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ~ ( member8440522571783428010at_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6265_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ ( insert_real @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ~ ( member_real @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6266_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ ( insert_o @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ~ ( member_o @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6267_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ~ ( member_nat @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6268_insert__disjoint_I1_J,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A3 ) @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ~ ( member_int @ A2 @ B2 )
% 9.00/5.08 & ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6269_Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ A3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6270_Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ A3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6271_Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ A3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6272_Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ A3 @ ( minus_minus_set_int @ B2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6273_Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6274_option_Ocollapse,axiom,
% 9.00/5.08 ! [Option: option_nat] :
% 9.00/5.08 ( ( Option != none_nat )
% 9.00/5.08 => ( ( some_nat @ ( the_nat @ Option ) )
% 9.00/5.08 = Option ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6275_option_Ocollapse,axiom,
% 9.00/5.08 ! [Option: option4927543243414619207at_nat] :
% 9.00/5.08 ( ( Option != none_P5556105721700978146at_nat )
% 9.00/5.08 => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 9.00/5.08 = Option ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6276_option_Ocollapse,axiom,
% 9.00/5.08 ! [Option: option_num] :
% 9.00/5.08 ( ( Option != none_num )
% 9.00/5.08 => ( ( some_num @ ( the_num @ Option ) )
% 9.00/5.08 = Option ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6277_Compl__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ ( uminus6524753893492686040at_nat @ A3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6278_Compl__disjoint,axiom,
% 9.00/5.08 ! [A3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ A3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6279_Compl__disjoint,axiom,
% 9.00/5.08 ! [A3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ A3 @ ( uminus_uminus_set_o @ A3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6280_Compl__disjoint,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ A3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6281_Compl__disjoint,axiom,
% 9.00/5.08 ! [A3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ A3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6282_Compl__disjoint2,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ A3 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6283_Compl__disjoint2,axiom,
% 9.00/5.08 ! [A3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( uminus612125837232591019t_real @ A3 ) @ A3 )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6284_Compl__disjoint2,axiom,
% 9.00/5.08 ! [A3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A3 ) @ A3 )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6285_Compl__disjoint2,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ A3 )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6286_Compl__disjoint2,axiom,
% 9.00/5.08 ! [A3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A3 ) @ A3 )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6287_Diff__Compl,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6288_Diff__Compl,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6289_Int__left__commute,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6290_Int__left__commute,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ B2 @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6291_Int__left__absorb,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6292_Int__left__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6293_Int__commute,axiom,
% 9.00/5.08 ( inf_inf_set_nat
% 9.00/5.08 = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ B6 @ A6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6294_Int__commute,axiom,
% 9.00/5.08 ( inf_in2572325071724192079at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B6 @ A6 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6295_Int__absorb,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ A3 )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6296_Int__absorb,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ A3 )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6297_Int__assoc,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6298_Int__assoc,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6299_IntD2,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.08 => ( member_real @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6300_IntD2,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.08 => ( member_o @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6301_IntD2,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member_set_nat @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6302_IntD2,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ( member_int @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6303_IntD2,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member_nat @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6304_IntD2,axiom,
% 9.00/5.08 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6305_IntD1,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.08 => ( member_real @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6306_IntD1,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.08 => ( member_o @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6307_IntD1,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member_set_nat @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6308_IntD1,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ( member_int @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6309_IntD1,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member_nat @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6310_IntD1,axiom,
% 9.00/5.08 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( member8440522571783428010at_nat @ C @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6311_IntE,axiom,
% 9.00/5.08 ! [C: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( member_real @ C @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member_real @ C @ A3 )
% 9.00/5.08 => ~ ( member_real @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6312_IntE,axiom,
% 9.00/5.08 ! [C: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( member_o @ C @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member_o @ C @ A3 )
% 9.00/5.08 => ~ ( member_o @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6313_IntE,axiom,
% 9.00/5.08 ! [C: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( member_set_nat @ C @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member_set_nat @ C @ A3 )
% 9.00/5.08 => ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6314_IntE,axiom,
% 9.00/5.08 ! [C: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( member_int @ C @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member_int @ C @ A3 )
% 9.00/5.08 => ~ ( member_int @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6315_IntE,axiom,
% 9.00/5.08 ! [C: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( member_nat @ C @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member_nat @ C @ A3 )
% 9.00/5.08 => ~ ( member_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6316_IntE,axiom,
% 9.00/5.08 ! [C: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ~ ( ( member8440522571783428010at_nat @ C @ A3 )
% 9.00/5.08 => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6317_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ! [X5: set_nat] :
% 9.00/5.08 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.08 => ~ ( member_set_nat @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_set_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6318_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ! [X5: product_prod_nat_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 9.00/5.08 => ~ ( member8440522571783428010at_nat @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6319_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ! [X5: real] :
% 9.00/5.08 ( ( member_real @ X5 @ A3 )
% 9.00/5.08 => ~ ( member_real @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6320_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ! [X5: $o] :
% 9.00/5.08 ( ( member_o @ X5 @ A3 )
% 9.00/5.08 => ~ ( member_o @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6321_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ! [X5: nat] :
% 9.00/5.08 ( ( member_nat @ X5 @ A3 )
% 9.00/5.08 => ~ ( member_nat @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6322_Int__emptyI,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ! [X5: int] :
% 9.00/5.08 ( ( member_int @ X5 @ A3 )
% 9.00/5.08 => ~ ( member_int @ X5 @ B2 ) )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6323_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_set_nat )
% 9.00/5.08 = ( ! [X: set_nat] :
% 9.00/5.08 ( ( member_set_nat @ X @ A3 )
% 9.00/5.08 => ~ ( member_set_nat @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6324_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ! [X: product_prod_nat_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ X @ A3 )
% 9.00/5.08 => ~ ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6325_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ! [X: real] :
% 9.00/5.08 ( ( member_real @ X @ A3 )
% 9.00/5.08 => ~ ( member_real @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6326_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ! [X: $o] :
% 9.00/5.08 ( ( member_o @ X @ A3 )
% 9.00/5.08 => ~ ( member_o @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6327_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ A3 )
% 9.00/5.08 => ~ ( member_nat @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6328_disjoint__iff,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ! [X: int] :
% 9.00/5.08 ( ( member_int @ X @ A3 )
% 9.00/5.08 => ~ ( member_int @ X @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6329_Int__empty__left,axiom,
% 9.00/5.08 ! [B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6330_Int__empty__left,axiom,
% 9.00/5.08 ! [B2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ bot_bot_set_real @ B2 )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6331_Int__empty__left,axiom,
% 9.00/5.08 ! [B2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ bot_bot_set_o @ B2 )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6332_Int__empty__left,axiom,
% 9.00/5.08 ! [B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ bot_bot_set_nat @ B2 )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6333_Int__empty__left,axiom,
% 9.00/5.08 ! [B2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ bot_bot_set_int @ B2 )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6334_Int__empty__right,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ bot_bo2099793752762293965at_nat )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6335_Int__empty__right,axiom,
% 9.00/5.08 ! [A3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ A3 @ bot_bot_set_real )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6336_Int__empty__right,axiom,
% 9.00/5.08 ! [A3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ A3 @ bot_bot_set_o )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6337_Int__empty__right,axiom,
% 9.00/5.08 ! [A3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ bot_bot_set_nat )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6338_Int__empty__right,axiom,
% 9.00/5.08 ! [A3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ A3 @ bot_bot_set_int )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6339_disjoint__iff__not__equal,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ! [X: product_prod_nat_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ X @ A3 )
% 9.00/5.08 => ! [Y: product_prod_nat_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ Y @ B2 )
% 9.00/5.08 => ( X != Y ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6340_disjoint__iff__not__equal,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ! [X: real] :
% 9.00/5.08 ( ( member_real @ X @ A3 )
% 9.00/5.08 => ! [Y: real] :
% 9.00/5.08 ( ( member_real @ Y @ B2 )
% 9.00/5.08 => ( X != Y ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6341_disjoint__iff__not__equal,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ! [X: $o] :
% 9.00/5.08 ( ( member_o @ X @ A3 )
% 9.00/5.08 => ! [Y: $o] :
% 9.00/5.08 ( ( member_o @ Y @ B2 )
% 9.00/5.08 => ( X = ~ Y ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6342_disjoint__iff__not__equal,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ! [X: nat] :
% 9.00/5.08 ( ( member_nat @ X @ A3 )
% 9.00/5.08 => ! [Y: nat] :
% 9.00/5.08 ( ( member_nat @ Y @ B2 )
% 9.00/5.08 => ( X != Y ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6343_disjoint__iff__not__equal,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ! [X: int] :
% 9.00/5.08 ( ( member_int @ X @ A3 )
% 9.00/5.08 => ! [Y: int] :
% 9.00/5.08 ( ( member_int @ Y @ B2 )
% 9.00/5.08 => ( X != Y ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6344_Int__mono,axiom,
% 9.00/5.08 ! [A3: set_nat,C2: set_nat,B2: set_nat,D4: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_nat @ B2 @ D4 )
% 9.00/5.08 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6345_Int__mono,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,D4: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_le3146513528884898305at_nat @ B2 @ D4 )
% 9.00/5.08 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6346_Int__mono,axiom,
% 9.00/5.08 ! [A3: set_int,C2: set_int,B2: set_int,D4: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ C2 )
% 9.00/5.08 => ( ( ord_less_eq_set_int @ B2 @ D4 )
% 9.00/5.08 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( inf_inf_set_int @ C2 @ D4 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6347_Int__lower1,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6348_Int__lower1,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6349_Int__lower1,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6350_Int__lower2,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6351_Int__lower2,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6352_Int__lower2,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ B2 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6353_Int__absorb1,axiom,
% 9.00/5.08 ! [B2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6354_Int__absorb1,axiom,
% 9.00/5.08 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6355_Int__absorb1,axiom,
% 9.00/5.08 ! [B2: set_int,A3: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6356_Int__absorb2,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6357_Int__absorb2,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6358_Int__absorb2,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6359_Int__greatest,axiom,
% 9.00/5.08 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ C2 @ A3 )
% 9.00/5.08 => ( ( ord_less_eq_set_nat @ C2 @ B2 )
% 9.00/5.08 => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6360_Int__greatest,axiom,
% 9.00/5.08 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ C2 @ A3 )
% 9.00/5.08 => ( ( ord_le3146513528884898305at_nat @ C2 @ B2 )
% 9.00/5.08 => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6361_Int__greatest,axiom,
% 9.00/5.08 ! [C2: set_int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ C2 @ A3 )
% 9.00/5.08 => ( ( ord_less_eq_set_int @ C2 @ B2 )
% 9.00/5.08 => ( ord_less_eq_set_int @ C2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6362_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o,P: $o > $o,Q: $o > $o] :
% 9.00/5.08 ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: $o] :
% 9.00/5.08 ( ( member_o @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B2 @ ( collect_o @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6363_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real,P: real > $o,Q: real > $o] :
% 9.00/5.08 ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: real] :
% 9.00/5.08 ( ( member_real @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B2 @ ( collect_real @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6364_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_list_nat,B2: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.08 ( ( ord_le6045566169113846134st_nat @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: list_nat] :
% 9.00/5.08 ( ( member_list_nat @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B2 @ ( collect_list_nat @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6365_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.08 ( ( ord_le6893508408891458716et_nat @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: set_nat] :
% 9.00/5.08 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B2 @ ( collect_set_nat @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6366_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,P: nat > $o,Q: nat > $o] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: nat] :
% 9.00/5.08 ( ( member_nat @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B2 @ ( collect_nat @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6367_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: product_prod_nat_nat] :
% 9.00/5.08 ( ( member8440522571783428010at_nat @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B2 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6368_Int__Collect__mono,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int,P: int > $o,Q: int > $o] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ A3 @ B2 )
% 9.00/5.08 => ( ! [X5: int] :
% 9.00/5.08 ( ( member_int @ X5 @ A3 )
% 9.00/5.08 => ( ( P @ X5 )
% 9.00/5.08 => ( Q @ X5 ) ) )
% 9.00/5.08 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B2 @ ( collect_int @ Q ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6369_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: real,A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( member_real @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_real @ A2 @ ( inf_inf_set_real @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_real @ A3 @ ( insert_real @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_real @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6370_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: $o,A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( member_o @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_o @ A2 @ ( inf_inf_set_o @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_o @ A3 @ ( insert_o @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_o @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6371_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: set_nat,A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_set_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ A3 @ ( insert_set_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6372_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: int,A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( member_int @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_int @ A2 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_int @ A3 @ ( insert_int @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6373_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( member_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ A3 @ ( insert_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6374_Int__insert__right,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) )
% 9.00/5.08 & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6375_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: real,C2: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( member_real @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_real @ A2 @ ( inf_inf_set_real @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_real @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_real @ ( insert_real @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_real @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6376_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: $o,C2: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( member_o @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_o @ A2 @ ( inf_inf_set_o @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_o @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_o @ ( insert_o @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_o @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6377_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: set_nat,C2: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( ( member_set_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_set_nat @ A2 @ ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_set_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_set_nat @ ( insert_set_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_set_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6378_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: int,C2: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( member_int @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_int @ A2 @ ( inf_inf_set_int @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_int @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_int @ ( insert_int @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_int @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6379_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: nat,C2: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( member_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6380_Int__insert__left,axiom,
% 9.00/5.08 ! [A2: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( member8440522571783428010at_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( insert8211810215607154385at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) )
% 9.00/5.08 & ( ~ ( member8440522571783428010at_nat @ A2 @ C2 )
% 9.00/5.08 => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A2 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6381_Un__Int__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ A3 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6382_Un__Int__distrib2,axiom,
% 9.00/5.08 ! [B2: set_nat,C2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ A3 ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6383_Un__Int__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ A3 ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6384_Un__Int__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ A3 ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6385_Int__Un__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6386_Int__Un__distrib2,axiom,
% 9.00/5.08 ! [B2: set_nat,C2: set_nat,A3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6387_Int__Un__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat,A3: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ B2 @ A3 ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6388_Int__Un__distrib2,axiom,
% 9.00/5.08 ! [B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A3: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) @ A3 )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B2 @ A3 ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6389_Un__Int__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6390_Un__Int__distrib,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6391_Un__Int__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6392_Un__Int__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6393_Int__Un__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6394_Int__Un__distrib,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6395_Int__Un__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6396_Int__Un__distrib,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6397_Un__Int__crazy,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6398_Un__Int__crazy,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A3 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A3 @ B2 ) @ ( sup_sup_set_nat @ B2 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6399_Un__Int__crazy,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) ) @ ( inf_in4302113700860409141at_nat @ C2 @ A3 ) )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) @ ( sup_su718114333110466843at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6400_Un__Int__crazy,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A3 ) )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6401_Diff__Int__distrib2,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6402_Diff__Int__distrib2,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6403_Diff__Int__distrib,axiom,
% 9.00/5.08 ! [C2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A3 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6404_Diff__Int__distrib,axiom,
% 9.00/5.08 ! [C2: set_nat,A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A3 ) @ ( inf_inf_set_nat @ C2 @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6405_Diff__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ A3 @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6406_Diff__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ A3 @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6407_Diff__Int2,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ C2 ) @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6408_Diff__Int2,axiom,
% 9.00/5.08 ! [A3: set_nat,C2: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ C2 ) @ B2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6409_Int__Diff,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ ( minus_1356011639430497352at_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6410_Int__Diff,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ ( minus_minus_set_nat @ B2 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6411_option_Osel,axiom,
% 9.00/5.08 ! [X22: nat] :
% 9.00/5.08 ( ( the_nat @ ( some_nat @ X22 ) )
% 9.00/5.08 = X22 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6412_option_Osel,axiom,
% 9.00/5.08 ! [X22: product_prod_nat_nat] :
% 9.00/5.08 ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.00/5.08 = X22 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6413_option_Osel,axiom,
% 9.00/5.08 ! [X22: num] :
% 9.00/5.08 ( ( the_num @ ( some_num @ X22 ) )
% 9.00/5.08 = X22 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6414_option_Oexpand,axiom,
% 9.00/5.08 ! [Option: option_nat,Option2: option_nat] :
% 9.00/5.08 ( ( ( Option = none_nat )
% 9.00/5.08 = ( Option2 = none_nat ) )
% 9.00/5.08 => ( ( ( Option != none_nat )
% 9.00/5.08 => ( ( Option2 != none_nat )
% 9.00/5.08 => ( ( the_nat @ Option )
% 9.00/5.08 = ( the_nat @ Option2 ) ) ) )
% 9.00/5.08 => ( Option = Option2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6415_option_Oexpand,axiom,
% 9.00/5.08 ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 9.00/5.08 ( ( ( Option = none_P5556105721700978146at_nat )
% 9.00/5.08 = ( Option2 = none_P5556105721700978146at_nat ) )
% 9.00/5.08 => ( ( ( Option != none_P5556105721700978146at_nat )
% 9.00/5.08 => ( ( Option2 != none_P5556105721700978146at_nat )
% 9.00/5.08 => ( ( the_Pr8591224930841456533at_nat @ Option )
% 9.00/5.08 = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 9.00/5.08 => ( Option = Option2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6416_option_Oexpand,axiom,
% 9.00/5.08 ! [Option: option_num,Option2: option_num] :
% 9.00/5.08 ( ( ( Option = none_num )
% 9.00/5.08 = ( Option2 = none_num ) )
% 9.00/5.08 => ( ( ( Option != none_num )
% 9.00/5.08 => ( ( Option2 != none_num )
% 9.00/5.08 => ( ( the_num @ Option )
% 9.00/5.08 = ( the_num @ Option2 ) ) ) )
% 9.00/5.08 => ( Option = Option2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6417_inf__cancel__left1,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ A2 ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ B3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6418_inf__cancel__left1,axiom,
% 9.00/5.08 ! [X2: set_real,A2: set_real,B3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( inf_inf_set_real @ X2 @ A2 ) @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ B3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6419_inf__cancel__left1,axiom,
% 9.00/5.08 ! [X2: set_o,A2: set_o,B3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( inf_inf_set_o @ X2 @ A2 ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ B3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6420_inf__cancel__left1,axiom,
% 9.00/5.08 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X2 @ A2 ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ B3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6421_inf__cancel__left1,axiom,
% 9.00/5.08 ! [X2: set_int,A2: set_int,B3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( inf_inf_set_int @ X2 @ A2 ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ B3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6422_inf__cancel__left2,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X2 ) @ A2 ) @ ( inf_in2572325071724192079at_nat @ X2 @ B3 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6423_inf__cancel__left2,axiom,
% 9.00/5.08 ! [X2: set_real,A2: set_real,B3: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( inf_inf_set_real @ ( uminus612125837232591019t_real @ X2 ) @ A2 ) @ ( inf_inf_set_real @ X2 @ B3 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6424_inf__cancel__left2,axiom,
% 9.00/5.08 ! [X2: set_o,A2: set_o,B3: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X2 ) @ A2 ) @ ( inf_inf_set_o @ X2 @ B3 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6425_inf__cancel__left2,axiom,
% 9.00/5.08 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X2 ) @ A2 ) @ ( inf_inf_set_nat @ X2 @ B3 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6426_inf__cancel__left2,axiom,
% 9.00/5.08 ! [X2: set_int,A2: set_int,B3: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X2 ) @ A2 ) @ ( inf_inf_set_int @ X2 @ B3 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6427_Diff__triv,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 => ( ( minus_1356011639430497352at_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6428_Diff__triv,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 => ( ( minus_minus_set_real @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6429_Diff__triv,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 => ( ( minus_minus_set_o @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6430_Diff__triv,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 => ( ( minus_minus_set_int @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6431_Diff__triv,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 => ( ( minus_minus_set_nat @ A3 @ B2 )
% 9.00/5.08 = A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6432_Int__Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.08 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6433_Int__Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( inf_inf_set_real @ ( inf_inf_set_real @ A3 @ B2 ) @ ( minus_minus_set_real @ A3 @ B2 ) )
% 9.00/5.08 = bot_bot_set_real ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6434_Int__Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( inf_inf_set_o @ ( inf_inf_set_o @ A3 @ B2 ) @ ( minus_minus_set_o @ A3 @ B2 ) )
% 9.00/5.08 = bot_bot_set_o ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6435_Int__Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( inf_inf_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.08 = bot_bot_set_int ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6436_Int__Diff__disjoint,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = bot_bot_set_nat ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6437_Un__Int__assoc__eq,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) ) )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ C2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6438_Un__Int__assoc__eq,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) ) )
% 9.00/5.08 = ( ord_less_eq_set_nat @ C2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6439_Un__Int__assoc__eq,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) ) )
% 9.00/5.08 = ( ord_le3000389064537975527at_nat @ C2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6440_Un__Int__assoc__eq,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) ) )
% 9.00/5.08 = ( ord_le1268244103169919719at_nat @ C2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6441_Un__Int__assoc__eq,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int,C2: set_int] :
% 9.00/5.08 ( ( ( sup_sup_set_int @ ( inf_inf_set_int @ A3 @ B2 ) @ C2 )
% 9.00/5.08 = ( inf_inf_set_int @ A3 @ ( sup_sup_set_int @ B2 @ C2 ) ) )
% 9.00/5.08 = ( ord_less_eq_set_int @ C2 @ A3 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6442_Un__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6443_Un__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6444_Un__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6445_Un__Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6446_Int__Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6447_Int__Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6448_Int__Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6449_Int__Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = A3 ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6450_Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ A3 @ ( inf_in2572325071724192079at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6451_Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( minus_8321449233255521966at_nat @ A3 @ ( inf_in4302113700860409141at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6452_Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( minus_3314409938677909166at_nat @ A3 @ ( inf_in7913087082777306421at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6453_Diff__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ A3 @ ( inf_inf_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6454_Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( minus_1356011639430497352at_nat @ A3 @ ( sup_su6327502436637775413at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) @ ( minus_1356011639430497352at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6455_Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( minus_8321449233255521966at_nat @ A3 @ ( sup_su718114333110466843at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ ( minus_8321449233255521966at_nat @ A3 @ B2 ) @ ( minus_8321449233255521966at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6456_Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( minus_3314409938677909166at_nat @ A3 @ ( sup_su5525570899277871387at_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A3 @ B2 ) @ ( minus_3314409938677909166at_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6457_Diff__Un,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat,C2: set_nat] :
% 9.00/5.08 ( ( minus_minus_set_nat @ A3 @ ( sup_sup_set_nat @ B2 @ C2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A3 @ B2 ) @ ( minus_minus_set_nat @ A3 @ C2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6458_Compl__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6459_Compl__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6460_Compl__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( uminus4384627049435823934at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6461_Compl__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6462_Compl__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A3 ) @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6463_Compl__Un,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A3 ) @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6464_Compl__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( uminus4384627049435823934at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_in4302113700860409141at_nat @ ( uminus4384627049435823934at_nat @ A3 ) @ ( uminus4384627049435823934at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6465_Compl__Un,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A3 ) @ ( uminus935396558254630718at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6466_Diff__eq,axiom,
% 9.00/5.08 ( minus_1356011639430497352at_nat
% 9.00/5.08 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ A6 @ ( uminus6524753893492686040at_nat @ B6 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6467_Diff__eq,axiom,
% 9.00/5.08 ( minus_minus_set_nat
% 9.00/5.08 = ( ^ [A6: set_nat,B6: set_nat] : ( inf_inf_set_nat @ A6 @ ( uminus5710092332889474511et_nat @ B6 ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6468_option_Oexhaust__sel,axiom,
% 9.00/5.08 ! [Option: option_nat] :
% 9.00/5.08 ( ( Option != none_nat )
% 9.00/5.08 => ( Option
% 9.00/5.08 = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6469_option_Oexhaust__sel,axiom,
% 9.00/5.08 ! [Option: option4927543243414619207at_nat] :
% 9.00/5.08 ( ( Option != none_P5556105721700978146at_nat )
% 9.00/5.08 => ( Option
% 9.00/5.08 = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6470_option_Oexhaust__sel,axiom,
% 9.00/5.08 ! [Option: option_num] :
% 9.00/5.08 ( ( Option != none_num )
% 9.00/5.08 => ( Option
% 9.00/5.08 = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6471_inf__shunt,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6472_inf__shunt,axiom,
% 9.00/5.08 ! [X2: set_real,Y3: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ X2 @ Y3 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ord_less_eq_set_real @ X2 @ ( uminus612125837232591019t_real @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6473_inf__shunt,axiom,
% 9.00/5.08 ! [X2: set_o,Y3: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ X2 @ Y3 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ord_less_eq_set_o @ X2 @ ( uminus_uminus_set_o @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6474_inf__shunt,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ X2 @ Y3 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ord_less_eq_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6475_inf__shunt,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ X2 @ Y3 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ord_less_eq_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6476_shunt1,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y3 ) @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6477_shunt1,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y3 ) @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6478_shunt1,axiom,
% 9.00/5.08 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.08 = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ Y3 ) @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6479_shunt1,axiom,
% 9.00/5.08 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.08 = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y3 ) @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6480_shunt1,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Z )
% 9.00/5.08 = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6481_shunt2,axiom,
% 9.00/5.08 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ ( uminus6524753893492686040at_nat @ Y3 ) ) @ Z )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6482_shunt2,axiom,
% 9.00/5.08 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ ( uminus5710092332889474511et_nat @ Y3 ) ) @ Z )
% 9.00/5.08 = ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6483_shunt2,axiom,
% 9.00/5.08 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ ( uminus4384627049435823934at_nat @ Y3 ) ) @ Z )
% 9.00/5.08 = ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6484_shunt2,axiom,
% 9.00/5.08 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ ( uminus935396558254630718at_nat @ Y3 ) ) @ Z )
% 9.00/5.08 = ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6485_shunt2,axiom,
% 9.00/5.08 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ ( uminus1532241313380277803et_int @ Y3 ) ) @ Z )
% 9.00/5.08 = ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6486_sup__neg__inf,axiom,
% 9.00/5.08 ! [P6: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ord_le3146513528884898305at_nat @ P6 @ ( sup_su6327502436637775413at_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P6 @ ( uminus6524753893492686040at_nat @ Q3 ) ) @ R2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6487_sup__neg__inf,axiom,
% 9.00/5.08 ! [P6: set_nat,Q3: set_nat,R2: set_nat] :
% 9.00/5.08 ( ( ord_less_eq_set_nat @ P6 @ ( sup_sup_set_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P6 @ ( uminus5710092332889474511et_nat @ Q3 ) ) @ R2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6488_sup__neg__inf,axiom,
% 9.00/5.08 ! [P6: set_Pr8693737435421807431at_nat,Q3: set_Pr8693737435421807431at_nat,R2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( ord_le3000389064537975527at_nat @ P6 @ ( sup_su718114333110466843at_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( ord_le3000389064537975527at_nat @ ( inf_in4302113700860409141at_nat @ P6 @ ( uminus4384627049435823934at_nat @ Q3 ) ) @ R2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6489_sup__neg__inf,axiom,
% 9.00/5.08 ! [P6: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( ord_le1268244103169919719at_nat @ P6 @ ( sup_su5525570899277871387at_nat @ Q3 @ R2 ) )
% 9.00/5.08 = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P6 @ ( uminus935396558254630718at_nat @ Q3 ) ) @ R2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6490_sup__neg__inf,axiom,
% 9.00/5.08 ! [P6: set_int,Q3: set_int,R2: set_int] :
% 9.00/5.08 ( ( ord_less_eq_set_int @ P6 @ ( sup_sup_set_int @ Q3 @ R2 ) )
% 9.00/5.08 = ( ord_less_eq_set_int @ ( inf_inf_set_int @ P6 @ ( uminus1532241313380277803et_int @ Q3 ) ) @ R2 ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6491_disjoint__eq__subset__Compl,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 = ( ord_le3146513528884898305at_nat @ A3 @ ( uminus6524753893492686040at_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6492_disjoint__eq__subset__Compl,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 = ( ord_less_eq_set_real @ A3 @ ( uminus612125837232591019t_real @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6493_disjoint__eq__subset__Compl,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 = ( ord_less_eq_set_o @ A3 @ ( uminus_uminus_set_o @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6494_disjoint__eq__subset__Compl,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 = ( ord_less_eq_set_nat @ A3 @ ( uminus5710092332889474511et_nat @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6495_disjoint__eq__subset__Compl,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 = ( ord_less_eq_set_int @ A3 @ ( uminus1532241313380277803et_int @ B2 ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6496_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.08 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.08 => ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6497_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.08 => ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6498_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( finite_finite_int @ A3 )
% 9.00/5.08 => ( ( finite_finite_int @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6499_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.08 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.08 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6500_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.08 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.08 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6501_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.08 => ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6502_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( finite_finite_nat @ A3 )
% 9.00/5.08 => ( ( finite_finite_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6503_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( finite4392333629123659920at_nat @ A3 )
% 9.00/5.08 => ( ( finite4392333629123659920at_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) ) @ ( finite1207074278014112911at_nat @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6504_card__Un__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.08 ( ( finite4343798906461161616at_nat @ A3 )
% 9.00/5.08 => ( ( finite4343798906461161616at_nat @ B2 )
% 9.00/5.08 => ( ( plus_plus_nat @ ( finite3771342082235030671at_nat @ A3 ) @ ( finite3771342082235030671at_nat @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite3771342082235030671at_nat @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) ) @ ( finite3771342082235030671at_nat @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6505_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.08 ( ( finite8100373058378681591st_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ ( inf_inf_set_list_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6506_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( finite1152437895449049373et_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ ( inf_inf_set_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6507_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( finite_finite_int @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite_card_int @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6508_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.08 ( ( finite3207457112153483333omplex @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite_card_complex @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6509_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.08 ( ( finite4001608067531595151d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite121521170596916366d_enat @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6510_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( finite6177210948735845034at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6511_card__Diff__subset__Int,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( finite_finite_nat @ ( inf_inf_set_nat @ A3 @ B2 ) )
% 9.00/5.08 => ( ( finite_card_nat @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( minus_minus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6512_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.08 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.08 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_complex )
% 9.00/5.08 => ( ( finite_card_complex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_complex @ A3 ) @ ( finite_card_complex @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6513_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.08 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.08 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.08 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 9.00/5.08 = bot_bo7653980558646680370d_enat )
% 9.00/5.08 => ( ( finite121521170596916366d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite121521170596916366d_enat @ A3 ) @ ( finite121521170596916366d_enat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6514_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_real,B2: set_real] :
% 9.00/5.08 ( ( finite_finite_real @ A3 )
% 9.00/5.08 => ( ( finite_finite_real @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_real )
% 9.00/5.08 => ( ( finite_card_real @ ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_real @ A3 ) @ ( finite_card_real @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6515_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_o,B2: set_o] :
% 9.00/5.08 ( ( finite_finite_o @ A3 )
% 9.00/5.08 => ( ( finite_finite_o @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_o @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_o )
% 9.00/5.08 => ( ( finite_card_o @ ( sup_sup_set_o @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_o @ A3 ) @ ( finite_card_o @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6516_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.08 ( ( finite_finite_nat @ A3 )
% 9.00/5.08 => ( ( finite_finite_nat @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_nat )
% 9.00/5.08 => ( ( finite_card_nat @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_nat @ A3 ) @ ( finite_card_nat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6517_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_int,B2: set_int] :
% 9.00/5.08 ( ( finite_finite_int @ A3 )
% 9.00/5.08 => ( ( finite_finite_int @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_int @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_int )
% 9.00/5.08 => ( ( finite_card_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_int @ A3 ) @ ( finite_card_int @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6518_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_list_nat,B2: set_list_nat] :
% 9.00/5.08 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.08 => ( ( finite8100373058378681591st_nat @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_list_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_list_nat )
% 9.00/5.08 => ( ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_list_nat @ A3 ) @ ( finite_card_list_nat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6519_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_set_nat,B2: set_set_nat] :
% 9.00/5.08 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.08 => ( ( finite1152437895449049373et_nat @ B2 )
% 9.00/5.08 => ( ( ( inf_inf_set_set_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bot_set_set_nat )
% 9.00/5.08 => ( ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite_card_set_nat @ A3 ) @ ( finite_card_set_nat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6520_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.08 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.08 => ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.08 => ( ( ( inf_in2572325071724192079at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo2099793752762293965at_nat )
% 9.00/5.08 => ( ( finite711546835091564841at_nat @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite711546835091564841at_nat @ A3 ) @ ( finite711546835091564841at_nat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6521_card__Un__disjoint,axiom,
% 9.00/5.08 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
% 9.00/5.08 ( ( finite4392333629123659920at_nat @ A3 )
% 9.00/5.08 => ( ( finite4392333629123659920at_nat @ B2 )
% 9.00/5.08 => ( ( ( inf_in4302113700860409141at_nat @ A3 @ B2 )
% 9.00/5.08 = bot_bo5327735625951526323at_nat )
% 9.00/5.08 => ( ( finite1207074278014112911at_nat @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.08 = ( plus_plus_nat @ ( finite1207074278014112911at_nat @ A3 ) @ ( finite1207074278014112911at_nat @ B2 ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6522_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 9.00/5.08 ! [X2: nat,N: nat,M: nat] :
% 9.00/5.08 ( ( ord_less_nat @ X2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M ) ) )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.08 => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.08 => ( ord_less_nat @ ( vEBT_VEBT_low @ X2 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6523_invar__vebt_Ointros_I4_J,axiom,
% 9.00/5.08 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.00/5.08 ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.00/5.08 => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.00/5.08 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( M = N )
% 9.00/5.08 => ( ( Deg
% 9.00/5.08 = ( plus_plus_nat @ N @ M ) )
% 9.00/5.08 => ( ! [I2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 9.00/5.08 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.00/5.08 => ( ( ( Mi = Ma )
% 9.00/5.08 => ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.00/5.08 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.00/5.08 => ( ( ( Mi != Ma )
% 9.00/5.08 => ! [I2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.00/5.08 = I2 )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.00/5.08 & ! [X5: nat] :
% 9.00/5.08 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 9.00/5.08 = I2 )
% 9.00/5.08 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 9.00/5.08 => ( ( ord_less_nat @ Mi @ X5 )
% 9.00/5.08 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6524_invar__vebt_Ointros_I5_J,axiom,
% 9.00/5.08 ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.00/5.08 ( ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X5 @ N ) )
% 9.00/5.08 => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.00/5.08 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( M
% 9.00/5.08 = ( suc @ N ) )
% 9.00/5.08 => ( ( Deg
% 9.00/5.08 = ( plus_plus_nat @ N @ M ) )
% 9.00/5.08 => ( ! [I2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
% 9.00/5.08 = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.00/5.08 => ( ( ( Mi = Ma )
% 9.00/5.08 => ! [X5: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.08 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
% 9.00/5.08 => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.00/5.08 => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.00/5.08 => ( ( ( Mi != Ma )
% 9.00/5.08 => ! [I2: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.00/5.08 => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
% 9.00/5.08 = I2 )
% 9.00/5.08 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
% 9.00/5.08 & ! [X5: nat] :
% 9.00/5.08 ( ( ( ( vEBT_VEBT_high @ X5 @ N )
% 9.00/5.08 = I2 )
% 9.00/5.08 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
% 9.00/5.08 => ( ( ord_less_nat @ Mi @ X5 )
% 9.00/5.08 & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.08
% 9.00/5.08 thf(fact_6525_invar__vebt_Osimps,axiom,
% 9.00/5.08 ( vEBT_invar_vebt
% 9.00/5.08 = ( ^ [A12: vEBT_VEBT,A23: nat] :
% 9.00/5.08 ( ( ? [A4: $o,B4: $o] :
% 9.00/5.08 ( A12
% 9.00/5.08 = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.00/5.08 & ( A23
% 9.00/5.08 = ( suc @ zero_zero_nat ) ) )
% 9.00/5.08 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
% 9.00/5.08 ( ( A12
% 9.00/5.08 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
% 9.00/5.08 & ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.00/5.08 & ( vEBT_invar_vebt @ Summary4 @ N2 )
% 9.00/5.08 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.00/5.08 & ( A23
% 9.00/5.08 = ( plus_plus_nat @ N2 @ N2 ) )
% 9.00/5.08 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
% 9.00/5.08 & ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.08 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT] :
% 9.00/5.08 ( ( A12
% 9.00/5.08 = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
% 9.00/5.08 & ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.00/5.08 & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
% 9.00/5.08 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.00/5.08 & ( A23
% 9.00/5.08 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.00/5.08 & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
% 9.00/5.08 & ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.08 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.00/5.08 ( ( A12
% 9.00/5.08 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
% 9.00/5.08 & ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.00/5.08 & ( vEBT_invar_vebt @ Summary4 @ N2 )
% 9.00/5.08 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 9.00/5.08 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.00/5.08 & ( A23
% 9.00/5.08 = ( plus_plus_nat @ N2 @ N2 ) )
% 9.00/5.08 & ! [I4: nat] :
% 9.00/5.08 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.00/5.08 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
% 9.00/5.08 = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
% 9.00/5.08 & ( ( Mi3 = Ma3 )
% 9.00/5.08 => ! [X: vEBT_VEBT] :
% 9.00/5.08 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.08 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.09 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.09 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 9.00/5.09 & ( ( Mi3 != Ma3 )
% 9.00/5.09 => ! [I4: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.00/5.09 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 9.00/5.09 = I4 )
% 9.00/5.09 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 9.00/5.09 & ! [X: nat] :
% 9.00/5.09 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 9.00/5.09 = I4 )
% 9.00/5.09 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 9.00/5.09 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.09 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 9.00/5.09 | ? [TreeList4: list_VEBT_VEBT,N2: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.00/5.09 ( ( A12
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
% 9.00/5.09 & ! [X: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.09 => ( vEBT_invar_vebt @ X @ N2 ) )
% 9.00/5.09 & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N2 ) )
% 9.00/5.09 & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
% 9.00/5.09 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.00/5.09 & ( A23
% 9.00/5.09 = ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) )
% 9.00/5.09 & ! [I4: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.00/5.09 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
% 9.00/5.09 = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
% 9.00/5.09 & ( ( Mi3 = Ma3 )
% 9.00/5.09 => ! [X: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList4 ) )
% 9.00/5.09 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.09 & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.09 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
% 9.00/5.09 & ( ( Mi3 != Ma3 )
% 9.00/5.09 => ! [I4: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) )
% 9.00/5.09 => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
% 9.00/5.09 = I4 )
% 9.00/5.09 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
% 9.00/5.09 & ! [X: nat] :
% 9.00/5.09 ( ( ( ( vEBT_VEBT_high @ X @ N2 )
% 9.00/5.09 = I4 )
% 9.00/5.09 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
% 9.00/5.09 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.09 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6526_invar__vebt_Ocases,axiom,
% 9.00/5.09 ! [A13: vEBT_VEBT,A24: nat] :
% 9.00/5.09 ( ( vEBT_invar_vebt @ A13 @ A24 )
% 9.00/5.09 => ( ( ? [A: $o,B: $o] :
% 9.00/5.09 ( A13
% 9.00/5.09 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.09 => ( A24
% 9.00/5.09 != ( suc @ zero_zero_nat ) ) )
% 9.00/5.09 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 9.00/5.09 ( ( A13
% 9.00/5.09 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.09 => ( ( A24 = Deg2 )
% 9.00/5.09 => ( ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.00/5.09 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.00/5.09 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.09 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( M4 = N3 )
% 9.00/5.09 => ( ( Deg2
% 9.00/5.09 = ( plus_plus_nat @ N3 @ M4 ) )
% 9.00/5.09 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.00/5.09 => ~ ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
% 9.00/5.09 ( ( A13
% 9.00/5.09 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.09 => ( ( A24 = Deg2 )
% 9.00/5.09 => ( ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.00/5.09 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.00/5.09 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.09 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( M4
% 9.00/5.09 = ( suc @ N3 ) )
% 9.00/5.09 => ( ( Deg2
% 9.00/5.09 = ( plus_plus_nat @ N3 @ M4 ) )
% 9.00/5.09 => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.00/5.09 => ~ ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.00/5.09 ( ( A13
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.09 => ( ( A24 = Deg2 )
% 9.00/5.09 => ( ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.00/5.09 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.00/5.09 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.09 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( M4 = N3 )
% 9.00/5.09 => ( ( Deg2
% 9.00/5.09 = ( plus_plus_nat @ N3 @ M4 ) )
% 9.00/5.09 => ( ! [I3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 9.00/5.09 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.00/5.09 => ( ( ( Mi2 = Ma2 )
% 9.00/5.09 => ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.00/5.09 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.09 => ~ ( ( Mi2 != Ma2 )
% 9.00/5.09 => ! [I3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.00/5.09 = I3 )
% 9.00/5.09 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.00/5.09 & ! [X4: nat] :
% 9.00/5.09 ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 9.00/5.09 = I3 )
% 9.00/5.09 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 9.00/5.09 => ( ( ord_less_nat @ Mi2 @ X4 )
% 9.00/5.09 & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 => ~ ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.00/5.09 ( ( A13
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.09 => ( ( A24 = Deg2 )
% 9.00/5.09 => ( ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ( vEBT_invar_vebt @ X4 @ N3 ) )
% 9.00/5.09 => ( ( vEBT_invar_vebt @ Summary2 @ M4 )
% 9.00/5.09 => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.09 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( M4
% 9.00/5.09 = ( suc @ N3 ) )
% 9.00/5.09 => ( ( Deg2
% 9.00/5.09 = ( plus_plus_nat @ N3 @ M4 ) )
% 9.00/5.09 => ( ! [I3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
% 9.00/5.09 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
% 9.00/5.09 => ( ( ( Mi2 = Ma2 )
% 9.00/5.09 => ! [X4: vEBT_VEBT] :
% 9.00/5.09 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.09 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.00/5.09 => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.09 => ~ ( ( Mi2 != Ma2 )
% 9.00/5.09 => ! [I3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M4 ) )
% 9.00/5.09 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.00/5.09 = I3 )
% 9.00/5.09 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.00/5.09 & ! [X4: nat] :
% 9.00/5.09 ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
% 9.00/5.09 = I3 )
% 9.00/5.09 & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
% 9.00/5.09 => ( ( ord_less_nat @ Mi2 @ X4 )
% 9.00/5.09 & ( ord_less_eq_nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6527_in__children__def,axiom,
% 9.00/5.09 ( vEBT_V5917875025757280293ildren
% 9.00/5.09 = ( ^ [N2: nat,TreeList4: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X @ N2 ) ) @ ( vEBT_VEBT_low @ X @ N2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6528_del__x__mi__lets__in__not__minNull,axiom,
% 9.00/5.09 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ( X2 = Mi )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( Xn
% 9.00/5.09 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6529_del__x__not__mi__newnode__not__nil,axiom,
% 9.00/5.09 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ( ord_less_nat @ Mi @ X2 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6530_set__bit__0,axiom,
% 9.00/5.09 ! [A2: code_integer] :
% 9.00/5.09 ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6531_set__bit__0,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6532_set__bit__0,axiom,
% 9.00/5.09 ! [A2: int] :
% 9.00/5.09 ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6533_enat__ord__number_I1_J,axiom,
% 9.00/5.09 ! [M: num,N: num] :
% 9.00/5.09 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.00/5.09 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6534_set__bit__nonnegative__int__iff,axiom,
% 9.00/5.09 ! [N: nat,K: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N @ K ) )
% 9.00/5.09 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6535_set__bit__negative__int__iff,axiom,
% 9.00/5.09 ! [N: nat,K: int] :
% 9.00/5.09 ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.09 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6536_list__update__beyond,axiom,
% 9.00/5.09 ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 9.00/5.09 => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
% 9.00/5.09 = Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6537_list__update__beyond,axiom,
% 9.00/5.09 ! [Xs: list_o,I: nat,X2: $o] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 9.00/5.09 => ( ( list_update_o @ Xs @ I @ X2 )
% 9.00/5.09 = Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6538_list__update__beyond,axiom,
% 9.00/5.09 ! [Xs: list_nat,I: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 9.00/5.09 => ( ( list_update_nat @ Xs @ I @ X2 )
% 9.00/5.09 = Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6539_enat__ord__number_I2_J,axiom,
% 9.00/5.09 ! [M: num,N: num] :
% 9.00/5.09 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N ) )
% 9.00/5.09 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6540_nth__list__update__eq,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_int,X2: int] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ I )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6541_nth__list__update__eq,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ I )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6542_nth__list__update__eq,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_o,X2: $o] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ I )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6543_nth__list__update__eq,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ I )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6544_set__swap,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_int,J: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 9.00/5.09 = ( set_int2 @ Xs ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6545_set__swap,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 9.00/5.09 = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6546_set__swap,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_o,J: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 9.00/5.09 = ( set_o2 @ Xs ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6547_set__swap,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_nat,J: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 9.00/5.09 = ( set_nat2 @ Xs ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6548_add__diff__assoc__enat,axiom,
% 9.00/5.09 ! [Z: extended_enat,Y3: extended_enat,X2: extended_enat] :
% 9.00/5.09 ( ( ord_le2932123472753598470d_enat @ Z @ Y3 )
% 9.00/5.09 => ( ( plus_p3455044024723400733d_enat @ X2 @ ( minus_3235023915231533773d_enat @ Y3 @ Z ) )
% 9.00/5.09 = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6549_ile0__eq,axiom,
% 9.00/5.09 ! [N: extended_enat] :
% 9.00/5.09 ( ( ord_le2932123472753598470d_enat @ N @ zero_z5237406670263579293d_enat )
% 9.00/5.09 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6550_i0__lb,axiom,
% 9.00/5.09 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6551_set__bit__greater__eq,axiom,
% 9.00/5.09 ! [K: int,N: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N @ K ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6552_list__update__code_I2_J,axiom,
% 9.00/5.09 ! [X2: int,Xs: list_int,Y3: int] :
% 9.00/5.09 ( ( list_update_int @ ( cons_int @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 9.00/5.09 = ( cons_int @ Y3 @ Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6553_list__update__code_I2_J,axiom,
% 9.00/5.09 ! [X2: nat,Xs: list_nat,Y3: nat] :
% 9.00/5.09 ( ( list_update_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 9.00/5.09 = ( cons_nat @ Y3 @ Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6554_list__update__code_I2_J,axiom,
% 9.00/5.09 ! [X2: vEBT_VEBT,Xs: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 9.00/5.09 ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X2 @ Xs ) @ zero_zero_nat @ Y3 )
% 9.00/5.09 = ( cons_VEBT_VEBT @ Y3 @ Xs ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6555_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_real,A3: set_real,X2: real,I: nat] :
% 9.00/5.09 ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_real @ X2 @ A3 )
% 9.00/5.09 => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6556_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_o,A3: set_o,X2: $o,I: nat] :
% 9.00/5.09 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_o @ X2 @ A3 )
% 9.00/5.09 => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6557_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_set_nat,A3: set_set_nat,X2: set_nat,I: nat] :
% 9.00/5.09 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.09 => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6558_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_nat,A3: set_nat,X2: nat,I: nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_nat @ X2 @ A3 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6559_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_VEBT_VEBT,A3: set_VEBT_VEBT,X2: vEBT_VEBT,I: nat] :
% 9.00/5.09 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_VEBT_VEBT @ X2 @ A3 )
% 9.00/5.09 => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6560_set__update__subsetI,axiom,
% 9.00/5.09 ! [Xs: list_int,A3: set_int,X2: int,I: nat] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A3 )
% 9.00/5.09 => ( ( member_int @ X2 @ A3 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ A3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6561_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_P6011104703257516679at_nat,I: nat,X2: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X2 ) ) @ ( insert8211810215607154385at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6562_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_real,I: nat,X2: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X2 ) ) @ ( insert_real @ X2 @ ( set_real2 @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6563_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_o,I: nat,X2: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs @ I @ X2 ) ) @ ( insert_o @ X2 @ ( set_o2 @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6564_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_nat,I: nat,X2: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X2 ) ) @ ( insert_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6565_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_VEBT_VEBT,I: nat,X2: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) ) @ ( insert_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6566_set__update__subset__insert,axiom,
% 9.00/5.09 ! [Xs: list_int,I: nat,X2: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X2 ) ) @ ( insert_int @ X2 @ ( set_int2 @ Xs ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6567_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_real,X2: real] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs ) )
% 9.00/5.09 => ( member_real @ X2 @ ( set_real2 @ ( list_update_real @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6568_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_set_nat,X2: set_nat] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.00/5.09 => ( member_set_nat @ X2 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6569_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_int,X2: int] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( member_int @ X2 @ ( set_int2 @ ( list_update_int @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6570_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6571_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_o,X2: $o] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( member_o @ X2 @ ( set_o2 @ ( list_update_o @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6572_set__update__memI,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( member_nat @ X2 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6573_nth__list__update,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_int,J: nat,X2: int] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( ( ( I = J )
% 9.00/5.09 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = X2 ) )
% 9.00/5.09 & ( ( I != J )
% 9.00/5.09 => ( ( nth_int @ ( list_update_int @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6574_nth__list__update,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X2: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( ( ( I = J )
% 9.00/5.09 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = X2 ) )
% 9.00/5.09 & ( ( I != J )
% 9.00/5.09 => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6575_nth__list__update,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_o,X2: $o,J: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( ( nth_o @ ( list_update_o @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = ( ( ( I = J )
% 9.00/5.09 => X2 )
% 9.00/5.09 & ( ( I != J )
% 9.00/5.09 => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6576_nth__list__update,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_nat,J: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( ( ( I = J )
% 9.00/5.09 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = X2 ) )
% 9.00/5.09 & ( ( I != J )
% 9.00/5.09 => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X2 ) @ J )
% 9.00/5.09 = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6577_list__update__same__conv,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_int,X2: int] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.00/5.09 => ( ( ( list_update_int @ Xs @ I @ X2 )
% 9.00/5.09 = Xs )
% 9.00/5.09 = ( ( nth_int @ Xs @ I )
% 9.00/5.09 = X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6578_list__update__same__conv,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_VEBT_VEBT,X2: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.00/5.09 => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X2 )
% 9.00/5.09 = Xs )
% 9.00/5.09 = ( ( nth_VEBT_VEBT @ Xs @ I )
% 9.00/5.09 = X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6579_list__update__same__conv,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_o,X2: $o] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.00/5.09 => ( ( ( list_update_o @ Xs @ I @ X2 )
% 9.00/5.09 = Xs )
% 9.00/5.09 = ( ( nth_o @ Xs @ I )
% 9.00/5.09 = X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6580_list__update__same__conv,axiom,
% 9.00/5.09 ! [I: nat,Xs: list_nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.00/5.09 => ( ( ( list_update_nat @ Xs @ I @ X2 )
% 9.00/5.09 = Xs )
% 9.00/5.09 = ( ( nth_nat @ Xs @ I )
% 9.00/5.09 = X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6581_insert__simp__norm,axiom,
% 9.00/5.09 ! [X2: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( ord_less_nat @ Mi @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( X2 != Ma )
% 9.00/5.09 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X2 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6582_insert__simp__excp,axiom,
% 9.00/5.09 ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X2: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( X2 != Ma )
% 9.00/5.09 => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X2 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6583_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6584_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6585_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( sup_sup_set_real @ bot_bot_set_real @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6586_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( sup_sup_set_o @ bot_bot_set_o @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6587_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ bot_bot_set_nat @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6588_sup__bot__left,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ bot_bot_set_int @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6589_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ X2 @ bot_bo5327735625951526323at_nat )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6590_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ X2 @ bot_bo228742789529271731at_nat )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6591_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( sup_sup_set_real @ X2 @ bot_bot_set_real )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6592_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( sup_sup_set_o @ X2 @ bot_bot_set_o )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6593_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6594_sup__bot__right,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ X2 @ bot_bot_set_int )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6595_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( bot_bo5327735625951526323at_nat
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6596_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( bot_bo228742789529271731at_nat
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bo228742789529271731at_nat )
% 9.00/5.09 & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6597_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_real,Y3: set_real] :
% 9.00/5.09 ( ( bot_bot_set_real
% 9.00/5.09 = ( sup_sup_set_real @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bot_set_real )
% 9.00/5.09 & ( Y3 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6598_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_o,Y3: set_o] :
% 9.00/5.09 ( ( bot_bot_set_o
% 9.00/5.09 = ( sup_sup_set_o @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bot_set_o )
% 9.00/5.09 & ( Y3 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6599_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ( bot_bot_set_nat
% 9.00/5.09 = ( sup_sup_set_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bot_set_nat )
% 9.00/5.09 & ( Y3 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6600_bot__eq__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ( bot_bot_set_int
% 9.00/5.09 = ( sup_sup_set_int @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( X2 = bot_bot_set_int )
% 9.00/5.09 & ( Y3 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6601_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 9.00/5.09 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 = ( ( X2 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 & ( Y3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6602_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 9.00/5.09 = bot_bo228742789529271731at_nat )
% 9.00/5.09 = ( ( X2 = bot_bo228742789529271731at_nat )
% 9.00/5.09 & ( Y3 = bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6603_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_real,Y3: set_real] :
% 9.00/5.09 ( ( ( sup_sup_set_real @ X2 @ Y3 )
% 9.00/5.09 = bot_bot_set_real )
% 9.00/5.09 = ( ( X2 = bot_bot_set_real )
% 9.00/5.09 & ( Y3 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6604_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_o,Y3: set_o] :
% 9.00/5.09 ( ( ( sup_sup_set_o @ X2 @ Y3 )
% 9.00/5.09 = bot_bot_set_o )
% 9.00/5.09 = ( ( X2 = bot_bot_set_o )
% 9.00/5.09 & ( Y3 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6605_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ( ( sup_sup_set_nat @ X2 @ Y3 )
% 9.00/5.09 = bot_bot_set_nat )
% 9.00/5.09 = ( ( X2 = bot_bot_set_nat )
% 9.00/5.09 & ( Y3 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6606_sup__eq__bot__iff,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ( ( sup_sup_set_int @ X2 @ Y3 )
% 9.00/5.09 = bot_bot_set_int )
% 9.00/5.09 = ( ( X2 = bot_bot_set_int )
% 9.00/5.09 & ( Y3 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6607_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 9.00/5.09 & ( ord_less_eq_set_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6608_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 9.00/5.09 & ( ord_le3146513528884898305at_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6609_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.09 & ( ord_less_eq_set_int @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6610_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.09 & ( ord_less_eq_rat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6611_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6612_le__inf__iff,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) )
% 9.00/5.09 = ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.09 & ( ord_less_eq_int @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6613_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 & ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6614_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 9.00/5.09 & ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6615_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 & ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6616_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 & ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6617_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 & ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6618_inf_Obounded__iff,axiom,
% 9.00/5.09 ! [A2: int,B3: int,C: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 9.00/5.09 = ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 & ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6619_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_eq_set_nat @ X2 @ Z )
% 9.00/5.09 & ( ord_less_eq_set_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6620_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_le3000389064537975527at_nat @ X2 @ Z )
% 9.00/5.09 & ( ord_le3000389064537975527at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6621_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_le1268244103169919719at_nat @ X2 @ Z )
% 9.00/5.09 & ( ord_le1268244103169919719at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6622_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_eq_set_int @ X2 @ Z )
% 9.00/5.09 & ( ord_less_eq_set_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6623_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_eq_rat @ X2 @ Z )
% 9.00/5.09 & ( ord_less_eq_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6624_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_eq_nat @ X2 @ Z )
% 9.00/5.09 & ( ord_less_eq_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6625_le__sup__iff,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( sup_sup_int @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_eq_int @ X2 @ Z )
% 9.00/5.09 & ( ord_less_eq_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6626_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6627_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 9.00/5.09 & ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6628_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 9.00/5.09 & ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6629_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: set_int,C: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6630_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6631_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6632_sup_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6633_inf__bot__left,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X2 )
% 9.00/5.09 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6634_inf__bot__left,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( inf_inf_set_real @ bot_bot_set_real @ X2 )
% 9.00/5.09 = bot_bot_set_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6635_inf__bot__left,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( inf_inf_set_o @ bot_bot_set_o @ X2 )
% 9.00/5.09 = bot_bot_set_o ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6636_inf__bot__left,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( inf_inf_set_nat @ bot_bot_set_nat @ X2 )
% 9.00/5.09 = bot_bot_set_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6637_inf__bot__left,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( inf_inf_set_int @ bot_bot_set_int @ X2 )
% 9.00/5.09 = bot_bot_set_int ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6638_inf__bot__right,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( inf_in2572325071724192079at_nat @ X2 @ bot_bo2099793752762293965at_nat )
% 9.00/5.09 = bot_bo2099793752762293965at_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6639_inf__bot__right,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( inf_inf_set_real @ X2 @ bot_bot_set_real )
% 9.00/5.09 = bot_bot_set_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6640_inf__bot__right,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( inf_inf_set_o @ X2 @ bot_bot_set_o )
% 9.00/5.09 = bot_bot_set_o ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6641_inf__bot__right,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( inf_inf_set_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.09 = bot_bot_set_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6642_inf__bot__right,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( inf_inf_set_int @ X2 @ bot_bot_set_int )
% 9.00/5.09 = bot_bot_set_int ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6643_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ A2 @ bot_bo5327735625951526323at_nat )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6644_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ A2 @ bot_bo228742789529271731at_nat )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6645_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_real] :
% 9.00/5.09 ( ( sup_sup_set_real @ A2 @ bot_bot_set_real )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6646_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_o] :
% 9.00/5.09 ( ( sup_sup_set_o @ A2 @ bot_bot_set_o )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6647_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6648_sup__bot_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6649_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( bot_bo5327735625951526323at_nat
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6650_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( bot_bo228742789529271731at_nat
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bo228742789529271731at_nat )
% 9.00/5.09 & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6651_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_real,B3: set_real] :
% 9.00/5.09 ( ( bot_bot_set_real
% 9.00/5.09 = ( sup_sup_set_real @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bot_set_real )
% 9.00/5.09 & ( B3 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6652_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_o,B3: set_o] :
% 9.00/5.09 ( ( bot_bot_set_o
% 9.00/5.09 = ( sup_sup_set_o @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bot_set_o )
% 9.00/5.09 & ( B3 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6653_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( bot_bot_set_nat
% 9.00/5.09 = ( sup_sup_set_nat @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bot_set_nat )
% 9.00/5.09 & ( B3 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6654_sup__bot_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( bot_bot_set_int
% 9.00/5.09 = ( sup_sup_set_int @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = bot_bot_set_int )
% 9.00/5.09 & ( B3 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6655_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ bot_bo5327735625951526323at_nat @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6656_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6657_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_real] :
% 9.00/5.09 ( ( sup_sup_set_real @ bot_bot_set_real @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6658_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_o] :
% 9.00/5.09 ( ( sup_sup_set_o @ bot_bot_set_o @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6659_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6660_sup__bot_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6661_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 9.00/5.09 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 = ( ( A2 = bot_bo5327735625951526323at_nat )
% 9.00/5.09 & ( B3 = bot_bo5327735625951526323at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6662_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 9.00/5.09 = bot_bo228742789529271731at_nat )
% 9.00/5.09 = ( ( A2 = bot_bo228742789529271731at_nat )
% 9.00/5.09 & ( B3 = bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6663_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_real,B3: set_real] :
% 9.00/5.09 ( ( ( sup_sup_set_real @ A2 @ B3 )
% 9.00/5.09 = bot_bot_set_real )
% 9.00/5.09 = ( ( A2 = bot_bot_set_real )
% 9.00/5.09 & ( B3 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6664_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_o,B3: set_o] :
% 9.00/5.09 ( ( ( sup_sup_set_o @ A2 @ B3 )
% 9.00/5.09 = bot_bot_set_o )
% 9.00/5.09 = ( ( A2 = bot_bot_set_o )
% 9.00/5.09 & ( B3 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6665_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ( sup_sup_set_nat @ A2 @ B3 )
% 9.00/5.09 = bot_bot_set_nat )
% 9.00/5.09 = ( ( A2 = bot_bot_set_nat )
% 9.00/5.09 & ( B3 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6666_sup__bot_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ( sup_sup_set_int @ A2 @ B3 )
% 9.00/5.09 = bot_bot_set_int )
% 9.00/5.09 = ( ( A2 = bot_bot_set_int )
% 9.00/5.09 & ( B3 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6667_max_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6668_max_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: num,C: num,A2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6669_max_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6670_max_Obounded__iff,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 9.00/5.09 = ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 & ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6671_max_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6672_max_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: num,B3: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_num @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6673_max_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6674_max_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6675_max_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6676_max_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: num,A2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_num @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6677_max_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6678_max_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6679_max__less__iff__conj,axiom,
% 9.00/5.09 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.09 ( ( ord_less_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_real @ X2 @ Z )
% 9.00/5.09 & ( ord_less_real @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6680_max__less__iff__conj,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( ord_less_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_rat @ X2 @ Z )
% 9.00/5.09 & ( ord_less_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6681_max__less__iff__conj,axiom,
% 9.00/5.09 ! [X2: num,Y3: num,Z: num] :
% 9.00/5.09 ( ( ord_less_num @ ( ord_max_num @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_num @ X2 @ Z )
% 9.00/5.09 & ( ord_less_num @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6682_max__less__iff__conj,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_nat @ X2 @ Z )
% 9.00/5.09 & ( ord_less_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6683_max__less__iff__conj,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( ord_less_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ( ord_less_int @ X2 @ Z )
% 9.00/5.09 & ( ord_less_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6684_max_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_real @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6685_max_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6686_max_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: num,B3: num] :
% 9.00/5.09 ( ( ord_less_num @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_num @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6687_max_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6688_max_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_max_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6689_max_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_real @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6690_max_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6691_max_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: num,A2: num] :
% 9.00/5.09 ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_num @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6692_max_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6693_max_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_max_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6694_max__bot,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( ord_max_set_real @ bot_bot_set_real @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6695_max__bot,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( ord_max_set_o @ bot_bot_set_o @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6696_max__bot,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( ord_max_set_nat @ bot_bot_set_nat @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6697_max__bot,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( ord_max_set_int @ bot_bot_set_int @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6698_max__bot,axiom,
% 9.00/5.09 ! [X2: nat] :
% 9.00/5.09 ( ( ord_max_nat @ bot_bot_nat @ X2 )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6699_max__bot2,axiom,
% 9.00/5.09 ! [X2: set_real] :
% 9.00/5.09 ( ( ord_max_set_real @ X2 @ bot_bot_set_real )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6700_max__bot2,axiom,
% 9.00/5.09 ! [X2: set_o] :
% 9.00/5.09 ( ( ord_max_set_o @ X2 @ bot_bot_set_o )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6701_max__bot2,axiom,
% 9.00/5.09 ! [X2: set_nat] :
% 9.00/5.09 ( ( ord_max_set_nat @ X2 @ bot_bot_set_nat )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6702_max__bot2,axiom,
% 9.00/5.09 ! [X2: set_int] :
% 9.00/5.09 ( ( ord_max_set_int @ X2 @ bot_bot_set_int )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6703_max__bot2,axiom,
% 9.00/5.09 ! [X2: nat] :
% 9.00/5.09 ( ( ord_max_nat @ X2 @ bot_bot_nat )
% 9.00/5.09 = X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6704_max__Suc__Suc,axiom,
% 9.00/5.09 ! [M: nat,N: nat] :
% 9.00/5.09 ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.00/5.09 = ( suc @ ( ord_max_nat @ M @ N ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6705_max__nat_Oeq__neutr__iff,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ( ord_max_nat @ A2 @ B3 )
% 9.00/5.09 = zero_zero_nat )
% 9.00/5.09 = ( ( A2 = zero_zero_nat )
% 9.00/5.09 & ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6706_max__nat_Oleft__neutral,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( ord_max_nat @ zero_zero_nat @ A2 )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6707_max__nat_Oneutr__eq__iff,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( zero_zero_nat
% 9.00/5.09 = ( ord_max_nat @ A2 @ B3 ) )
% 9.00/5.09 = ( ( A2 = zero_zero_nat )
% 9.00/5.09 & ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6708_max__nat_Oright__neutral,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( ord_max_nat @ A2 @ zero_zero_nat )
% 9.00/5.09 = A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6709_max__0L,axiom,
% 9.00/5.09 ! [N: nat] :
% 9.00/5.09 ( ( ord_max_nat @ zero_zero_nat @ N )
% 9.00/5.09 = N ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6710_max__0R,axiom,
% 9.00/5.09 ! [N: nat] :
% 9.00/5.09 ( ( ord_max_nat @ N @ zero_zero_nat )
% 9.00/5.09 = N ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6711_i0__less,axiom,
% 9.00/5.09 ! [N: extended_enat] :
% 9.00/5.09 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.00/5.09 = ( N != zero_z5237406670263579293d_enat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6712_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 = ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6713_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.00/5.09 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.00/5.09 = ( numera1916890842035813515d_enat @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.00/5.09 => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.00/5.09 = ( numera1916890842035813515d_enat @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6714_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 = ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6715_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 = ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6716_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.00/5.09 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.00/5.09 = ( numeral_numeral_nat @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.00/5.09 => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V ) )
% 9.00/5.09 = ( numeral_numeral_nat @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6717_max__number__of_I1_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 = ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6718_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_rat @ ( numeral_numeral_rat @ X2 ) @ zero_zero_rat )
% 9.00/5.09 = ( numeral_numeral_rat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6719_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_nat @ ( numeral_numeral_nat @ X2 ) @ zero_zero_nat )
% 9.00/5.09 = ( numeral_numeral_nat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6720_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_real @ ( numeral_numeral_real @ X2 ) @ zero_zero_real )
% 9.00/5.09 = ( numeral_numeral_real @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6721_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_int @ ( numeral_numeral_int @ X2 ) @ zero_zero_int )
% 9.00/5.09 = ( numeral_numeral_int @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6722_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X2 ) @ zero_z5237406670263579293d_enat )
% 9.00/5.09 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6723_max__0__1_I4_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X2 ) @ zero_z3403309356797280102nteger )
% 9.00/5.09 = ( numera6620942414471956472nteger @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6724_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X2 ) )
% 9.00/5.09 = ( numeral_numeral_rat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6725_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X2 ) )
% 9.00/5.09 = ( numeral_numeral_nat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6726_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X2 ) )
% 9.00/5.09 = ( numeral_numeral_real @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6727_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X2 ) )
% 9.00/5.09 = ( numeral_numeral_int @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6728_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X2 ) )
% 9.00/5.09 = ( numera1916890842035813515d_enat @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6729_max__0__1_I3_J,axiom,
% 9.00/5.09 ! [X2: num] :
% 9.00/5.09 ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ X2 ) )
% 9.00/5.09 = ( numera6620942414471956472nteger @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6730_max__0__1_I2_J,axiom,
% 9.00/5.09 ( ( ord_max_real @ one_one_real @ zero_zero_real )
% 9.00/5.09 = one_one_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6731_max__0__1_I2_J,axiom,
% 9.00/5.09 ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
% 9.00/5.09 = one_one_rat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6732_max__0__1_I2_J,axiom,
% 9.00/5.09 ( ( ord_max_int @ one_one_int @ zero_zero_int )
% 9.00/5.09 = one_one_int ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6733_max__0__1_I2_J,axiom,
% 9.00/5.09 ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
% 9.00/5.09 = one_one_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6734_max__0__1_I1_J,axiom,
% 9.00/5.09 ( ( ord_max_real @ zero_zero_real @ one_one_real )
% 9.00/5.09 = one_one_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6735_max__0__1_I1_J,axiom,
% 9.00/5.09 ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
% 9.00/5.09 = one_one_rat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6736_max__0__1_I1_J,axiom,
% 9.00/5.09 ( ( ord_max_int @ zero_zero_int @ one_one_int )
% 9.00/5.09 = one_one_int ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6737_max__0__1_I1_J,axiom,
% 9.00/5.09 ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
% 9.00/5.09 = one_one_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6738_max__number__of_I2_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 = ( numeral_numeral_real @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6739_max__number__of_I2_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 = ( numera6620942414471956472nteger @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6740_max__number__of_I2_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 = ( numeral_numeral_rat @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6741_max__number__of_I2_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 = ( numeral_numeral_int @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6742_max__number__of_I3_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 = ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V ) )
% 9.00/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6743_max__number__of_I3_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 = ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V ) )
% 9.00/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6744_max__number__of_I3_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 = ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V ) )
% 9.00/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6745_max__number__of_I3_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 = ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V ) )
% 9.00/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6746_max__number__of_I4_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6747_max__number__of_I4_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) )
% 9.00/5.09 = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6748_max__number__of_I4_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6749_max__number__of_I4_J,axiom,
% 9.00/5.09 ! [U: num,V: num] :
% 9.00/5.09 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) )
% 9.00/5.09 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.00/5.09 = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6750_max_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6751_max_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: num,B3: num,A2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6752_max_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6753_max_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6754_max_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6755_max_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: num,A2: num,B3: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6756_max_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6757_max_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6758_max_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.09 ( ( ord_max_rat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6759_max_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_num
% 9.00/5.09 = ( ^ [A4: num,B4: num] :
% 9.00/5.09 ( ( ord_max_num @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6760_max_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.09 ( ( ord_max_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6761_max_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] :
% 9.00/5.09 ( ( ord_max_int @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6762_max_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( ( ord_max_rat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6763_max_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_num
% 9.00/5.09 = ( ^ [B4: num,A4: num] :
% 9.00/5.09 ( ( ord_max_num @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6764_max_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( ( ord_max_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6765_max_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( ( ord_max_int @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6766_le__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_eq_rat @ Z @ X2 )
% 9.00/5.09 | ( ord_less_eq_rat @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6767_le__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: num,X2: num,Y3: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_eq_num @ Z @ X2 )
% 9.00/5.09 | ( ord_less_eq_num @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6768_le__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: nat,X2: nat,Y3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_eq_nat @ Z @ X2 )
% 9.00/5.09 | ( ord_less_eq_nat @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6769_le__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: int,X2: int,Y3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_eq_int @ Z @ X2 )
% 9.00/5.09 | ( ord_less_eq_int @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6770_max_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( ord_max_rat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6771_max_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: num,A2: num] : ( ord_less_eq_num @ B3 @ ( ord_max_num @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6772_max_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6773_max_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( ord_max_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6774_max_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( ord_max_rat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6775_max_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: num,B3: num] : ( ord_less_eq_num @ A2 @ ( ord_max_num @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6776_max_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6777_max_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( ord_max_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6778_max_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( ord_max_rat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6779_max_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_num
% 9.00/5.09 = ( ^ [B4: num,A4: num] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( ord_max_num @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6780_max_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( ord_max_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6781_max_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( ord_max_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6782_max_OboundedI,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6783_max_OboundedI,axiom,
% 9.00/5.09 ! [B3: num,A2: num,C: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_num @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6784_max_OboundedI,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6785_max_OboundedI,axiom,
% 9.00/5.09 ! [B3: int,A2: int,C: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6786_max_OboundedE,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6787_max_OboundedE,axiom,
% 9.00/5.09 ! [B3: num,C: num,A2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_num @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6788_max_OboundedE,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6789_max_OboundedE,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6790_max_OorderI,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( ord_max_rat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6791_max_OorderI,axiom,
% 9.00/5.09 ! [A2: num,B3: num] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( ord_max_num @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_num @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6792_max_OorderI,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( ord_max_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6793_max_OorderI,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( ord_max_int @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6794_max_OorderE,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6795_max_OorderE,axiom,
% 9.00/5.09 ! [B3: num,A2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( ord_max_num @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6796_max_OorderE,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6797_max_OorderE,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( ord_max_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6798_max_Omono,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,D: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6799_max_Omono,axiom,
% 9.00/5.09 ! [C: num,A2: num,D: num,B3: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_num @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6800_max_Omono,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,D: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6801_max_Omono,axiom,
% 9.00/5.09 ! [C: int,A2: int,D: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6802_not__iless0,axiom,
% 9.00/5.09 ! [N: extended_enat] :
% 9.00/5.09 ~ ( ord_le72135733267957522d_enat @ N @ zero_z5237406670263579293d_enat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6803_enat__less__induct,axiom,
% 9.00/5.09 ! [P: extended_enat > $o,N: extended_enat] :
% 9.00/5.09 ( ! [N3: extended_enat] :
% 9.00/5.09 ( ! [M3: extended_enat] :
% 9.00/5.09 ( ( ord_le72135733267957522d_enat @ M3 @ N3 )
% 9.00/5.09 => ( P @ M3 ) )
% 9.00/5.09 => ( P @ N3 ) )
% 9.00/5.09 => ( P @ N ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6804_enat__0__less__mult__iff,axiom,
% 9.00/5.09 ! [M: extended_enat,N: extended_enat] :
% 9.00/5.09 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N ) )
% 9.00/5.09 = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 9.00/5.09 & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6805_bot__enat__def,axiom,
% 9.00/5.09 bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6806_of__nat__max,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6807_of__nat__max,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6808_of__nat__max,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ord_max_real @ ( semiri5074537144036343181t_real @ X2 ) @ ( semiri5074537144036343181t_real @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6809_less__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: real,X2: real,Y3: real] :
% 9.00/5.09 ( ( ord_less_real @ Z @ ( ord_max_real @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_real @ Z @ X2 )
% 9.00/5.09 | ( ord_less_real @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6810_less__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: rat,X2: rat,Y3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ Z @ ( ord_max_rat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_rat @ Z @ X2 )
% 9.00/5.09 | ( ord_less_rat @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6811_less__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: num,X2: num,Y3: num] :
% 9.00/5.09 ( ( ord_less_num @ Z @ ( ord_max_num @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_num @ Z @ X2 )
% 9.00/5.09 | ( ord_less_num @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6812_less__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: nat,X2: nat,Y3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ Z @ ( ord_max_nat @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_nat @ Z @ X2 )
% 9.00/5.09 | ( ord_less_nat @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6813_less__max__iff__disj,axiom,
% 9.00/5.09 ! [Z: int,X2: int,Y3: int] :
% 9.00/5.09 ( ( ord_less_int @ Z @ ( ord_max_int @ X2 @ Y3 ) )
% 9.00/5.09 = ( ( ord_less_int @ Z @ X2 )
% 9.00/5.09 | ( ord_less_int @ Z @ Y3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6814_max_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: real,C: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ ( ord_max_real @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_real @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6815_max_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ ( ord_max_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6816_max_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: num,C: num,A2: num] :
% 9.00/5.09 ( ( ord_less_num @ ( ord_max_num @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_num @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_num @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6817_max_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ ( ord_max_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6818_max_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ ( ord_max_int @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6819_max_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_real
% 9.00/5.09 = ( ^ [B4: real,A4: real] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( ord_max_real @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6820_max_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( ord_max_rat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6821_max_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_num
% 9.00/5.09 = ( ^ [B4: num,A4: num] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( ord_max_num @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6822_max_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( ord_max_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6823_max_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( ord_max_int @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6824_max_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: real,A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ C @ A2 )
% 9.00/5.09 => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6825_max_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6826_max_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: num,A2: num,B3: num] :
% 9.00/5.09 ( ( ord_less_num @ C @ A2 )
% 9.00/5.09 => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6827_max_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6828_max_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6829_max_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: real,B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ C @ B3 )
% 9.00/5.09 => ( ord_less_real @ C @ ( ord_max_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6830_max_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ C @ B3 )
% 9.00/5.09 => ( ord_less_rat @ C @ ( ord_max_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6831_max_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: num,B3: num,A2: num] :
% 9.00/5.09 ( ( ord_less_num @ C @ B3 )
% 9.00/5.09 => ( ord_less_num @ C @ ( ord_max_num @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6832_max_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6833_max_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ C @ B3 )
% 9.00/5.09 => ( ord_less_int @ C @ ( ord_max_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6834_sup__nat__def,axiom,
% 9.00/5.09 sup_sup_nat = ord_max_nat ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6835_max__def,axiom,
% 9.00/5.09 ( ord_max_set_int
% 9.00/5.09 = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6836_max__def,axiom,
% 9.00/5.09 ( ord_max_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6837_max__def,axiom,
% 9.00/5.09 ( ord_max_num
% 9.00/5.09 = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6838_max__def,axiom,
% 9.00/5.09 ( ord_max_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6839_max__def,axiom,
% 9.00/5.09 ( ord_max_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6840_max__absorb1,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_max_set_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6841_max__absorb1,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_max_rat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6842_max__absorb1,axiom,
% 9.00/5.09 ! [Y3: num,X2: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_max_num @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6843_max__absorb1,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_max_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6844_max__absorb1,axiom,
% 9.00/5.09 ! [Y3: int,X2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_max_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6845_max__absorb2,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_max_set_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6846_max__absorb2,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_max_rat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6847_max__absorb2,axiom,
% 9.00/5.09 ! [X2: num,Y3: num] :
% 9.00/5.09 ( ( ord_less_eq_num @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_max_num @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6848_max__absorb2,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_max_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6849_max__absorb2,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_max_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6850_max__add__distrib__right,axiom,
% 9.00/5.09 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.09 ( ( plus_plus_real @ X2 @ ( ord_max_real @ Y3 @ Z ) )
% 9.00/5.09 = ( ord_max_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( plus_plus_real @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6851_max__add__distrib__right,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( plus_plus_rat @ X2 @ ( ord_max_rat @ Y3 @ Z ) )
% 9.00/5.09 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Y3 ) @ ( plus_plus_rat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6852_max__add__distrib__right,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( plus_plus_int @ X2 @ ( ord_max_int @ Y3 @ Z ) )
% 9.00/5.09 = ( ord_max_int @ ( plus_plus_int @ X2 @ Y3 ) @ ( plus_plus_int @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6853_max__add__distrib__right,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( plus_plus_nat @ X2 @ ( ord_max_nat @ Y3 @ Z ) )
% 9.00/5.09 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Y3 ) @ ( plus_plus_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6854_max__add__distrib__left,axiom,
% 9.00/5.09 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.09 ( ( plus_plus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_real @ ( plus_plus_real @ X2 @ Z ) @ ( plus_plus_real @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6855_max__add__distrib__left,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( plus_plus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_rat @ ( plus_plus_rat @ X2 @ Z ) @ ( plus_plus_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6856_max__add__distrib__left,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( plus_plus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_int @ ( plus_plus_int @ X2 @ Z ) @ ( plus_plus_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6857_max__add__distrib__left,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( plus_plus_nat @ ( ord_max_nat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_nat @ ( plus_plus_nat @ X2 @ Z ) @ ( plus_plus_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6858_max__diff__distrib__left,axiom,
% 9.00/5.09 ! [X2: real,Y3: real,Z: real] :
% 9.00/5.09 ( ( minus_minus_real @ ( ord_max_real @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_real @ ( minus_minus_real @ X2 @ Z ) @ ( minus_minus_real @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6859_max__diff__distrib__left,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( minus_minus_rat @ ( ord_max_rat @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_rat @ ( minus_minus_rat @ X2 @ Z ) @ ( minus_minus_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6860_max__diff__distrib__left,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( minus_minus_int @ ( ord_max_int @ X2 @ Y3 ) @ Z )
% 9.00/5.09 = ( ord_max_int @ ( minus_minus_int @ X2 @ Z ) @ ( minus_minus_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6861_nat__add__max__right,axiom,
% 9.00/5.09 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.09 ( ( plus_plus_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 9.00/5.09 = ( ord_max_nat @ ( plus_plus_nat @ M @ N ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6862_nat__add__max__left,axiom,
% 9.00/5.09 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.09 ( ( plus_plus_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 9.00/5.09 = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6863_nat__mult__max__left,axiom,
% 9.00/5.09 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.09 ( ( times_times_nat @ ( ord_max_nat @ M @ N ) @ Q3 )
% 9.00/5.09 = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6864_nat__mult__max__right,axiom,
% 9.00/5.09 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.09 ( ( times_times_nat @ M @ ( ord_max_nat @ N @ Q3 ) )
% 9.00/5.09 = ( ord_max_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6865_nat__minus__add__max,axiom,
% 9.00/5.09 ! [N: nat,M: nat] :
% 9.00/5.09 ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M ) @ M )
% 9.00/5.09 = ( ord_max_nat @ N @ M ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6866_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6867_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6868_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6869_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6870_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6871_inf__sup__ord_I2_J,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6872_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6873_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6874_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6875_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6876_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6877_inf__sup__ord_I1_J,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6878_inf__le1,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6879_inf__le1,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6880_inf__le1,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6881_inf__le1,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6882_inf__le1,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6883_inf__le1,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ X2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6884_inf__le2,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6885_inf__le2,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6886_inf__le2,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6887_inf__le2,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6888_inf__le2,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6889_inf__le2,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ ( inf_inf_int @ X2 @ Y3 ) @ Y3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6890_le__infE,axiom,
% 9.00/5.09 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_nat @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6891_le__infE,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_le3146513528884898305at_nat @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6892_le__infE,axiom,
% 9.00/5.09 ! [X2: set_int,A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_int @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_int @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6893_le__infE,axiom,
% 9.00/5.09 ! [X2: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_rat @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_rat @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6894_le__infE,axiom,
% 9.00/5.09 ! [X2: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_nat @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_nat @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6895_le__infE,axiom,
% 9.00/5.09 ! [X2: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_int @ X2 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_int @ X2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6896_le__infI,axiom,
% 9.00/5.09 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6897_le__infI,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_le3146513528884898305at_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6898_le__infI,axiom,
% 9.00/5.09 ! [X2: set_int,A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6899_le__infI,axiom,
% 9.00/5.09 ! [X2: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6900_le__infI,axiom,
% 9.00/5.09 ! [X2: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6901_le__infI,axiom,
% 9.00/5.09 ! [X2: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6902_inf__mono,axiom,
% 9.00/5.09 ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6903_inf__mono,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_le3146513528884898305at_nat @ B3 @ D )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ ( inf_in2572325071724192079at_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6904_inf__mono,axiom,
% 9.00/5.09 ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ ( inf_inf_set_int @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6905_inf__mono,axiom,
% 9.00/5.09 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_rat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ ( inf_inf_rat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6906_inf__mono,axiom,
% 9.00/5.09 ! [A2: nat,C: nat,B3: nat,D: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_nat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6907_inf__mono,axiom,
% 9.00/5.09 ! [A2: int,C: int,B3: int,D: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_int @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ ( inf_inf_int @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6908_le__infI1,axiom,
% 9.00/5.09 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6909_le__infI1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6910_le__infI1,axiom,
% 9.00/5.09 ! [A2: set_int,X2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6911_le__infI1,axiom,
% 9.00/5.09 ! [A2: rat,X2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6912_le__infI1,axiom,
% 9.00/5.09 ! [A2: nat,X2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6913_le__infI1,axiom,
% 9.00/5.09 ! [A2: int,X2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6914_le__infI2,axiom,
% 9.00/5.09 ! [B3: set_nat,X2: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6915_le__infI2,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6916_le__infI2,axiom,
% 9.00/5.09 ! [B3: set_int,X2: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6917_le__infI2,axiom,
% 9.00/5.09 ! [B3: rat,X2: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6918_le__infI2,axiom,
% 9.00/5.09 ! [B3: nat,X2: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6919_le__infI2,axiom,
% 9.00/5.09 ! [B3: int,X2: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6920_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_inf_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6921_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6922_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_inf_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6923_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_inf_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6924_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_inf_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6925_inf_OorderE,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( inf_inf_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6926_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_inf_set_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6927_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_in2572325071724192079at_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6928_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_inf_set_int @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6929_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_inf_rat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_rat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6930_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_inf_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6931_inf_OorderI,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( inf_inf_int @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6932_inf__unique,axiom,
% 9.00/5.09 ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ X5 @ Z4 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6933_inf__unique,axiom,
% 9.00/5.09 ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z4: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_le3146513528884898305at_nat @ X5 @ Z4 )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6934_inf__unique,axiom,
% 9.00/5.09 ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ X5 @ Z4 )
% 9.00/5.09 => ( ord_less_eq_set_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6935_inf__unique,axiom,
% 9.00/5.09 ! [F: rat > rat > rat,X2: rat,Y3: rat] :
% 9.00/5.09 ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: rat,Y4: rat,Z4: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ X5 @ Z4 )
% 9.00/5.09 => ( ord_less_eq_rat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_inf_rat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6936_inf__unique,axiom,
% 9.00/5.09 ! [F: nat > nat > nat,X2: nat,Y3: nat] :
% 9.00/5.09 ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ X5 @ Z4 )
% 9.00/5.09 => ( ord_less_eq_nat @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_inf_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6937_inf__unique,axiom,
% 9.00/5.09 ! [F: int > int > int,X2: int,Y3: int] :
% 9.00/5.09 ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ X5 )
% 9.00/5.09 => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ Y4 )
% 9.00/5.09 => ( ! [X5: int,Y4: int,Z4: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X5 @ Y4 )
% 9.00/5.09 => ( ( ord_less_eq_int @ X5 @ Z4 )
% 9.00/5.09 => ( ord_less_eq_int @ X5 @ ( F @ Y4 @ Z4 ) ) ) )
% 9.00/5.09 => ( ( inf_inf_int @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6938_le__iff__inf,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [X: set_nat,Y: set_nat] :
% 9.00/5.09 ( ( inf_inf_set_nat @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6939_le__iff__inf,axiom,
% 9.00/5.09 ( ord_le3146513528884898305at_nat
% 9.00/5.09 = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( inf_in2572325071724192079at_nat @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6940_le__iff__inf,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.09 ( ( inf_inf_set_int @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6941_le__iff__inf,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [X: rat,Y: rat] :
% 9.00/5.09 ( ( inf_inf_rat @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6942_le__iff__inf,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [X: nat,Y: nat] :
% 9.00/5.09 ( ( inf_inf_nat @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6943_le__iff__inf,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [X: int,Y: int] :
% 9.00/5.09 ( ( inf_inf_int @ X @ Y )
% 9.00/5.09 = X ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6944_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6945_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6946_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_set_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6947_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6948_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6949_inf_Oabsorb1,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6950_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6951_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6952_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_set_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6953_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6954_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6955_inf_Oabsorb2,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6956_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6957_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6958_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6959_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_inf_rat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6960_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_inf_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6961_inf__absorb1,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( inf_inf_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6962_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: set_nat,X2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6963_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6964_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_inf_set_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6965_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_inf_rat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6966_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_inf_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6967_inf__absorb2,axiom,
% 9.00/5.09 ! [Y3: int,X2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( inf_inf_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6968_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6969_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_le3146513528884898305at_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6970_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_eq_set_int @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6971_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_eq_rat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6972_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6973_inf_OboundedE,axiom,
% 9.00/5.09 ! [A2: int,B3: int,C: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_eq_int @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6974_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6975_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6976_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int,C: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_set_int @ A2 @ ( inf_inf_set_int @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6977_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6978_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6979_inf_OboundedI,axiom,
% 9.00/5.09 ! [A2: int,B3: int,C: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ( ( ord_less_eq_int @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_int @ A2 @ ( inf_inf_int @ B3 @ C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6980_inf__greatest,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat,Z: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ X2 @ Z )
% 9.00/5.09 => ( ord_less_eq_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6981_inf__greatest,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_le3146513528884898305at_nat @ X2 @ Z )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6982_inf__greatest,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int,Z: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ X2 @ Z )
% 9.00/5.09 => ( ord_less_eq_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6983_inf__greatest,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ X2 @ Z )
% 9.00/5.09 => ( ord_less_eq_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6984_inf__greatest,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ X2 @ Z )
% 9.00/5.09 => ( ord_less_eq_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6985_inf__greatest,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( ord_less_eq_int @ X2 @ Z )
% 9.00/5.09 => ( ord_less_eq_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6986_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat,B4: set_nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6987_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_le3146513528884898305at_nat
% 9.00/5.09 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6988_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6989_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_inf_rat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6990_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_inf_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6991_inf_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( inf_inf_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6992_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6993_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6994_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6995_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6996_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6997_inf_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ A2 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6998_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_6999_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7000_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7001_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7002_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7003_inf_Ocobounded2,axiom,
% 9.00/5.09 ! [A2: int,B3: int] : ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ B3 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7004_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat,B4: set_nat] :
% 9.00/5.09 ( ( inf_inf_set_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7005_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_le3146513528884898305at_nat
% 9.00/5.09 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7006_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.09 ( ( inf_inf_set_int @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7007_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.09 ( ( inf_inf_rat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7008_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.09 ( ( inf_inf_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7009_inf_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] :
% 9.00/5.09 ( ( inf_inf_int @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7010_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [B4: set_nat,A4: set_nat] :
% 9.00/5.09 ( ( inf_inf_set_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7011_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_le3146513528884898305at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7012_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.09 ( ( inf_inf_set_int @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7013_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( ( inf_inf_rat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7014_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( ( inf_inf_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7015_inf_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( ( inf_inf_int @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7016_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: set_nat,C: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7017_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ A2 @ C )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7018_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: set_int,C: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7019_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7020_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7021_inf_OcoboundedI1,axiom,
% 9.00/5.09 ! [A2: int,C: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ C )
% 9.00/5.09 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7022_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ C )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7023_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le3146513528884898305at_nat @ B3 @ C )
% 9.00/5.09 => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7024_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: set_int,C: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ C )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7025_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ C )
% 9.00/5.09 => ( ord_less_eq_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7026_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ C )
% 9.00/5.09 => ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7027_inf_OcoboundedI2,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ C )
% 9.00/5.09 => ( ord_less_eq_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7028_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7029_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7030_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7031_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7032_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7033_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7034_inf__sup__ord_I4_J,axiom,
% 9.00/5.09 ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7035_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7036_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7037_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7038_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7039_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7040_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7041_inf__sup__ord_I3_J,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7042_le__supE,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat,X2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_nat @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7043_le__supE,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_le3000389064537975527at_nat @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7044_le__supE,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_le1268244103169919719at_nat @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7045_le__supE,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int,X2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_int @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_int @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7046_le__supE,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat,X2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_rat @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_less_eq_rat @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7047_le__supE,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_nat @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_less_eq_nat @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7048_le__supE,axiom,
% 9.00/5.09 ! [A2: int,B3: int,X2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_int @ A2 @ X2 )
% 9.00/5.09 => ~ ( ord_less_eq_int @ B3 @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7049_le__supI,axiom,
% 9.00/5.09 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7050_le__supI,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7051_le__supI,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7052_le__supI,axiom,
% 9.00/5.09 ! [A2: set_int,X2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7053_le__supI,axiom,
% 9.00/5.09 ! [A2: rat,X2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7054_le__supI,axiom,
% 9.00/5.09 ! [A2: nat,X2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7055_le__supI,axiom,
% 9.00/5.09 ! [A2: int,X2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7056_sup__ge1,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7057_sup__ge1,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7058_sup__ge1,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7059_sup__ge1,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] : ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7060_sup__ge1,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] : ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7061_sup__ge1,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7062_sup__ge1,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7063_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: set_nat,X2: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( sup_sup_set_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7064_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y3 @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7065_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y3 @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7066_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int] : ( ord_less_eq_set_int @ Y3 @ ( sup_sup_set_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7067_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat] : ( ord_less_eq_rat @ Y3 @ ( sup_sup_rat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7068_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat] : ( ord_less_eq_nat @ Y3 @ ( sup_sup_nat @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7069_sup__ge2,axiom,
% 9.00/5.09 ! [Y3: int,X2: int] : ( ord_less_eq_int @ Y3 @ ( sup_sup_int @ X2 @ Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7070_le__supI1,axiom,
% 9.00/5.09 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7071_le__supI1,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7072_le__supI1,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7073_le__supI1,axiom,
% 9.00/5.09 ! [X2: set_int,A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7074_le__supI1,axiom,
% 9.00/5.09 ! [X2: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7075_le__supI1,axiom,
% 9.00/5.09 ! [X2: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7076_le__supI1,axiom,
% 9.00/5.09 ! [X2: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7077_le__supI2,axiom,
% 9.00/5.09 ! [X2: set_nat,B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7078_le__supI2,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7079_le__supI2,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7080_le__supI2,axiom,
% 9.00/5.09 ! [X2: set_int,B3: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_int @ X2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7081_le__supI2,axiom,
% 9.00/5.09 ! [X2: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7082_le__supI2,axiom,
% 9.00/5.09 ! [X2: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7083_le__supI2,axiom,
% 9.00/5.09 ! [X2: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7084_sup_Omono,axiom,
% 9.00/5.09 ! [C: set_nat,A2: set_nat,D: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7085_sup_Omono,axiom,
% 9.00/5.09 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ D @ B3 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ C @ D ) @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7086_sup_Omono,axiom,
% 9.00/5.09 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ D @ B3 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D ) @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7087_sup_Omono,axiom,
% 9.00/5.09 ! [C: set_int,A2: set_int,D: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ C @ D ) @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7088_sup_Omono,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,D: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7089_sup_Omono,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,D: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7090_sup_Omono,axiom,
% 9.00/5.09 ! [C: int,A2: int,D: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ D @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A2 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7091_sup__mono,axiom,
% 9.00/5.09 ! [A2: set_nat,C: set_nat,B3: set_nat,D: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B3 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7092_sup__mono,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,D: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ B3 @ D )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) @ ( sup_su718114333110466843at_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7093_sup__mono,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ B3 @ D )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) @ ( sup_su5525570899277871387at_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7094_sup__mono,axiom,
% 9.00/5.09 ! [A2: set_int,C: set_int,B3: set_int,D: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ A2 @ B3 ) @ ( sup_sup_set_int @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7095_sup__mono,axiom,
% 9.00/5.09 ! [A2: rat,C: rat,B3: rat,D: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_rat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_rat @ ( sup_sup_rat @ A2 @ B3 ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7096_sup__mono,axiom,
% 9.00/5.09 ! [A2: nat,C: nat,B3: nat,D: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_nat @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B3 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7097_sup__mono,axiom,
% 9.00/5.09 ! [A2: int,C: int,B3: int,D: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ C )
% 9.00/5.09 => ( ( ord_less_eq_int @ B3 @ D )
% 9.00/5.09 => ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B3 ) @ ( sup_sup_int @ C @ D ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7098_sup__least,axiom,
% 9.00/5.09 ! [Y3: set_nat,X2: set_nat,Z: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ Z @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7099_sup__least,axiom,
% 9.00/5.09 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ Z @ X2 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7100_sup__least,axiom,
% 9.00/5.09 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ Z @ X2 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7101_sup__least,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int,Z: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ Z @ X2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7102_sup__least,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat,Z: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ Z @ X2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( sup_sup_rat @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7103_sup__least,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat,Z: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Z @ X2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( sup_sup_nat @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7104_sup__least,axiom,
% 9.00/5.09 ! [Y3: int,X2: int,Z: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ Z @ X2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( sup_sup_int @ Y3 @ Z ) @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7105_le__iff__sup,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [X: set_nat,Y: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7106_le__iff__sup,axiom,
% 9.00/5.09 ( ord_le3000389064537975527at_nat
% 9.00/5.09 = ( ^ [X: set_Pr8693737435421807431at_nat,Y: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7107_le__iff__sup,axiom,
% 9.00/5.09 ( ord_le1268244103169919719at_nat
% 9.00/5.09 = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7108_le__iff__sup,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [X: set_int,Y: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7109_le__iff__sup,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [X: rat,Y: rat] :
% 9.00/5.09 ( ( sup_sup_rat @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7110_le__iff__sup,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [X: nat,Y: nat] :
% 9.00/5.09 ( ( sup_sup_nat @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7111_le__iff__sup,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [X: int,Y: int] :
% 9.00/5.09 ( ( sup_sup_int @ X @ Y )
% 9.00/5.09 = Y ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7112_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7113_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7114_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7115_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7116_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7117_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7118_sup_OorderE,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( A2
% 9.00/5.09 = ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7119_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_sup_set_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_set_nat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7120_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7121_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7122_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_sup_set_int @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7123_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_sup_rat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_rat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7124_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_sup_nat @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_nat @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7125_sup_OorderI,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( A2
% 9.00/5.09 = ( sup_sup_int @ A2 @ B3 ) )
% 9.00/5.09 => ( ord_less_eq_int @ B3 @ A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7126_sup__unique,axiom,
% 9.00/5.09 ! [F: set_nat > set_nat > set_nat,X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_nat,Y4: set_nat,Z4: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ Z4 @ X5 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7127_sup__unique,axiom,
% 9.00/5.09 ! [F: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_Pr8693737435421807431at_nat,Y4: set_Pr8693737435421807431at_nat,Z4: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ Z4 @ X5 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7128_sup__unique,axiom,
% 9.00/5.09 ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,Z4: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ Z4 @ X5 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7129_sup__unique,axiom,
% 9.00/5.09 ! [F: set_int > set_int > set_int,X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_int,Y4: set_int] : ( ord_less_eq_set_int @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: set_int,Y4: set_int,Z4: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ Z4 @ X5 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7130_sup__unique,axiom,
% 9.00/5.09 ! [F: rat > rat > rat,X2: rat,Y3: rat] :
% 9.00/5.09 ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: rat,Y4: rat,Z4: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ Z4 @ X5 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_sup_rat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7131_sup__unique,axiom,
% 9.00/5.09 ! [F: nat > nat > nat,X2: nat,Y3: nat] :
% 9.00/5.09 ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: nat,Y4: nat,Z4: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Z4 @ X5 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_sup_nat @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7132_sup__unique,axiom,
% 9.00/5.09 ! [F: int > int > int,X2: int,Y3: int] :
% 9.00/5.09 ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ X5 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ Y4 @ ( F @ X5 @ Y4 ) )
% 9.00/5.09 => ( ! [X5: int,Y4: int,Z4: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Y4 @ X5 )
% 9.00/5.09 => ( ( ord_less_eq_int @ Z4 @ X5 )
% 9.00/5.09 => ( ord_less_eq_int @ ( F @ Y4 @ Z4 ) @ X5 ) ) )
% 9.00/5.09 => ( ( sup_sup_int @ X2 @ Y3 )
% 9.00/5.09 = ( F @ X2 @ Y3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7133_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7134_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7135_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7136_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_set_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7137_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7138_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7139_sup_Oabsorb1,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7140_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7141_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7142_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7143_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_set_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7144_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7145_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7146_sup_Oabsorb2,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7147_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: set_nat,X2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7148_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: set_Pr8693737435421807431at_nat,X2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7149_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: set_Pr4329608150637261639at_nat,X2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7150_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: set_int,X2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7151_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: rat,X2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_sup_rat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7152_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: nat,X2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_sup_nat @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7153_sup__absorb1,axiom,
% 9.00/5.09 ! [Y3: int,X2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ Y3 @ X2 )
% 9.00/5.09 => ( ( sup_sup_int @ X2 @ Y3 )
% 9.00/5.09 = X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7154_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7155_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7156_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7157_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_sup_set_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7158_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_sup_rat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7159_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_sup_nat @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7160_sup__absorb2,axiom,
% 9.00/5.09 ! [X2: int,Y3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.09 => ( ( sup_sup_int @ X2 @ Y3 )
% 9.00/5.09 = Y3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7161_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7162_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_le3000389064537975527at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7163_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_le1268244103169919719at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7164_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: set_int,C: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_set_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7165_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7166_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7167_sup_OboundedE,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7168_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat,C: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7169_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7170_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7171_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: set_int,A2: set_int,C: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ ( sup_sup_set_int @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7172_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7173_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7174_sup_OboundedI,axiom,
% 9.00/5.09 ! [B3: int,A2: int,C: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ B3 @ A2 )
% 9.00/5.09 => ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_int @ ( sup_sup_int @ B3 @ C ) @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7175_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [B4: set_nat,A4: set_nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_sup_set_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7176_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_le3000389064537975527at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7177_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_le1268244103169919719at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7178_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_sup_set_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7179_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_sup_rat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7180_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_sup_nat @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7181_sup_Oorder__iff,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( A4
% 9.00/5.09 = ( sup_sup_int @ A4 @ B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7182_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7183_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ A2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7184_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7185_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7186_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] : ( ord_less_eq_rat @ A2 @ ( sup_sup_rat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7187_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7188_sup_Ocobounded1,axiom,
% 9.00/5.09 ! [A2: int,B3: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7189_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B3 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7190_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ B3 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7191_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B3 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7192_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: set_int,A2: set_int] : ( ord_less_eq_set_int @ B3 @ ( sup_sup_set_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7193_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] : ( ord_less_eq_rat @ B3 @ ( sup_sup_rat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7194_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( sup_sup_nat @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7195_sup_Ocobounded2,axiom,
% 9.00/5.09 ! [B3: int,A2: int] : ( ord_less_eq_int @ B3 @ ( sup_sup_int @ A2 @ B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7196_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [B4: set_nat,A4: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7197_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_le3000389064537975527at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7198_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_le1268244103169919719at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7199_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [B4: set_int,A4: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7200_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( ( sup_sup_rat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7201_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( ( sup_sup_nat @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7202_sup_Oabsorb__iff1,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( ( sup_sup_int @ A4 @ B4 )
% 9.00/5.09 = A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7203_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat,B4: set_nat] :
% 9.00/5.09 ( ( sup_sup_set_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7204_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_le3000389064537975527at_nat
% 9.00/5.09 = ( ^ [A4: set_Pr8693737435421807431at_nat,B4: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( sup_su718114333110466843at_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7205_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_le1268244103169919719at_nat
% 9.00/5.09 = ( ^ [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7206_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 = ( ^ [A4: set_int,B4: set_int] :
% 9.00/5.09 ( ( sup_sup_set_int @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7207_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.09 ( ( sup_sup_rat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7208_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.09 ( ( sup_sup_nat @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7209_sup_Oabsorb__iff2,axiom,
% 9.00/5.09 ( ord_less_eq_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] :
% 9.00/5.09 ( ( sup_sup_int @ A4 @ B4 )
% 9.00/5.09 = B4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7210_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7211_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7212_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7213_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: set_int,A2: set_int,B3: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7214_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7215_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7216_sup_OcoboundedI1,axiom,
% 9.00/5.09 ! [C: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7217_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: set_nat,B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_set_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7218_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le3000389064537975527at_nat @ C @ B3 )
% 9.00/5.09 => ( ord_le3000389064537975527at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7219_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le1268244103169919719at_nat @ C @ B3 )
% 9.00/5.09 => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7220_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: set_int,B3: set_int,A2: set_int] :
% 9.00/5.09 ( ( ord_less_eq_set_int @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_set_int @ C @ ( sup_sup_set_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7221_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_eq_rat @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7222_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_eq_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7223_sup_OcoboundedI2,axiom,
% 9.00/5.09 ! [C: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ C @ B3 )
% 9.00/5.09 => ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7224_less__infI1,axiom,
% 9.00/5.09 ! [A2: set_nat,X2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7225_less__infI1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7226_less__infI1,axiom,
% 9.00/5.09 ! [A2: real,X2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7227_less__infI1,axiom,
% 9.00/5.09 ! [A2: rat,X2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7228_less__infI1,axiom,
% 9.00/5.09 ! [A2: nat,X2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7229_less__infI1,axiom,
% 9.00/5.09 ! [A2: int,X2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ X2 )
% 9.00/5.09 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7230_less__infI2,axiom,
% 9.00/5.09 ! [B3: set_nat,X2: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7231_less__infI2,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7232_less__infI2,axiom,
% 9.00/5.09 ! [B3: real,X2: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7233_less__infI2,axiom,
% 9.00/5.09 ! [B3: rat,X2: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7234_less__infI2,axiom,
% 9.00/5.09 ! [B3: nat,X2: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7235_less__infI2,axiom,
% 9.00/5.09 ! [B3: int,X2: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ B3 @ X2 )
% 9.00/5.09 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ X2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7236_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7237_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7238_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_real @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7239_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7240_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7241_inf_Oabsorb3,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.09 => ( ( inf_inf_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7242_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_set_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7243_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_in2572325071724192079at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7244_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_real @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7245_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7246_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7247_inf_Oabsorb4,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.09 => ( ( inf_inf_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7248_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat,C: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ A2 @ ( inf_inf_set_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_set_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_set_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7249_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ A2 @ ( inf_in2572325071724192079at_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_le7866589430770878221at_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7250_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: real,B3: real,C: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ ( inf_inf_real @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_real @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7251_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat,C: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ ( inf_inf_rat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_rat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7252_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat,C: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ ( inf_inf_nat @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_nat @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7253_inf_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [A2: int,B3: int,C: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ ( inf_inf_int @ B3 @ C ) )
% 9.00/5.09 => ~ ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.09 => ~ ( ord_less_int @ A2 @ C ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7254_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat,B4: set_nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_inf_set_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7255_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_le7866589430770878221at_nat
% 9.00/5.09 = ( ^ [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7256_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_real
% 9.00/5.09 = ( ^ [A4: real,B4: real] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_inf_real @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7257_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_inf_rat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7258_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_inf_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7259_inf_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( inf_inf_int @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7260_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: set_nat,C: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7261_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ A2 @ C )
% 9.00/5.09 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7262_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: real,C: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ C )
% 9.00/5.09 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7263_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: rat,C: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ C )
% 9.00/5.09 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7264_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: nat,C: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ C )
% 9.00/5.09 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7265_inf_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [A2: int,C: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ C )
% 9.00/5.09 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7266_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ B3 @ C )
% 9.00/5.09 => ( ord_less_set_nat @ ( inf_inf_set_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7267_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( ord_le7866589430770878221at_nat @ B3 @ C )
% 9.00/5.09 => ( ord_le7866589430770878221at_nat @ ( inf_in2572325071724192079at_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7268_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: real,C: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ B3 @ C )
% 9.00/5.09 => ( ord_less_real @ ( inf_inf_real @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7269_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ B3 @ C )
% 9.00/5.09 => ( ord_less_rat @ ( inf_inf_rat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7270_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ B3 @ C )
% 9.00/5.09 => ( ord_less_nat @ ( inf_inf_nat @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7271_inf_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ B3 @ C )
% 9.00/5.09 => ( ord_less_int @ ( inf_inf_int @ A2 @ B3 ) @ C ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7272_less__supI1,axiom,
% 9.00/5.09 ! [X2: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7273_less__supI1,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7274_less__supI1,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7275_less__supI1,axiom,
% 9.00/5.09 ! [X2: real,A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7276_less__supI1,axiom,
% 9.00/5.09 ! [X2: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7277_less__supI1,axiom,
% 9.00/5.09 ! [X2: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7278_less__supI1,axiom,
% 9.00/5.09 ! [X2: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ X2 @ A2 )
% 9.00/5.09 => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7279_less__supI2,axiom,
% 9.00/5.09 ! [X2: set_nat,B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_set_nat @ X2 @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7280_less__supI2,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_le6428140832669894131at_nat @ X2 @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7281_less__supI2,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_le2604355607129572851at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7282_less__supI2,axiom,
% 9.00/5.09 ! [X2: real,B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_real @ X2 @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7283_less__supI2,axiom,
% 9.00/5.09 ! [X2: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_rat @ X2 @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7284_less__supI2,axiom,
% 9.00/5.09 ! [X2: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7285_less__supI2,axiom,
% 9.00/5.09 ! [X2: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ X2 @ B3 )
% 9.00/5.09 => ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7286_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7287_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7288_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7289_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_real @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7290_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_rat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7291_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_nat @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7292_sup_Oabsorb3,axiom,
% 9.00/5.09 ! [B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.09 => ( ( sup_sup_int @ A2 @ B3 )
% 9.00/5.09 = A2 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7293_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_set_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7294_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_su718114333110466843at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7295_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_su5525570899277871387at_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7296_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_real @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7297_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_rat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7298_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_nat @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7299_sup_Oabsorb4,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ B3 )
% 9.00/5.09 => ( ( sup_sup_int @ A2 @ B3 )
% 9.00/5.09 = B3 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7300_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: set_nat,C: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_set_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7301_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: set_Pr8693737435421807431at_nat,C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ ( sup_su718114333110466843at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_le6428140832669894131at_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_le6428140832669894131at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7302_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ ( sup_su5525570899277871387at_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_le2604355607129572851at_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_le2604355607129572851at_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7303_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: real,C: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ ( sup_sup_real @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_real @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7304_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: rat,C: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ ( sup_sup_rat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_rat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_rat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7305_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: nat,C: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ ( sup_sup_nat @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7306_sup_Ostrict__boundedE,axiom,
% 9.00/5.09 ! [B3: int,C: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ ( sup_sup_int @ B3 @ C ) @ A2 )
% 9.00/5.09 => ~ ( ( ord_less_int @ B3 @ A2 )
% 9.00/5.09 => ~ ( ord_less_int @ C @ A2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7307_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_set_nat
% 9.00/5.09 = ( ^ [B4: set_nat,A4: set_nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_sup_set_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7308_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_le6428140832669894131at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr8693737435421807431at_nat,A4: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7309_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_le2604355607129572851at_nat
% 9.00/5.09 = ( ^ [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7310_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_real
% 9.00/5.09 = ( ^ [B4: real,A4: real] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_sup_real @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7311_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_rat
% 9.00/5.09 = ( ^ [B4: rat,A4: rat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_sup_rat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7312_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_nat
% 9.00/5.09 = ( ^ [B4: nat,A4: nat] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_sup_nat @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7313_sup_Ostrict__order__iff,axiom,
% 9.00/5.09 ( ord_less_int
% 9.00/5.09 = ( ^ [B4: int,A4: int] :
% 9.00/5.09 ( ( A4
% 9.00/5.09 = ( sup_sup_int @ A4 @ B4 ) )
% 9.00/5.09 & ( A4 != B4 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7314_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: set_nat,A2: set_nat,B3: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7315_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7316_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ C @ A2 )
% 9.00/5.09 => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7317_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: real,A2: real,B3: real] :
% 9.00/5.09 ( ( ord_less_real @ C @ A2 )
% 9.00/5.09 => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7318_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: rat,A2: rat,B3: rat] :
% 9.00/5.09 ( ( ord_less_rat @ C @ A2 )
% 9.00/5.09 => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7319_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: nat,A2: nat,B3: nat] :
% 9.00/5.09 ( ( ord_less_nat @ C @ A2 )
% 9.00/5.09 => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7320_sup_Ostrict__coboundedI1,axiom,
% 9.00/5.09 ! [C: int,A2: int,B3: int] :
% 9.00/5.09 ( ( ord_less_int @ C @ A2 )
% 9.00/5.09 => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7321_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: set_nat,B3: set_nat,A2: set_nat] :
% 9.00/5.09 ( ( ord_less_set_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7322_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: set_Pr8693737435421807431at_nat,B3: set_Pr8693737435421807431at_nat,A2: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( ( ord_le6428140832669894131at_nat @ C @ B3 )
% 9.00/5.09 => ( ord_le6428140832669894131at_nat @ C @ ( sup_su718114333110466843at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7323_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: set_Pr4329608150637261639at_nat,B3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( ( ord_le2604355607129572851at_nat @ C @ B3 )
% 9.00/5.09 => ( ord_le2604355607129572851at_nat @ C @ ( sup_su5525570899277871387at_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7324_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: real,B3: real,A2: real] :
% 9.00/5.09 ( ( ord_less_real @ C @ B3 )
% 9.00/5.09 => ( ord_less_real @ C @ ( sup_sup_real @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7325_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: rat,B3: rat,A2: rat] :
% 9.00/5.09 ( ( ord_less_rat @ C @ B3 )
% 9.00/5.09 => ( ord_less_rat @ C @ ( sup_sup_rat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7326_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: nat,B3: nat,A2: nat] :
% 9.00/5.09 ( ( ord_less_nat @ C @ B3 )
% 9.00/5.09 => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7327_sup_Ostrict__coboundedI2,axiom,
% 9.00/5.09 ! [C: int,B3: int,A2: int] :
% 9.00/5.09 ( ( ord_less_int @ C @ B3 )
% 9.00/5.09 => ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B3 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7328_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ Y3 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X2 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7329_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y3 @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y3 ) @ ( sup_sup_set_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7330_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ X2 @ ( inf_in4302113700860409141at_nat @ Y3 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ ( sup_su718114333110466843at_nat @ X2 @ Y3 ) @ ( sup_su718114333110466843at_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7331_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ ( inf_in7913087082777306421at_nat @ Y3 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X2 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7332_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ X2 @ ( inf_inf_set_int @ Y3 @ Z ) ) @ ( inf_inf_set_int @ ( sup_sup_set_int @ X2 @ Y3 ) @ ( sup_sup_set_int @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7333_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X2 @ ( inf_inf_rat @ Y3 @ Z ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X2 @ Y3 ) @ ( sup_sup_rat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7334_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y3 @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y3 ) @ ( sup_sup_nat @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7335_distrib__sup__le,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ X2 @ ( inf_inf_int @ Y3 @ Z ) ) @ ( inf_inf_int @ ( sup_sup_int @ X2 @ Y3 ) @ ( sup_sup_int @ X2 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7336_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X2 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ X2 @ Z ) ) @ ( inf_in2572325071724192079at_nat @ X2 @ ( sup_su6327502436637775413at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7337_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: set_nat,Y3: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y3 ) @ ( inf_inf_set_nat @ X2 @ Z ) ) @ ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7338_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: set_Pr8693737435421807431at_nat,Y3: set_Pr8693737435421807431at_nat,Z: set_Pr8693737435421807431at_nat] : ( ord_le3000389064537975527at_nat @ ( sup_su718114333110466843at_nat @ ( inf_in4302113700860409141at_nat @ X2 @ Y3 ) @ ( inf_in4302113700860409141at_nat @ X2 @ Z ) ) @ ( inf_in4302113700860409141at_nat @ X2 @ ( sup_su718114333110466843at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7339_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,Z: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X2 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ X2 @ Z ) ) @ ( inf_in7913087082777306421at_nat @ X2 @ ( sup_su5525570899277871387at_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7340_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: set_int,Y3: set_int,Z: set_int] : ( ord_less_eq_set_int @ ( sup_sup_set_int @ ( inf_inf_set_int @ X2 @ Y3 ) @ ( inf_inf_set_int @ X2 @ Z ) ) @ ( inf_inf_set_int @ X2 @ ( sup_sup_set_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7341_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: rat,Y3: rat,Z: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X2 @ Y3 ) @ ( inf_inf_rat @ X2 @ Z ) ) @ ( inf_inf_rat @ X2 @ ( sup_sup_rat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7342_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: nat,Y3: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y3 ) @ ( inf_inf_nat @ X2 @ Z ) ) @ ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7343_distrib__inf__le,axiom,
% 9.00/5.09 ! [X2: int,Y3: int,Z: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X2 @ Y3 ) @ ( inf_inf_int @ X2 @ Z ) ) @ ( inf_inf_int @ X2 @ ( sup_sup_int @ Y3 @ Z ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7344_lemma__termdiff3,axiom,
% 9.00/5.09 ! [H2: real,Z: real,K4: real,N: nat] :
% 9.00/5.09 ( ( H2 != zero_zero_real )
% 9.00/5.09 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K4 )
% 9.00/5.09 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H2 ) ) @ K4 )
% 9.00/5.09 => ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7345_lemma__termdiff3,axiom,
% 9.00/5.09 ! [H2: complex,Z: complex,K4: real,N: nat] :
% 9.00/5.09 ( ( H2 != zero_zero_complex )
% 9.00/5.09 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K4 )
% 9.00/5.09 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H2 ) ) @ K4 )
% 9.00/5.09 => ( 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 @ K4 @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7346_set__encode__insert,axiom,
% 9.00/5.09 ! [A3: set_nat,N: nat] :
% 9.00/5.09 ( ( finite_finite_nat @ A3 )
% 9.00/5.09 => ( ~ ( member_nat @ N @ A3 )
% 9.00/5.09 => ( ( nat_set_encode @ ( insert_nat @ N @ A3 ) )
% 9.00/5.09 = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7347_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_int,Ys2: list_int] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr4439495888332055232nt_int @ ( product_int_int @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( product_Pair_int_int @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7348_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_int,Ys2: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr3474266648193625910T_VEBT @ ( produc662631939642741121T_VEBT @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc3329399203697025711T_VEBT @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7349_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_int,Ys2: list_o] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr7514405829937366042_int_o @ ( product_int_o @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( product_Pair_int_o @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7350_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_int,Ys2: list_nat] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr8617346907841251940nt_nat @ ( product_int_nat @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( product_Pair_int_nat @ ( nth_int @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7351_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_int] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7352_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7353_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_o] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) @ ( nth_o @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_o @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7354_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_VEBT_VEBT,Ys2: list_nat] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) @ ( nth_nat @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_nat @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7355_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_o,Ys2: list_int] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) @ ( nth_int @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7356_product__nth,axiom,
% 9.00/5.09 ! [N: nat,Xs: list_o,Ys2: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) )
% 9.00/5.09 => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys2 ) @ N )
% 9.00/5.09 = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) @ ( nth_VEBT_VEBT @ Ys2 @ ( modulo_modulo_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys2 ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7357_unset__bit__0,axiom,
% 9.00/5.09 ! [A2: code_integer] :
% 9.00/5.09 ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7358_unset__bit__0,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7359_unset__bit__0,axiom,
% 9.00/5.09 ! [A2: int] :
% 9.00/5.09 ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A2 )
% 9.00/5.09 = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7360_succ__less__length__list,axiom,
% 9.00/5.09 ! [Deg: nat,Mi: nat,X2: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_option_nat
% 9.00/5.09 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 != none_nat )
% 9.00/5.09 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ none_nat
% 9.00/5.09 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7361_succ__greatereq__min,axiom,
% 9.00/5.09 ! [Deg: nat,Mi: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ord_less_eq_nat @ Mi @ X2 )
% 9.00/5.09 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 != none_nat )
% 9.00/5.09 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ none_nat
% 9.00/5.09 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 @ none_nat ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7362_set__vebt_H__def,axiom,
% 9.00/5.09 ( vEBT_VEBT_set_vebt
% 9.00/5.09 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7363_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: real > $o,Q: real > $o] :
% 9.00/5.09 ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.00/5.09 | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 9.00/5.09 => ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7364_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.09 ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.00/5.09 | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 9.00/5.09 => ( finite8100373058378681591st_nat
% 9.00/5.09 @ ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7365_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.09 ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.00/5.09 | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 9.00/5.09 => ( finite1152437895449049373et_nat
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7366_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.09 ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.09 | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 9.00/5.09 => ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7367_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: int > $o,Q: int > $o] :
% 9.00/5.09 ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.00/5.09 | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 9.00/5.09 => ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7368_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: complex > $o,Q: complex > $o] :
% 9.00/5.09 ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.00/5.09 | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 9.00/5.09 => ( finite3207457112153483333omplex
% 9.00/5.09 @ ( collect_complex
% 9.00/5.09 @ ^ [X: complex] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7369_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.00/5.09 ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.00/5.09 | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 9.00/5.09 => ( finite6177210948735845034at_nat
% 9.00/5.09 @ ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7370_finite__Collect__conjI,axiom,
% 9.00/5.09 ! [P: extended_enat > $o,Q: extended_enat > $o] :
% 9.00/5.09 ( ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 9.00/5.09 | ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) )
% 9.00/5.09 => ( finite4001608067531595151d_enat
% 9.00/5.09 @ ( collec4429806609662206161d_enat
% 9.00/5.09 @ ^ [X: extended_enat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 & ( Q @ X ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7371_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: real > $o,Q: real > $o] :
% 9.00/5.09 ( ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.00/5.09 & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7372_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.09 ( ( finite8100373058378681591st_nat
% 9.00/5.09 @ ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.00/5.09 & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7373_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.09 ( ( finite1152437895449049373et_nat
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.00/5.09 & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7374_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.09 ( ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.09 & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7375_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: int > $o,Q: int > $o] :
% 9.00/5.09 ( ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.00/5.09 & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7376_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: complex > $o,Q: complex > $o] :
% 9.00/5.09 ( ( finite3207457112153483333omplex
% 9.00/5.09 @ ( collect_complex
% 9.00/5.09 @ ^ [X: complex] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.00/5.09 & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7377_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.00/5.09 ( ( finite6177210948735845034at_nat
% 9.00/5.09 @ ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.00/5.09 & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7378_finite__Collect__disjI,axiom,
% 9.00/5.09 ! [P: extended_enat > $o,Q: extended_enat > $o] :
% 9.00/5.09 ( ( finite4001608067531595151d_enat
% 9.00/5.09 @ ( collec4429806609662206161d_enat
% 9.00/5.09 @ ^ [X: extended_enat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) ) )
% 9.00/5.09 = ( ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 9.00/5.09 & ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ Q ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7379_succ__empty,axiom,
% 9.00/5.09 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.09 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.09 => ( ( ( vEBT_vebt_succ @ T @ X2 )
% 9.00/5.09 = none_nat )
% 9.00/5.09 = ( ( collect_nat
% 9.00/5.09 @ ^ [Y: nat] :
% 9.00/5.09 ( ( vEBT_vebt_member @ T @ Y )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Y ) ) )
% 9.00/5.09 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7380_pred__empty,axiom,
% 9.00/5.09 ! [T: vEBT_VEBT,N: nat,X2: nat] :
% 9.00/5.09 ( ( vEBT_invar_vebt @ T @ N )
% 9.00/5.09 => ( ( ( vEBT_vebt_pred @ T @ X2 )
% 9.00/5.09 = none_nat )
% 9.00/5.09 = ( ( collect_nat
% 9.00/5.09 @ ^ [Y: nat] :
% 9.00/5.09 ( ( vEBT_vebt_member @ T @ Y )
% 9.00/5.09 & ( ord_less_nat @ Y @ X2 ) ) )
% 9.00/5.09 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7381_unset__bit__nonnegative__int__iff,axiom,
% 9.00/5.09 ! [N: nat,K: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N @ K ) )
% 9.00/5.09 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7382_unset__bit__negative__int__iff,axiom,
% 9.00/5.09 ! [N: nat,K: int] :
% 9.00/5.09 ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.09 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7383_finite__nth__roots,axiom,
% 9.00/5.09 ! [N: nat,C: complex] :
% 9.00/5.09 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.09 => ( finite3207457112153483333omplex
% 9.00/5.09 @ ( collect_complex
% 9.00/5.09 @ ^ [Z2: complex] :
% 9.00/5.09 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.09 = C ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7384_finite__Collect__subsets,axiom,
% 9.00/5.09 ! [A3: set_nat] :
% 9.00/5.09 ( ( finite_finite_nat @ A3 )
% 9.00/5.09 => ( finite1152437895449049373et_nat
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [B6: set_nat] : ( ord_less_eq_set_nat @ B6 @ A3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7385_finite__Collect__subsets,axiom,
% 9.00/5.09 ! [A3: set_complex] :
% 9.00/5.09 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.09 => ( finite6551019134538273531omplex
% 9.00/5.09 @ ( collect_set_complex
% 9.00/5.09 @ ^ [B6: set_complex] : ( ord_le211207098394363844omplex @ B6 @ A3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7386_finite__Collect__subsets,axiom,
% 9.00/5.09 ! [A3: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.09 => ( finite9047747110432174090at_nat
% 9.00/5.09 @ ( collec5514110066124741708at_nat
% 9.00/5.09 @ ^ [B6: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B6 @ A3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7387_finite__Collect__subsets,axiom,
% 9.00/5.09 ! [A3: set_Extended_enat] :
% 9.00/5.09 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.09 => ( finite5468666774076196335d_enat
% 9.00/5.09 @ ( collec2260605976452661553d_enat
% 9.00/5.09 @ ^ [B6: set_Extended_enat] : ( ord_le7203529160286727270d_enat @ B6 @ A3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7388_finite__Collect__subsets,axiom,
% 9.00/5.09 ! [A3: set_int] :
% 9.00/5.09 ( ( finite_finite_int @ A3 )
% 9.00/5.09 => ( finite6197958912794628473et_int
% 9.00/5.09 @ ( collect_set_int
% 9.00/5.09 @ ^ [B6: set_int] : ( ord_less_eq_set_int @ B6 @ A3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7389_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: product_prod_nat_nat] :
% 9.00/5.09 ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ( ^ [Y6: product_prod_nat_nat,Z3: product_prod_nat_nat] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7390_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: list_nat] :
% 9.00/5.09 ( ( collect_list_nat
% 9.00/5.09 @ ( ^ [Y6: list_nat,Z3: list_nat] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7391_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: set_nat] :
% 9.00/5.09 ( ( collect_set_nat
% 9.00/5.09 @ ( ^ [Y6: set_nat,Z3: set_nat] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7392_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: real] :
% 9.00/5.09 ( ( collect_real
% 9.00/5.09 @ ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7393_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: $o] :
% 9.00/5.09 ( ( collect_o
% 9.00/5.09 @ ( ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7394_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( collect_nat
% 9.00/5.09 @ ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7395_singleton__conv2,axiom,
% 9.00/5.09 ! [A2: int] :
% 9.00/5.09 ( ( collect_int
% 9.00/5.09 @ ( ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 9.00/5.09 @ A2 ) )
% 9.00/5.09 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7396_singleton__conv,axiom,
% 9.00/5.09 ! [A2: product_prod_nat_nat] :
% 9.00/5.09 ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] : ( X = A2 ) )
% 9.00/5.09 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7397_singleton__conv,axiom,
% 9.00/5.09 ! [A2: list_nat] :
% 9.00/5.09 ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] : ( X = A2 ) )
% 9.00/5.09 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7398_singleton__conv,axiom,
% 9.00/5.09 ! [A2: set_nat] :
% 9.00/5.09 ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] : ( X = A2 ) )
% 9.00/5.09 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7399_singleton__conv,axiom,
% 9.00/5.09 ! [A2: real] :
% 9.00/5.09 ( ( collect_real
% 9.00/5.09 @ ^ [X: real] : ( X = A2 ) )
% 9.00/5.09 = ( insert_real @ A2 @ bot_bot_set_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7400_singleton__conv,axiom,
% 9.00/5.09 ! [A2: $o] :
% 9.00/5.09 ( ( collect_o
% 9.00/5.09 @ ^ [X: $o] : ( X = A2 ) )
% 9.00/5.09 = ( insert_o @ A2 @ bot_bot_set_o ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7401_singleton__conv,axiom,
% 9.00/5.09 ! [A2: nat] :
% 9.00/5.09 ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] : ( X = A2 ) )
% 9.00/5.09 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7402_singleton__conv,axiom,
% 9.00/5.09 ! [A2: int] :
% 9.00/5.09 ( ( collect_int
% 9.00/5.09 @ ^ [X: int] : ( X = A2 ) )
% 9.00/5.09 = ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7403_finite__Collect__less__nat,axiom,
% 9.00/5.09 ! [K: nat] :
% 9.00/5.09 ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [N2: nat] : ( ord_less_nat @ N2 @ K ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7404_finite__Collect__le__nat,axiom,
% 9.00/5.09 ! [K: nat] :
% 9.00/5.09 ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [N2: nat] : ( ord_less_eq_nat @ N2 @ K ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7405_card__Collect__less__nat,axiom,
% 9.00/5.09 ! [N: nat] :
% 9.00/5.09 ( ( finite_card_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [I4: nat] : ( ord_less_nat @ I4 @ N ) ) )
% 9.00/5.09 = N ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7406_finite__interval__int1,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [I4: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ I4 )
% 9.00/5.09 & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7407_finite__interval__int4,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [I4: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ I4 )
% 9.00/5.09 & ( ord_less_int @ I4 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7408_norm__eq__zero,axiom,
% 9.00/5.09 ! [X2: real] :
% 9.00/5.09 ( ( ( real_V7735802525324610683m_real @ X2 )
% 9.00/5.09 = zero_zero_real )
% 9.00/5.09 = ( X2 = zero_zero_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7409_norm__eq__zero,axiom,
% 9.00/5.09 ! [X2: complex] :
% 9.00/5.09 ( ( ( real_V1022390504157884413omplex @ X2 )
% 9.00/5.09 = zero_zero_real )
% 9.00/5.09 = ( X2 = zero_zero_complex ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7410_norm__zero,axiom,
% 9.00/5.09 ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 9.00/5.09 = zero_zero_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7411_norm__zero,axiom,
% 9.00/5.09 ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 9.00/5.09 = zero_zero_real ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7412_set__encode__empty,axiom,
% 9.00/5.09 ( ( nat_set_encode @ bot_bot_set_nat )
% 9.00/5.09 = zero_zero_nat ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7413_card__Collect__le__nat,axiom,
% 9.00/5.09 ! [N: nat] :
% 9.00/5.09 ( ( finite_card_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [I4: nat] : ( ord_less_eq_nat @ I4 @ N ) ) )
% 9.00/5.09 = ( suc @ N ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7414_finite__interval__int2,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [I4: int] :
% 9.00/5.09 ( ( ord_less_eq_int @ A2 @ I4 )
% 9.00/5.09 & ( ord_less_int @ I4 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7415_finite__interval__int3,axiom,
% 9.00/5.09 ! [A2: int,B3: int] :
% 9.00/5.09 ( finite_finite_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [I4: int] :
% 9.00/5.09 ( ( ord_less_int @ A2 @ I4 )
% 9.00/5.09 & ( ord_less_eq_int @ I4 @ B3 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7416_zero__less__norm__iff,axiom,
% 9.00/5.09 ! [X2: real] :
% 9.00/5.09 ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X2 ) )
% 9.00/5.09 = ( X2 != zero_zero_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7417_zero__less__norm__iff,axiom,
% 9.00/5.09 ! [X2: complex] :
% 9.00/5.09 ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) )
% 9.00/5.09 = ( X2 != zero_zero_complex ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7418_norm__le__zero__iff,axiom,
% 9.00/5.09 ! [X2: real] :
% 9.00/5.09 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ zero_zero_real )
% 9.00/5.09 = ( X2 = zero_zero_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7419_norm__le__zero__iff,axiom,
% 9.00/5.09 ! [X2: complex] :
% 9.00/5.09 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real )
% 9.00/5.09 = ( X2 = zero_zero_complex ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7420_del__x__not__mia,axiom,
% 9.00/5.09 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ( ord_less_nat @ Mi @ X2 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Mi
% 9.00/5.09 @ ( if_nat @ ( X2 = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Mi
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7421_del__x__not__mi__new__node__nil,axiom,
% 9.00/5.09 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ( ord_less_nat @ Mi @ X2 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( Sn
% 9.00/5.09 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Mi
% 9.00/5.09 @ ( if_nat @ ( X2 = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ Sn )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Mi
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ Newlist
% 9.00/5.09 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7422_del__x__not__mi,axiom,
% 9.00/5.09 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,H2: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ( ord_less_nat @ Mi @ X2 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Mi
% 9.00/5.09 @ ( if_nat @ ( X2 = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Mi
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ Newlist
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 9.00/5.09 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X2 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7423_del__in__range,axiom,
% 9.00/5.09 ! [Mi: nat,X2: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ( ord_less_eq_nat @ Mi @ X2 )
% 9.00/5.09 & ( ord_less_eq_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( ( X2 = Mi )
% 9.00/5.09 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 = Ma ) )
% 9.00/5.09 & ( ( X2 != Mi )
% 9.00/5.09 => ( X2 = Ma ) ) )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( ( X2 = Mi )
% 9.00/5.09 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 = Ma ) )
% 9.00/5.09 & ( ( X2 != Mi )
% 9.00/5.09 => ( X2 = Ma ) ) )
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ Summary ) )
% 9.00/5.09 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7424_del__x__mi,axiom,
% 9.00/5.09 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
% 9.00/5.09 ( ( ( X2 = Mi )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( Xn
% 9.00/5.09 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Xn
% 9.00/5.09 @ ( if_nat @ ( Xn = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Xn
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7425_del__x__mi__lets__in,axiom,
% 9.00/5.09 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
% 9.00/5.09 ( ( ( X2 = Mi )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( Xn
% 9.00/5.09 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Xn
% 9.00/5.09 @ ( if_nat @ ( Xn = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Xn
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ Newlist
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ H2 ) ) ) )
% 9.00/5.09 & ( ~ ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H2 ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7426_del__x__mi__lets__in__minNull,axiom,
% 9.00/5.09 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H2: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
% 9.00/5.09 ( ( ( X2 = Mi )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = H2 )
% 9.00/5.09 => ( ( Xn
% 9.00/5.09 = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
% 9.00/5.09 => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.09 = L )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( Newnode
% 9.00/5.09 = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H2 ) @ L ) )
% 9.00/5.09 => ( ( Newlist
% 9.00/5.09 = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H2 @ Newnode ) )
% 9.00/5.09 => ( ( vEBT_VEBT_minNull @ Newnode )
% 9.00/5.09 => ( ( Sn
% 9.00/5.09 = ( vEBT_vebt_delete @ Summary @ H2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ Xn
% 9.00/5.09 @ ( if_nat @ ( Xn = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ Sn )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ Xn
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ Newlist
% 9.00/5.09 @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7427_del__x__mia,axiom,
% 9.00/5.09 ! [X2: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ( X2 = Mi )
% 9.00/5.09 & ( ord_less_nat @ X2 @ Ma ) )
% 9.00/5.09 => ( ( Mi != Ma )
% 9.00/5.09 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 = Ma )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( vEBT_Node
% 9.00/5.09 @ ( some_P7363390416028606310at_nat
% 9.00/5.09 @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 @ ( if_nat
% 9.00/5.09 @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.09 = Ma )
% 9.00/5.09 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ Ma ) ) )
% 9.00/5.09 @ Deg
% 9.00/5.09 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ Summary ) )
% 9.00/5.09 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7428_pred__less__length__list,axiom,
% 9.00/5.09 ! [Deg: nat,X2: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 9.00/5.09 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_option_nat
% 9.00/5.09 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 != none_nat )
% 9.00/5.09 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.00/5.09 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7429_pred__lesseq__max,axiom,
% 9.00/5.09 ! [Deg: nat,X2: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.09 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.00/5.09 => ( ( ord_less_eq_nat @ X2 @ Ma )
% 9.00/5.09 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.09 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 != none_nat )
% 9.00/5.09 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.09 @ ( 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 @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.09 @ ( if_option_nat
% 9.00/5.09 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.09 = none_nat )
% 9.00/5.09 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.00/5.09 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.09 @ none_nat ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7430_max__def__raw,axiom,
% 9.00/5.09 ( ord_max_set_int
% 9.00/5.09 = ( ^ [A4: set_int,B4: set_int] : ( if_set_int @ ( ord_less_eq_set_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7431_max__def__raw,axiom,
% 9.00/5.09 ( ord_max_rat
% 9.00/5.09 = ( ^ [A4: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7432_max__def__raw,axiom,
% 9.00/5.09 ( ord_max_num
% 9.00/5.09 = ( ^ [A4: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7433_max__def__raw,axiom,
% 9.00/5.09 ( ord_max_nat
% 9.00/5.09 = ( ^ [A4: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7434_max__def__raw,axiom,
% 9.00/5.09 ( ord_max_int
% 9.00/5.09 = ( ^ [A4: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B4 ) @ B4 @ A4 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7435_finite__M__bounded__by__nat,axiom,
% 9.00/5.09 ! [P: nat > $o,I: nat] :
% 9.00/5.09 ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [K3: nat] :
% 9.00/5.09 ( ( P @ K3 )
% 9.00/5.09 & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7436_card__nth__roots,axiom,
% 9.00/5.09 ! [C: complex,N: nat] :
% 9.00/5.09 ( ( C != zero_zero_complex )
% 9.00/5.09 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.09 => ( ( finite_card_complex
% 9.00/5.09 @ ( collect_complex
% 9.00/5.09 @ ^ [Z2: complex] :
% 9.00/5.09 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.09 = C ) ) )
% 9.00/5.09 = N ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7437_card__roots__unity__eq,axiom,
% 9.00/5.09 ! [N: nat] :
% 9.00/5.09 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.09 => ( ( finite_card_complex
% 9.00/5.09 @ ( collect_complex
% 9.00/5.09 @ ^ [Z2: complex] :
% 9.00/5.09 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.09 = one_one_complex ) ) )
% 9.00/5.09 = N ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7438_minus__set__def,axiom,
% 9.00/5.09 ( minus_minus_set_o
% 9.00/5.09 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ( minus_minus_o_o
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7439_minus__set__def,axiom,
% 9.00/5.09 ( minus_minus_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ( minus_minus_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7440_minus__set__def,axiom,
% 9.00/5.09 ( minus_7954133019191499631st_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ( minus_1139252259498527702_nat_o
% 9.00/5.09 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7441_minus__set__def,axiom,
% 9.00/5.09 ( minus_2163939370556025621et_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ( minus_6910147592129066416_nat_o
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7442_minus__set__def,axiom,
% 9.00/5.09 ( minus_minus_set_int
% 9.00/5.09 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ( minus_minus_int_o
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7443_minus__set__def,axiom,
% 9.00/5.09 ( minus_minus_set_nat
% 9.00/5.09 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ( minus_minus_nat_o
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7444_set__diff__eq,axiom,
% 9.00/5.09 ( minus_minus_set_o
% 9.00/5.09 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( member_o @ X @ A6 )
% 9.00/5.09 & ~ ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7445_set__diff__eq,axiom,
% 9.00/5.09 ( minus_minus_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( member_real @ X @ A6 )
% 9.00/5.09 & ~ ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7446_set__diff__eq,axiom,
% 9.00/5.09 ( minus_7954133019191499631st_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( member_list_nat @ X @ A6 )
% 9.00/5.09 & ~ ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7447_set__diff__eq,axiom,
% 9.00/5.09 ( minus_2163939370556025621et_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( member_set_nat @ X @ A6 )
% 9.00/5.09 & ~ ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7448_set__diff__eq,axiom,
% 9.00/5.09 ( minus_minus_set_int
% 9.00/5.09 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( member_int @ X @ A6 )
% 9.00/5.09 & ~ ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7449_set__diff__eq,axiom,
% 9.00/5.09 ( minus_minus_set_nat
% 9.00/5.09 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( member_nat @ X @ A6 )
% 9.00/5.09 & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7450_lambda__zero,axiom,
% 9.00/5.09 ( ( ^ [H: complex] : zero_zero_complex )
% 9.00/5.09 = ( times_times_complex @ zero_zero_complex ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7451_lambda__zero,axiom,
% 9.00/5.09 ( ( ^ [H: real] : zero_zero_real )
% 9.00/5.09 = ( times_times_real @ zero_zero_real ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7452_lambda__zero,axiom,
% 9.00/5.09 ( ( ^ [H: rat] : zero_zero_rat )
% 9.00/5.09 = ( times_times_rat @ zero_zero_rat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7453_lambda__zero,axiom,
% 9.00/5.09 ( ( ^ [H: nat] : zero_zero_nat )
% 9.00/5.09 = ( times_times_nat @ zero_zero_nat ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7454_lambda__zero,axiom,
% 9.00/5.09 ( ( ^ [H: int] : zero_zero_int )
% 9.00/5.09 = ( times_times_int @ zero_zero_int ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7455_less__set__def,axiom,
% 9.00/5.09 ( ord_less_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( ord_less_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7456_less__set__def,axiom,
% 9.00/5.09 ( ord_less_set_o
% 9.00/5.09 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.09 ( ord_less_o_o
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7457_less__set__def,axiom,
% 9.00/5.09 ( ord_less_set_set_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.09 ( ord_less_set_nat_o
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7458_less__set__def,axiom,
% 9.00/5.09 ( ord_less_set_nat
% 9.00/5.09 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.09 ( ord_less_nat_o
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7459_less__set__def,axiom,
% 9.00/5.09 ( ord_less_set_int
% 9.00/5.09 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.09 ( ord_less_int_o
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7460_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_o
% 9.00/5.09 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( member_o @ X @ A6 )
% 9.00/5.09 | ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7461_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( member_real @ X @ A6 )
% 9.00/5.09 | ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7462_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_list_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( member_list_nat @ X @ A6 )
% 9.00/5.09 | ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7463_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_set_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( member_set_nat @ X @ A6 )
% 9.00/5.09 | ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7464_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_int
% 9.00/5.09 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( member_int @ X @ A6 )
% 9.00/5.09 | ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7465_Un__def,axiom,
% 9.00/5.09 ( sup_sup_set_nat
% 9.00/5.09 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( member_nat @ X @ A6 )
% 9.00/5.09 | ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7466_Un__def,axiom,
% 9.00/5.09 ( sup_su718114333110466843at_nat
% 9.00/5.09 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( collec7088162979684241874at_nat
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] :
% 9.00/5.09 ( ( member8206827879206165904at_nat @ X @ A6 )
% 9.00/5.09 | ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7467_Un__def,axiom,
% 9.00/5.09 ( sup_su5525570899277871387at_nat
% 9.00/5.09 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( collec6321179662152712658at_nat
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] :
% 9.00/5.09 ( ( member8757157785044589968at_nat @ X @ A6 )
% 9.00/5.09 | ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7468_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_o
% 9.00/5.09 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ( sup_sup_o_o
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7469_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ( sup_sup_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7470_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_list_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ( sup_sup_list_nat_o
% 9.00/5.09 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7471_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_set_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ( sup_sup_set_nat_o
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7472_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_int
% 9.00/5.09 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ( sup_sup_int_o
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7473_sup__set__def,axiom,
% 9.00/5.09 ( sup_sup_set_nat
% 9.00/5.09 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ( sup_sup_nat_o
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7474_sup__set__def,axiom,
% 9.00/5.09 ( sup_su718114333110466843at_nat
% 9.00/5.09 = ( ^ [A6: set_Pr8693737435421807431at_nat,B6: set_Pr8693737435421807431at_nat] :
% 9.00/5.09 ( collec7088162979684241874at_nat
% 9.00/5.09 @ ( sup_su8986005896011022210_nat_o
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7475_sup__set__def,axiom,
% 9.00/5.09 ( sup_su5525570899277871387at_nat
% 9.00/5.09 = ( ^ [A6: set_Pr4329608150637261639at_nat,B6: set_Pr4329608150637261639at_nat] :
% 9.00/5.09 ( collec6321179662152712658at_nat
% 9.00/5.09 @ ( sup_su2080679670758317954_nat_o
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ A6 )
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7476_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: real > $o,Q: real > $o] :
% 9.00/5.09 ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7477_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.09 ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7478_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.09 ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7479_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: int > $o,Q: int > $o] :
% 9.00/5.09 ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7480_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.09 ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7481_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
% 9.00/5.09 ( ( collec7088162979684241874at_nat
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ ( collec7088162979684241874at_nat @ P ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7482_Collect__disj__eq,axiom,
% 9.00/5.09 ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
% 9.00/5.09 ( ( collec6321179662152712658at_nat
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 | ( Q @ X ) ) )
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7483_insert__def,axiom,
% 9.00/5.09 ( insert8211810215607154385at_nat
% 9.00/5.09 = ( ^ [A4: product_prod_nat_nat] :
% 9.00/5.09 ( sup_su6327502436637775413at_nat
% 9.00/5.09 @ ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7484_insert__def,axiom,
% 9.00/5.09 ( insert_o
% 9.00/5.09 = ( ^ [A4: $o] :
% 9.00/5.09 ( sup_sup_set_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [X: $o] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7485_insert__def,axiom,
% 9.00/5.09 ( insert_real
% 9.00/5.09 = ( ^ [A4: real] :
% 9.00/5.09 ( sup_sup_set_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [X: real] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7486_insert__def,axiom,
% 9.00/5.09 ( insert_list_nat
% 9.00/5.09 = ( ^ [A4: list_nat] :
% 9.00/5.09 ( sup_sup_set_list_nat
% 9.00/5.09 @ ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7487_insert__def,axiom,
% 9.00/5.09 ( insert_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat] :
% 9.00/5.09 ( sup_sup_set_set_nat
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7488_insert__def,axiom,
% 9.00/5.09 ( insert_int
% 9.00/5.09 = ( ^ [A4: int] :
% 9.00/5.09 ( sup_sup_set_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [X: int] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7489_insert__def,axiom,
% 9.00/5.09 ( insert_nat
% 9.00/5.09 = ( ^ [A4: nat] :
% 9.00/5.09 ( sup_sup_set_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [X: nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7490_insert__def,axiom,
% 9.00/5.09 ( insert5050368324300391991at_nat
% 9.00/5.09 = ( ^ [A4: produc859450856879609959at_nat] :
% 9.00/5.09 ( sup_su718114333110466843at_nat
% 9.00/5.09 @ ( collec7088162979684241874at_nat
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7491_insert__def,axiom,
% 9.00/5.09 ( insert9069300056098147895at_nat
% 9.00/5.09 = ( ^ [A4: produc3843707927480180839at_nat] :
% 9.00/5.09 ( sup_su5525570899277871387at_nat
% 9.00/5.09 @ ( collec6321179662152712658at_nat
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] : ( X = A4 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7492_Compl__eq,axiom,
% 9.00/5.09 ( uminus_uminus_set_o
% 9.00/5.09 = ( ^ [A6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ~ ( member_o @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7493_Compl__eq,axiom,
% 9.00/5.09 ( uminus612125837232591019t_real
% 9.00/5.09 = ( ^ [A6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ~ ( member_real @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7494_Compl__eq,axiom,
% 9.00/5.09 ( uminus3195874150345416415st_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ~ ( member_list_nat @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7495_Compl__eq,axiom,
% 9.00/5.09 ( uminus613421341184616069et_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ~ ( member_set_nat @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7496_Compl__eq,axiom,
% 9.00/5.09 ( uminus5710092332889474511et_nat
% 9.00/5.09 = ( ^ [A6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ~ ( member_nat @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7497_Compl__eq,axiom,
% 9.00/5.09 ( uminus1532241313380277803et_int
% 9.00/5.09 = ( ^ [A6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ~ ( member_int @ X @ A6 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7498_Collect__neg__eq,axiom,
% 9.00/5.09 ! [P: real > $o] :
% 9.00/5.09 ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ~ ( P @ X ) )
% 9.00/5.09 = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7499_Collect__neg__eq,axiom,
% 9.00/5.09 ! [P: list_nat > $o] :
% 9.00/5.09 ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ~ ( P @ X ) )
% 9.00/5.09 = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7500_Collect__neg__eq,axiom,
% 9.00/5.09 ! [P: set_nat > $o] :
% 9.00/5.09 ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ~ ( P @ X ) )
% 9.00/5.09 = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7501_Collect__neg__eq,axiom,
% 9.00/5.09 ! [P: nat > $o] :
% 9.00/5.09 ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ~ ( P @ X ) )
% 9.00/5.09 = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7502_Collect__neg__eq,axiom,
% 9.00/5.09 ! [P: int > $o] :
% 9.00/5.09 ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ~ ( P @ X ) )
% 9.00/5.09 = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7503_uminus__set__def,axiom,
% 9.00/5.09 ( uminus_uminus_set_o
% 9.00/5.09 = ( ^ [A6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ( uminus_uminus_o_o
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7504_uminus__set__def,axiom,
% 9.00/5.09 ( uminus612125837232591019t_real
% 9.00/5.09 = ( ^ [A6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ( uminus_uminus_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7505_uminus__set__def,axiom,
% 9.00/5.09 ( uminus3195874150345416415st_nat
% 9.00/5.09 = ( ^ [A6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ( uminus5770388063884162150_nat_o
% 9.00/5.09 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7506_uminus__set__def,axiom,
% 9.00/5.09 ( uminus613421341184616069et_nat
% 9.00/5.09 = ( ^ [A6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ( uminus6401447641752708672_nat_o
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7507_uminus__set__def,axiom,
% 9.00/5.09 ( uminus5710092332889474511et_nat
% 9.00/5.09 = ( ^ [A6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ( uminus_uminus_nat_o
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7508_uminus__set__def,axiom,
% 9.00/5.09 ( uminus1532241313380277803et_int
% 9.00/5.09 = ( ^ [A6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ( uminus_uminus_int_o
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ A6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7509_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: real > $o] :
% 9.00/5.09 ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 9.00/5.09 => ? [X_1: real] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7510_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: list_nat > $o] :
% 9.00/5.09 ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.00/5.09 => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7511_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: set_nat > $o] :
% 9.00/5.09 ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.00/5.09 => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7512_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: nat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.09 => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7513_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: int > $o] :
% 9.00/5.09 ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 9.00/5.09 => ? [X_1: int] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7514_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: complex > $o] :
% 9.00/5.09 ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.00/5.09 => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7515_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: product_prod_nat_nat > $o] :
% 9.00/5.09 ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.00/5.09 => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7516_not__finite__existsD,axiom,
% 9.00/5.09 ! [P: extended_enat > $o] :
% 9.00/5.09 ( ~ ( finite4001608067531595151d_enat @ ( collec4429806609662206161d_enat @ P ) )
% 9.00/5.09 => ? [X_1: extended_enat] : ( P @ X_1 ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7517_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_o,B2: set_nat,R: $o > nat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_o @ A3 )
% 9.00/5.09 => ( ( finite_finite_nat @ B2 )
% 9.00/5.09 => ( ! [X5: $o] :
% 9.00/5.09 ( ( member_o @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: nat] :
% 9.00/5.09 ( ( member_nat @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: nat] :
% 9.00/5.09 ( ( member_nat @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [A4: $o] :
% 9.00/5.09 ( ( member_o @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7518_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_real,B2: set_nat,R: real > nat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_real @ A3 )
% 9.00/5.09 => ( ( finite_finite_nat @ B2 )
% 9.00/5.09 => ( ! [X5: real] :
% 9.00/5.09 ( ( member_real @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: nat] :
% 9.00/5.09 ( ( member_nat @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: nat] :
% 9.00/5.09 ( ( member_nat @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [A4: real] :
% 9.00/5.09 ( ( member_real @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7519_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_o,B2: set_int,R: $o > int > $o] :
% 9.00/5.09 ( ~ ( finite_finite_o @ A3 )
% 9.00/5.09 => ( ( finite_finite_int @ B2 )
% 9.00/5.09 => ( ! [X5: $o] :
% 9.00/5.09 ( ( member_o @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: int] :
% 9.00/5.09 ( ( member_int @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: int] :
% 9.00/5.09 ( ( member_int @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [A4: $o] :
% 9.00/5.09 ( ( member_o @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7520_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_real,B2: set_int,R: real > int > $o] :
% 9.00/5.09 ( ~ ( finite_finite_real @ A3 )
% 9.00/5.09 => ( ( finite_finite_int @ B2 )
% 9.00/5.09 => ( ! [X5: real] :
% 9.00/5.09 ( ( member_real @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: int] :
% 9.00/5.09 ( ( member_int @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: int] :
% 9.00/5.09 ( ( member_int @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [A4: real] :
% 9.00/5.09 ( ( member_real @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7521_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_o,B2: set_complex,R: $o > complex > $o] :
% 9.00/5.09 ( ~ ( finite_finite_o @ A3 )
% 9.00/5.09 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.09 => ( ! [X5: $o] :
% 9.00/5.09 ( ( member_o @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: complex] :
% 9.00/5.09 ( ( member_complex @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: complex] :
% 9.00/5.09 ( ( member_complex @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [A4: $o] :
% 9.00/5.09 ( ( member_o @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7522_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_real,B2: set_complex,R: real > complex > $o] :
% 9.00/5.09 ( ~ ( finite_finite_real @ A3 )
% 9.00/5.09 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.09 => ( ! [X5: real] :
% 9.00/5.09 ( ( member_real @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: complex] :
% 9.00/5.09 ( ( member_complex @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: complex] :
% 9.00/5.09 ( ( member_complex @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [A4: real] :
% 9.00/5.09 ( ( member_real @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7523_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_o,B2: set_Extended_enat,R: $o > extended_enat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_o @ A3 )
% 9.00/5.09 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.09 => ( ! [X5: $o] :
% 9.00/5.09 ( ( member_o @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: extended_enat] :
% 9.00/5.09 ( ( member_Extended_enat @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: extended_enat] :
% 9.00/5.09 ( ( member_Extended_enat @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [A4: $o] :
% 9.00/5.09 ( ( member_o @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7524_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_real,B2: set_Extended_enat,R: real > extended_enat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_real @ A3 )
% 9.00/5.09 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.09 => ( ! [X5: real] :
% 9.00/5.09 ( ( member_real @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: extended_enat] :
% 9.00/5.09 ( ( member_Extended_enat @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: extended_enat] :
% 9.00/5.09 ( ( member_Extended_enat @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [A4: real] :
% 9.00/5.09 ( ( member_real @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7525_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_nat,B2: set_nat,R: nat > nat > $o] :
% 9.00/5.09 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.09 => ( ( finite_finite_nat @ B2 )
% 9.00/5.09 => ( ! [X5: nat] :
% 9.00/5.09 ( ( member_nat @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: nat] :
% 9.00/5.09 ( ( member_nat @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: nat] :
% 9.00/5.09 ( ( member_nat @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [A4: nat] :
% 9.00/5.09 ( ( member_nat @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7526_pigeonhole__infinite__rel,axiom,
% 9.00/5.09 ! [A3: set_nat,B2: set_int,R: nat > int > $o] :
% 9.00/5.09 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.09 => ( ( finite_finite_int @ B2 )
% 9.00/5.09 => ( ! [X5: nat] :
% 9.00/5.09 ( ( member_nat @ X5 @ A3 )
% 9.00/5.09 => ? [Xa: int] :
% 9.00/5.09 ( ( member_int @ Xa @ B2 )
% 9.00/5.09 & ( R @ X5 @ Xa ) ) )
% 9.00/5.09 => ? [X5: int] :
% 9.00/5.09 ( ( member_int @ X5 @ B2 )
% 9.00/5.09 & ~ ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [A4: nat] :
% 9.00/5.09 ( ( member_nat @ A4 @ A3 )
% 9.00/5.09 & ( R @ A4 @ X5 ) ) ) ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7527_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7528_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: list_nat > $o,A2: list_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_list_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7529_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: set_nat > $o,A2: set_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7530_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: real > $o,A2: real] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7531_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: $o > $o,A2: $o] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7532_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: nat > $o,A2: nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7533_Collect__conv__if,axiom,
% 9.00/5.09 ! [P: int > $o,A2: int] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( X = A2 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7534_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: product_prod_nat_nat > $o,A2: product_prod_nat_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7535_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: list_nat > $o,A2: list_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_list_nat @ A2 @ bot_bot_set_list_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_list_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7536_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: set_nat > $o,A2: set_nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7537_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: real > $o,A2: real] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_real ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7538_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: $o > $o,A2: $o] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_o ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7539_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: nat > $o,A2: nat] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7540_Collect__conv__if2,axiom,
% 9.00/5.09 ! [P: int > $o,A2: int] :
% 9.00/5.09 ( ( ( P @ A2 )
% 9.00/5.09 => ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.09 & ( ~ ( P @ A2 )
% 9.00/5.09 => ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( A2 = X )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 = bot_bot_set_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7541_insert__Collect,axiom,
% 9.00/5.09 ! [A2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.00/5.09 ( ( insert8211810215607154385at_nat @ A2 @ ( collec3392354462482085612at_nat @ P ) )
% 9.00/5.09 = ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [U2: product_prod_nat_nat] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7542_insert__Collect,axiom,
% 9.00/5.09 ! [A2: $o,P: $o > $o] :
% 9.00/5.09 ( ( insert_o @ A2 @ ( collect_o @ P ) )
% 9.00/5.09 = ( collect_o
% 9.00/5.09 @ ^ [U2: $o] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7543_insert__Collect,axiom,
% 9.00/5.09 ! [A2: real,P: real > $o] :
% 9.00/5.09 ( ( insert_real @ A2 @ ( collect_real @ P ) )
% 9.00/5.09 = ( collect_real
% 9.00/5.09 @ ^ [U2: real] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7544_insert__Collect,axiom,
% 9.00/5.09 ! [A2: list_nat,P: list_nat > $o] :
% 9.00/5.09 ( ( insert_list_nat @ A2 @ ( collect_list_nat @ P ) )
% 9.00/5.09 = ( collect_list_nat
% 9.00/5.09 @ ^ [U2: list_nat] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7545_insert__Collect,axiom,
% 9.00/5.09 ! [A2: set_nat,P: set_nat > $o] :
% 9.00/5.09 ( ( insert_set_nat @ A2 @ ( collect_set_nat @ P ) )
% 9.00/5.09 = ( collect_set_nat
% 9.00/5.09 @ ^ [U2: set_nat] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7546_insert__Collect,axiom,
% 9.00/5.09 ! [A2: nat,P: nat > $o] :
% 9.00/5.09 ( ( insert_nat @ A2 @ ( collect_nat @ P ) )
% 9.00/5.09 = ( collect_nat
% 9.00/5.09 @ ^ [U2: nat] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7547_insert__Collect,axiom,
% 9.00/5.09 ! [A2: int,P: int > $o] :
% 9.00/5.09 ( ( insert_int @ A2 @ ( collect_int @ P ) )
% 9.00/5.09 = ( collect_int
% 9.00/5.09 @ ^ [U2: int] :
% 9.00/5.09 ( ( U2 != A2 )
% 9.00/5.09 => ( P @ U2 ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7548_insert__compr,axiom,
% 9.00/5.09 ( insert8211810215607154385at_nat
% 9.00/5.09 = ( ^ [A4: product_prod_nat_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.09 ( collec3392354462482085612at_nat
% 9.00/5.09 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7549_insert__compr,axiom,
% 9.00/5.09 ( insert_o
% 9.00/5.09 = ( ^ [A4: $o,B6: set_o] :
% 9.00/5.09 ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7550_insert__compr,axiom,
% 9.00/5.09 ( insert_real
% 9.00/5.09 = ( ^ [A4: real,B6: set_real] :
% 9.00/5.09 ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7551_insert__compr,axiom,
% 9.00/5.09 ( insert_list_nat
% 9.00/5.09 = ( ^ [A4: list_nat,B6: set_list_nat] :
% 9.00/5.09 ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7552_insert__compr,axiom,
% 9.00/5.09 ( insert_set_nat
% 9.00/5.09 = ( ^ [A4: set_nat,B6: set_set_nat] :
% 9.00/5.09 ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7553_insert__compr,axiom,
% 9.00/5.09 ( insert_nat
% 9.00/5.09 = ( ^ [A4: nat,B6: set_nat] :
% 9.00/5.09 ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7554_insert__compr,axiom,
% 9.00/5.09 ( insert_int
% 9.00/5.09 = ( ^ [A4: int,B6: set_int] :
% 9.00/5.09 ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( X = A4 )
% 9.00/5.09 | ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7555_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_list_nat
% 9.00/5.09 = ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7556_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_set_nat
% 9.00/5.09 = ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7557_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_real
% 9.00/5.09 = ( collect_real
% 9.00/5.09 @ ^ [X: real] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7558_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_o
% 9.00/5.09 = ( collect_o
% 9.00/5.09 @ ^ [X: $o] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7559_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_nat
% 9.00/5.09 = ( collect_nat
% 9.00/5.09 @ ^ [X: nat] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7560_empty__def,axiom,
% 9.00/5.09 ( bot_bot_set_int
% 9.00/5.09 = ( collect_int
% 9.00/5.09 @ ^ [X: int] : $false ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7561_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: real > $o,Q: real > $o] :
% 9.00/5.09 ( ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_real @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) @ ( collect_real @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7562_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.09 ( ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7563_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.09 ( ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7564_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: int > $o,Q: int > $o] :
% 9.00/5.09 ( ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_int @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) @ ( collect_int @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7565_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.09 ( ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7566_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: produc859450856879609959at_nat > $o,Q: produc859450856879609959at_nat > $o] :
% 9.00/5.09 ( ( collec7088162979684241874at_nat
% 9.00/5.09 @ ^ [X: produc859450856879609959at_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_su718114333110466843at_nat @ ( uminus4384627049435823934at_nat @ ( collec7088162979684241874at_nat @ P ) ) @ ( collec7088162979684241874at_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7567_Collect__imp__eq,axiom,
% 9.00/5.09 ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
% 9.00/5.09 ( ( collec6321179662152712658at_nat
% 9.00/5.09 @ ^ [X: produc3843707927480180839at_nat] :
% 9.00/5.09 ( ( P @ X )
% 9.00/5.09 => ( Q @ X ) ) )
% 9.00/5.09 = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7568_finite__less__ub,axiom,
% 9.00/5.09 ! [F: nat > nat,U: nat] :
% 9.00/5.09 ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 9.00/5.09 => ( finite_finite_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ U ) ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7569_bot__empty__eq2,axiom,
% 9.00/5.09 ( bot_bo5043116465536727218_nat_o
% 9.00/5.09 = ( ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ bot_bo232370072503712749on_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7570_bot__empty__eq2,axiom,
% 9.00/5.09 ( bot_bo3364206721330744218_nat_o
% 9.00/5.09 = ( ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ bot_bo4948859079157340979at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7571_bot__empty__eq2,axiom,
% 9.00/5.09 ( bot_bo394778441745866138_nat_o
% 9.00/5.09 = ( ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7572_bot__empty__eq2,axiom,
% 9.00/5.09 ( bot_bot_nat_nat_o
% 9.00/5.09 = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7573_bot__empty__eq2,axiom,
% 9.00/5.09 ( bot_bot_int_int_o
% 9.00/5.09 = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7574_pred__subset__eq,axiom,
% 9.00/5.09 ! [R: set_real,S: set_real] :
% 9.00/5.09 ( ( ord_less_eq_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ R )
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ S ) )
% 9.00/5.09 = ( ord_less_eq_set_real @ R @ S ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7575_pred__subset__eq,axiom,
% 9.00/5.09 ! [R: set_o,S: set_o] :
% 9.00/5.09 ( ( ord_less_eq_o_o
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ R )
% 9.00/5.09 @ ^ [X: $o] : ( member_o @ X @ S ) )
% 9.00/5.09 = ( ord_less_eq_set_o @ R @ S ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7576_pred__subset__eq,axiom,
% 9.00/5.09 ! [R: set_set_nat,S: set_set_nat] :
% 9.00/5.09 ( ( ord_le3964352015994296041_nat_o
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
% 9.00/5.09 @ ^ [X: set_nat] : ( member_set_nat @ X @ S ) )
% 9.00/5.09 = ( ord_le6893508408891458716et_nat @ R @ S ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7577_pred__subset__eq,axiom,
% 9.00/5.09 ! [R: set_nat,S: set_nat] :
% 9.00/5.09 ( ( ord_less_eq_nat_o
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ R )
% 9.00/5.09 @ ^ [X: nat] : ( member_nat @ X @ S ) )
% 9.00/5.09 = ( ord_less_eq_set_nat @ R @ S ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7578_pred__subset__eq,axiom,
% 9.00/5.09 ! [R: set_int,S: set_int] :
% 9.00/5.09 ( ( ord_less_eq_int_o
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ R )
% 9.00/5.09 @ ^ [X: int] : ( member_int @ X @ S ) )
% 9.00/5.09 = ( ord_less_eq_set_int @ R @ S ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7579_prop__restrict,axiom,
% 9.00/5.09 ! [X2: $o,Z6: set_o,X6: set_o,P: $o > $o] :
% 9.00/5.09 ( ( member_o @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_less_eq_set_o @ Z6
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( member_o @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7580_prop__restrict,axiom,
% 9.00/5.09 ! [X2: real,Z6: set_real,X6: set_real,P: real > $o] :
% 9.00/5.09 ( ( member_real @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_less_eq_set_real @ Z6
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( member_real @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7581_prop__restrict,axiom,
% 9.00/5.09 ! [X2: list_nat,Z6: set_list_nat,X6: set_list_nat,P: list_nat > $o] :
% 9.00/5.09 ( ( member_list_nat @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_le6045566169113846134st_nat @ Z6
% 9.00/5.09 @ ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( member_list_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7582_prop__restrict,axiom,
% 9.00/5.09 ! [X2: set_nat,Z6: set_set_nat,X6: set_set_nat,P: set_nat > $o] :
% 9.00/5.09 ( ( member_set_nat @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_le6893508408891458716et_nat @ Z6
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( member_set_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7583_prop__restrict,axiom,
% 9.00/5.09 ! [X2: nat,Z6: set_nat,X6: set_nat,P: nat > $o] :
% 9.00/5.09 ( ( member_nat @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_less_eq_set_nat @ Z6
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( member_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7584_prop__restrict,axiom,
% 9.00/5.09 ! [X2: int,Z6: set_int,X6: set_int,P: int > $o] :
% 9.00/5.09 ( ( member_int @ X2 @ Z6 )
% 9.00/5.09 => ( ( ord_less_eq_set_int @ Z6
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( member_int @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) ) )
% 9.00/5.09 => ( P @ X2 ) ) ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7585_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_o,P: $o > $o] :
% 9.00/5.09 ( ord_less_eq_set_o
% 9.00/5.09 @ ( collect_o
% 9.00/5.09 @ ^ [X: $o] :
% 9.00/5.09 ( ( member_o @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7586_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_real,P: real > $o] :
% 9.00/5.09 ( ord_less_eq_set_real
% 9.00/5.09 @ ( collect_real
% 9.00/5.09 @ ^ [X: real] :
% 9.00/5.09 ( ( member_real @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7587_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_list_nat,P: list_nat > $o] :
% 9.00/5.09 ( ord_le6045566169113846134st_nat
% 9.00/5.09 @ ( collect_list_nat
% 9.00/5.09 @ ^ [X: list_nat] :
% 9.00/5.09 ( ( member_list_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7588_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_set_nat,P: set_nat > $o] :
% 9.00/5.09 ( ord_le6893508408891458716et_nat
% 9.00/5.09 @ ( collect_set_nat
% 9.00/5.09 @ ^ [X: set_nat] :
% 9.00/5.09 ( ( member_set_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7589_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_nat,P: nat > $o] :
% 9.00/5.09 ( ord_less_eq_set_nat
% 9.00/5.09 @ ( collect_nat
% 9.00/5.09 @ ^ [X: nat] :
% 9.00/5.09 ( ( member_nat @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7590_Collect__restrict,axiom,
% 9.00/5.09 ! [X6: set_int,P: int > $o] :
% 9.00/5.09 ( ord_less_eq_set_int
% 9.00/5.09 @ ( collect_int
% 9.00/5.09 @ ^ [X: int] :
% 9.00/5.09 ( ( member_int @ X @ X6 )
% 9.00/5.09 & ( P @ X ) ) )
% 9.00/5.09 @ X6 ) ).
% 9.00/5.09
% 9.00/5.09 thf(fact_7591_less__eq__set__def,axiom,
% 9.00/5.09 ( ord_less_eq_set_real
% 9.00/5.09 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.09 ( ord_less_eq_real_o
% 9.00/5.09 @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.00/5.10 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7592_less__eq__set__def,axiom,
% 9.00/5.10 ( ord_less_eq_set_o
% 9.00/5.10 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.10 ( ord_less_eq_o_o
% 9.00/5.10 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 9.00/5.10 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7593_less__eq__set__def,axiom,
% 9.00/5.10 ( ord_le6893508408891458716et_nat
% 9.00/5.10 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.10 ( ord_le3964352015994296041_nat_o
% 9.00/5.10 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7594_less__eq__set__def,axiom,
% 9.00/5.10 ( ord_less_eq_set_nat
% 9.00/5.10 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.10 ( ord_less_eq_nat_o
% 9.00/5.10 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7595_less__eq__set__def,axiom,
% 9.00/5.10 ( ord_less_eq_set_int
% 9.00/5.10 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.10 ( ord_less_eq_int_o
% 9.00/5.10 @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.00/5.10 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7596_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_o,P: $o > $o] :
% 9.00/5.10 ( ord_less_eq_set_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7597_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_real,P: real > $o] :
% 9.00/5.10 ( ord_less_eq_set_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7598_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_list_nat,P: list_nat > $o] :
% 9.00/5.10 ( ord_le6045566169113846134st_nat
% 9.00/5.10 @ ( collect_list_nat
% 9.00/5.10 @ ^ [X: list_nat] :
% 9.00/5.10 ( ( member_list_nat @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7599_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_set_nat,P: set_nat > $o] :
% 9.00/5.10 ( ord_le6893508408891458716et_nat
% 9.00/5.10 @ ( collect_set_nat
% 9.00/5.10 @ ^ [X: set_nat] :
% 9.00/5.10 ( ( member_set_nat @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7600_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_nat,P: nat > $o] :
% 9.00/5.10 ( ord_less_eq_set_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7601_Collect__subset,axiom,
% 9.00/5.10 ! [A3: set_int,P: int > $o] :
% 9.00/5.10 ( ord_less_eq_set_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) )
% 9.00/5.10 @ A3 ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7602_pred__subset__eq2,axiom,
% 9.00/5.10 ! [R: set_Pr6588086440996610945on_nat,S: set_Pr6588086440996610945on_nat] :
% 9.00/5.10 ( ( ord_le8905833333647802342_nat_o
% 9.00/5.10 @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ R )
% 9.00/5.10 @ ^ [X: option_nat,Y: option_nat] : ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X @ Y ) @ S ) )
% 9.00/5.10 = ( ord_le6406482658798684961on_nat @ R @ S ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7603_pred__subset__eq2,axiom,
% 9.00/5.10 ! [R: set_Pr7459493094073627847at_nat,S: set_Pr7459493094073627847at_nat] :
% 9.00/5.10 ( ( ord_le3072208448688395470_nat_o
% 9.00/5.10 @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ R )
% 9.00/5.10 @ ^ [X: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X @ Y ) @ S ) )
% 9.00/5.10 = ( ord_le5997549366648089703at_nat @ R @ S ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7604_pred__subset__eq2,axiom,
% 9.00/5.10 ! [R: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
% 9.00/5.10 ( ( ord_le3935385432712749774_nat_o
% 9.00/5.10 @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R )
% 9.00/5.10 @ ^ [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ S ) )
% 9.00/5.10 = ( ord_le1268244103169919719at_nat @ R @ S ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7605_pred__subset__eq2,axiom,
% 9.00/5.10 ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.00/5.10 ( ( ord_le2646555220125990790_nat_o
% 9.00/5.10 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 9.00/5.10 @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) )
% 9.00/5.10 = ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7606_pred__subset__eq2,axiom,
% 9.00/5.10 ! [R: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 9.00/5.10 ( ( ord_le6741204236512500942_int_o
% 9.00/5.10 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 9.00/5.10 @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S ) )
% 9.00/5.10 = ( ord_le2843351958646193337nt_int @ R @ S ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7607_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_o
% 9.00/5.10 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.10 ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A6 )
% 9.00/5.10 & ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7608_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_real
% 9.00/5.10 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.10 ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A6 )
% 9.00/5.10 & ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7609_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_list_nat
% 9.00/5.10 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.10 ( collect_list_nat
% 9.00/5.10 @ ^ [X: list_nat] :
% 9.00/5.10 ( ( member_list_nat @ X @ A6 )
% 9.00/5.10 & ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7610_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_set_nat
% 9.00/5.10 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.10 ( collect_set_nat
% 9.00/5.10 @ ^ [X: set_nat] :
% 9.00/5.10 ( ( member_set_nat @ X @ A6 )
% 9.00/5.10 & ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7611_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_int
% 9.00/5.10 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.10 ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A6 )
% 9.00/5.10 & ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7612_Int__def,axiom,
% 9.00/5.10 ( inf_inf_set_nat
% 9.00/5.10 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.10 ( collect_nat
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ A6 )
% 9.00/5.10 & ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7613_Int__def,axiom,
% 9.00/5.10 ( inf_in2572325071724192079at_nat
% 9.00/5.10 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.10 ( collec3392354462482085612at_nat
% 9.00/5.10 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.10 ( ( member8440522571783428010at_nat @ X @ A6 )
% 9.00/5.10 & ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7614_Int__Collect,axiom,
% 9.00/5.10 ! [X2: $o,A3: set_o,P: $o > $o] :
% 9.00/5.10 ( ( member_o @ X2 @ ( inf_inf_set_o @ A3 @ ( collect_o @ P ) ) )
% 9.00/5.10 = ( ( member_o @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7615_Int__Collect,axiom,
% 9.00/5.10 ! [X2: real,A3: set_real,P: real > $o] :
% 9.00/5.10 ( ( member_real @ X2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) )
% 9.00/5.10 = ( ( member_real @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7616_Int__Collect,axiom,
% 9.00/5.10 ! [X2: list_nat,A3: set_list_nat,P: list_nat > $o] :
% 9.00/5.10 ( ( member_list_nat @ X2 @ ( inf_inf_set_list_nat @ A3 @ ( collect_list_nat @ P ) ) )
% 9.00/5.10 = ( ( member_list_nat @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7617_Int__Collect,axiom,
% 9.00/5.10 ! [X2: set_nat,A3: set_set_nat,P: set_nat > $o] :
% 9.00/5.10 ( ( member_set_nat @ X2 @ ( inf_inf_set_set_nat @ A3 @ ( collect_set_nat @ P ) ) )
% 9.00/5.10 = ( ( member_set_nat @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7618_Int__Collect,axiom,
% 9.00/5.10 ! [X2: int,A3: set_int,P: int > $o] :
% 9.00/5.10 ( ( member_int @ X2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) )
% 9.00/5.10 = ( ( member_int @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7619_Int__Collect,axiom,
% 9.00/5.10 ! [X2: nat,A3: set_nat,P: nat > $o] :
% 9.00/5.10 ( ( member_nat @ X2 @ ( inf_inf_set_nat @ A3 @ ( collect_nat @ P ) ) )
% 9.00/5.10 = ( ( member_nat @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7620_Int__Collect,axiom,
% 9.00/5.10 ! [X2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.00/5.10 ( ( member8440522571783428010at_nat @ X2 @ ( inf_in2572325071724192079at_nat @ A3 @ ( collec3392354462482085612at_nat @ P ) ) )
% 9.00/5.10 = ( ( member8440522571783428010at_nat @ X2 @ A3 )
% 9.00/5.10 & ( P @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7621_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_o
% 9.00/5.10 = ( ^ [A6: set_o,B6: set_o] :
% 9.00/5.10 ( collect_o
% 9.00/5.10 @ ( inf_inf_o_o
% 9.00/5.10 @ ^ [X: $o] : ( member_o @ X @ A6 )
% 9.00/5.10 @ ^ [X: $o] : ( member_o @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7622_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_real
% 9.00/5.10 = ( ^ [A6: set_real,B6: set_real] :
% 9.00/5.10 ( collect_real
% 9.00/5.10 @ ( inf_inf_real_o
% 9.00/5.10 @ ^ [X: real] : ( member_real @ X @ A6 )
% 9.00/5.10 @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7623_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_list_nat
% 9.00/5.10 = ( ^ [A6: set_list_nat,B6: set_list_nat] :
% 9.00/5.10 ( collect_list_nat
% 9.00/5.10 @ ( inf_inf_list_nat_o
% 9.00/5.10 @ ^ [X: list_nat] : ( member_list_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: list_nat] : ( member_list_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7624_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_set_nat
% 9.00/5.10 = ( ^ [A6: set_set_nat,B6: set_set_nat] :
% 9.00/5.10 ( collect_set_nat
% 9.00/5.10 @ ( inf_inf_set_nat_o
% 9.00/5.10 @ ^ [X: set_nat] : ( member_set_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: set_nat] : ( member_set_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7625_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_int
% 9.00/5.10 = ( ^ [A6: set_int,B6: set_int] :
% 9.00/5.10 ( collect_int
% 9.00/5.10 @ ( inf_inf_int_o
% 9.00/5.10 @ ^ [X: int] : ( member_int @ X @ A6 )
% 9.00/5.10 @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7626_inf__set__def,axiom,
% 9.00/5.10 ( inf_inf_set_nat
% 9.00/5.10 = ( ^ [A6: set_nat,B6: set_nat] :
% 9.00/5.10 ( collect_nat
% 9.00/5.10 @ ( inf_inf_nat_o
% 9.00/5.10 @ ^ [X: nat] : ( member_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7627_inf__set__def,axiom,
% 9.00/5.10 ( inf_in2572325071724192079at_nat
% 9.00/5.10 = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
% 9.00/5.10 ( collec3392354462482085612at_nat
% 9.00/5.10 @ ( inf_in5163264567034779214_nat_o
% 9.00/5.10 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
% 9.00/5.10 @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7628_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: real > $o,Q: real > $o] :
% 9.00/5.10 ( ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_inf_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7629_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.00/5.10 ( ( collect_list_nat
% 9.00/5.10 @ ^ [X: list_nat] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7630_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.00/5.10 ( ( collect_set_nat
% 9.00/5.10 @ ^ [X: set_nat] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7631_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: int > $o,Q: int > $o] :
% 9.00/5.10 ( ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_inf_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7632_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: nat > $o,Q: nat > $o] :
% 9.00/5.10 ( ( collect_nat
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7633_Collect__conj__eq,axiom,
% 9.00/5.10 ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.00/5.10 ( ( collec3392354462482085612at_nat
% 9.00/5.10 @ ^ [X: product_prod_nat_nat] :
% 9.00/5.10 ( ( P @ X )
% 9.00/5.10 & ( Q @ X ) ) )
% 9.00/5.10 = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7634_set__vebt__def,axiom,
% 9.00/5.10 ( vEBT_set_vebt
% 9.00/5.10 = ( ^ [T3: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7635_finite__int__segment,axiom,
% 9.00/5.10 ! [A2: real,B3: real] :
% 9.00/5.10 ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ ring_1_Ints_real )
% 9.00/5.10 & ( ord_less_eq_real @ A2 @ X )
% 9.00/5.10 & ( ord_less_eq_real @ X @ B3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7636_finite__int__segment,axiom,
% 9.00/5.10 ! [A2: rat,B3: rat] :
% 9.00/5.10 ( finite_finite_rat
% 9.00/5.10 @ ( collect_rat
% 9.00/5.10 @ ^ [X: rat] :
% 9.00/5.10 ( ( member_rat @ X @ ring_1_Ints_rat )
% 9.00/5.10 & ( ord_less_eq_rat @ A2 @ X )
% 9.00/5.10 & ( ord_less_eq_rat @ X @ B3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7637_nat__less__as__int,axiom,
% 9.00/5.10 ( ord_less_nat
% 9.00/5.10 = ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7638_nat__leq__as__int,axiom,
% 9.00/5.10 ( ord_less_eq_nat
% 9.00/5.10 = ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7639_unset__bit__less__eq,axiom,
% 9.00/5.10 ! [N: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N @ K ) @ K ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7640_finite__abs__int__segment,axiom,
% 9.00/5.10 ! [A2: real] :
% 9.00/5.10 ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [K3: real] :
% 9.00/5.10 ( ( member_real @ K3 @ ring_1_Ints_real )
% 9.00/5.10 & ( ord_less_eq_real @ ( abs_abs_real @ K3 ) @ A2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7641_finite__abs__int__segment,axiom,
% 9.00/5.10 ! [A2: rat] :
% 9.00/5.10 ( finite_finite_rat
% 9.00/5.10 @ ( collect_rat
% 9.00/5.10 @ ^ [K3: rat] :
% 9.00/5.10 ( ( member_rat @ K3 @ ring_1_Ints_rat )
% 9.00/5.10 & ( ord_less_eq_rat @ ( abs_abs_rat @ K3 ) @ A2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7642_card__less,axiom,
% 9.00/5.10 ! [M5: set_nat,I: nat] :
% 9.00/5.10 ( ( member_nat @ zero_zero_nat @ M5 )
% 9.00/5.10 => ( ( finite_card_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [K3: nat] :
% 9.00/5.10 ( ( member_nat @ K3 @ M5 )
% 9.00/5.10 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 9.00/5.10 != zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7643_card__less__Suc,axiom,
% 9.00/5.10 ! [M5: set_nat,I: nat] :
% 9.00/5.10 ( ( member_nat @ zero_zero_nat @ M5 )
% 9.00/5.10 => ( ( suc
% 9.00/5.10 @ ( finite_card_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [K3: nat] :
% 9.00/5.10 ( ( member_nat @ ( suc @ K3 ) @ M5 )
% 9.00/5.10 & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 9.00/5.10 = ( finite_card_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [K3: nat] :
% 9.00/5.10 ( ( member_nat @ K3 @ M5 )
% 9.00/5.10 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7644_card__less__Suc2,axiom,
% 9.00/5.10 ! [M5: set_nat,I: nat] :
% 9.00/5.10 ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 9.00/5.10 => ( ( finite_card_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [K3: nat] :
% 9.00/5.10 ( ( member_nat @ ( suc @ K3 ) @ M5 )
% 9.00/5.10 & ( ord_less_nat @ K3 @ I ) ) ) )
% 9.00/5.10 = ( finite_card_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [K3: nat] :
% 9.00/5.10 ( ( member_nat @ K3 @ M5 )
% 9.00/5.10 & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7645_norm__not__less__zero,axiom,
% 9.00/5.10 ! [X2: complex] :
% 9.00/5.10 ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7646_norm__ge__zero,axiom,
% 9.00/5.10 ! [X2: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7647_complex__mod__minus__le__complex__mod,axiom,
% 9.00/5.10 ! [X2: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7648_complex__mod__triangle__ineq2,axiom,
% 9.00/5.10 ! [B3: complex,A2: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B3 @ A2 ) ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ A2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7649_set__encode__eq,axiom,
% 9.00/5.10 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( ( nat_set_encode @ A3 )
% 9.00/5.10 = ( nat_set_encode @ B2 ) )
% 9.00/5.10 = ( A3 = B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7650_finite__roots__unity,axiom,
% 9.00/5.10 ! [N: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.00/5.10 => ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [Z2: real] :
% 9.00/5.10 ( ( power_power_real @ Z2 @ N )
% 9.00/5.10 = one_one_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7651_finite__roots__unity,axiom,
% 9.00/5.10 ! [N: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.00/5.10 => ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Z2: complex] :
% 9.00/5.10 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.10 = one_one_complex ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7652_card__roots__unity,axiom,
% 9.00/5.10 ! [N: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.00/5.10 => ( ord_less_eq_nat
% 9.00/5.10 @ ( finite_card_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [Z2: real] :
% 9.00/5.10 ( ( power_power_real @ Z2 @ N )
% 9.00/5.10 = one_one_real ) ) )
% 9.00/5.10 @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7653_card__roots__unity,axiom,
% 9.00/5.10 ! [N: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ one_one_nat @ N )
% 9.00/5.10 => ( ord_less_eq_nat
% 9.00/5.10 @ ( finite_card_complex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Z2: complex] :
% 9.00/5.10 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.10 = one_one_complex ) ) )
% 9.00/5.10 @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7654_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_complex,N: nat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( finite8712137658972009173omplex
% 9.00/5.10 @ ( collect_list_complex
% 9.00/5.10 @ ^ [Xs2: list_complex] :
% 9.00/5.10 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7655_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 9.00/5.10 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.10 => ( finite500796754983035824at_nat
% 9.00/5.10 @ ( collec3343600615725829874at_nat
% 9.00/5.10 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.00/5.10 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7656_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,N: nat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( finite1862508098717546133d_enat
% 9.00/5.10 @ ( collec8433460942617342167d_enat
% 9.00/5.10 @ ^ [Xs2: list_Extended_enat] :
% 9.00/5.10 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3941691890525107288d_enat @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7657_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_VEBT_VEBT,N: nat] :
% 9.00/5.10 ( ( finite5795047828879050333T_VEBT @ A3 )
% 9.00/5.10 => ( finite3004134309566078307T_VEBT
% 9.00/5.10 @ ( collec5608196760682091941T_VEBT
% 9.00/5.10 @ ^ [Xs2: list_VEBT_VEBT] :
% 9.00/5.10 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7658_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_o,N: nat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( finite_finite_list_o
% 9.00/5.10 @ ( collect_list_o
% 9.00/5.10 @ ^ [Xs2: list_o] :
% 9.00/5.10 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_o @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7659_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_nat,N: nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( finite8100373058378681591st_nat
% 9.00/5.10 @ ( collect_list_nat
% 9.00/5.10 @ ^ [Xs2: list_nat] :
% 9.00/5.10 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7660_finite__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_int,N: nat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( finite3922522038869484883st_int
% 9.00/5.10 @ ( collect_list_int
% 9.00/5.10 @ ^ [Xs2: list_int] :
% 9.00/5.10 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_int @ Xs2 )
% 9.00/5.10 = N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7661_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_list_nat,N: nat] :
% 9.00/5.10 ( ( finite8100373058378681591st_nat @ A3 )
% 9.00/5.10 => ( ( finite7325466520557071688st_nat
% 9.00/5.10 @ ( collec5989764272469232197st_nat
% 9.00/5.10 @ ^ [Xs2: list_list_nat] :
% 9.00/5.10 ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3023201423986296836st_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_list_nat @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7662_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_set_nat,N: nat] :
% 9.00/5.10 ( ( finite1152437895449049373et_nat @ A3 )
% 9.00/5.10 => ( ( finite5631907774883551598et_nat
% 9.00/5.10 @ ( collect_list_set_nat
% 9.00/5.10 @ ^ [Xs2: list_set_nat] :
% 9.00/5.10 ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3254054031482475050et_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_set_nat @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7663_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_complex,N: nat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( finite5120063068150530198omplex
% 9.00/5.10 @ ( collect_list_complex
% 9.00/5.10 @ ^ [Xs2: list_complex] :
% 9.00/5.10 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3451745648224563538omplex @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_complex @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7664_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 9.00/5.10 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.10 => ( ( finite249151656366948015at_nat
% 9.00/5.10 @ ( collec3343600615725829874at_nat
% 9.00/5.10 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.00/5.10 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite711546835091564841at_nat @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7665_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,N: nat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( finite7441382602597825044d_enat
% 9.00/5.10 @ ( collec8433460942617342167d_enat
% 9.00/5.10 @ ^ [Xs2: list_Extended_enat] :
% 9.00/5.10 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s3941691890525107288d_enat @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite121521170596916366d_enat @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7666_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_VEBT_VEBT,N: nat] :
% 9.00/5.10 ( ( finite5795047828879050333T_VEBT @ A3 )
% 9.00/5.10 => ( ( finite5915292604075114978T_VEBT
% 9.00/5.10 @ ( collec5608196760682091941T_VEBT
% 9.00/5.10 @ ^ [Xs2: list_VEBT_VEBT] :
% 9.00/5.10 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7667_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_o,N: nat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( finite_card_list_o
% 9.00/5.10 @ ( collect_list_o
% 9.00/5.10 @ ^ [Xs2: list_o] :
% 9.00/5.10 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_o @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_o @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7668_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_nat,N: nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( finite_card_list_nat
% 9.00/5.10 @ ( collect_list_nat
% 9.00/5.10 @ ^ [Xs2: list_nat] :
% 9.00/5.10 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_nat @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_nat @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7669_card__lists__length__eq,axiom,
% 9.00/5.10 ! [A3: set_int,N: nat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( finite_card_list_int
% 9.00/5.10 @ ( collect_list_int
% 9.00/5.10 @ ^ [Xs2: list_int] :
% 9.00/5.10 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ( size_size_list_int @ Xs2 )
% 9.00/5.10 = N ) ) ) )
% 9.00/5.10 = ( power_power_nat @ ( finite_card_int @ A3 ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7670_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_complex,N: nat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( finite8712137658972009173omplex
% 9.00/5.10 @ ( collect_list_complex
% 9.00/5.10 @ ^ [Xs2: list_complex] :
% 9.00/5.10 ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7671_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_Pr1261947904930325089at_nat,N: nat] :
% 9.00/5.10 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.10 => ( finite500796754983035824at_nat
% 9.00/5.10 @ ( collec3343600615725829874at_nat
% 9.00/5.10 @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.00/5.10 ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7672_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,N: nat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( finite1862508098717546133d_enat
% 9.00/5.10 @ ( collec8433460942617342167d_enat
% 9.00/5.10 @ ^ [Xs2: list_Extended_enat] :
% 9.00/5.10 ( ( ord_le7203529160286727270d_enat @ ( set_Extended_enat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_s3941691890525107288d_enat @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7673_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_VEBT_VEBT,N: nat] :
% 9.00/5.10 ( ( finite5795047828879050333T_VEBT @ A3 )
% 9.00/5.10 => ( finite3004134309566078307T_VEBT
% 9.00/5.10 @ ( collec5608196760682091941T_VEBT
% 9.00/5.10 @ ^ [Xs2: list_VEBT_VEBT] :
% 9.00/5.10 ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7674_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_o,N: nat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( finite_finite_list_o
% 9.00/5.10 @ ( collect_list_o
% 9.00/5.10 @ ^ [Xs2: list_o] :
% 9.00/5.10 ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7675_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_nat,N: nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( finite8100373058378681591st_nat
% 9.00/5.10 @ ( collect_list_nat
% 9.00/5.10 @ ^ [Xs2: list_nat] :
% 9.00/5.10 ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7676_finite__lists__length__le,axiom,
% 9.00/5.10 ! [A3: set_int,N: nat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( finite3922522038869484883st_int
% 9.00/5.10 @ ( collect_list_int
% 9.00/5.10 @ ^ [Xs2: list_int] :
% 9.00/5.10 ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A3 )
% 9.00/5.10 & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7677_nonzero__norm__divide,axiom,
% 9.00/5.10 ! [B3: real,A2: real] :
% 9.00/5.10 ( ( B3 != zero_zero_real )
% 9.00/5.10 => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A2 @ B3 ) )
% 9.00/5.10 = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7678_nonzero__norm__divide,axiom,
% 9.00/5.10 ! [B3: complex,A2: complex] :
% 9.00/5.10 ( ( B3 != zero_zero_complex )
% 9.00/5.10 => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A2 @ B3 ) )
% 9.00/5.10 = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7679_power__eq__imp__eq__norm,axiom,
% 9.00/5.10 ! [W2: real,N: nat,Z: real] :
% 9.00/5.10 ( ( ( power_power_real @ W2 @ N )
% 9.00/5.10 = ( power_power_real @ Z @ N ) )
% 9.00/5.10 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.10 => ( ( real_V7735802525324610683m_real @ W2 )
% 9.00/5.10 = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7680_power__eq__imp__eq__norm,axiom,
% 9.00/5.10 ! [W2: complex,N: nat,Z: complex] :
% 9.00/5.10 ( ( ( power_power_complex @ W2 @ N )
% 9.00/5.10 = ( power_power_complex @ Z @ N ) )
% 9.00/5.10 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.10 => ( ( real_V1022390504157884413omplex @ W2 )
% 9.00/5.10 = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7681_norm__mult__less,axiom,
% 9.00/5.10 ! [X2: real,R2: real,Y3: real,S2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7682_norm__mult__less,axiom,
% 9.00/5.10 ! [X2: complex,R2: real,Y3: complex,S2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7683_norm__mult__ineq,axiom,
% 9.00/5.10 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7684_norm__mult__ineq,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X2 @ Y3 ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7685_norm__add__less,axiom,
% 9.00/5.10 ! [X2: real,R2: real,Y3: real,S2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7686_norm__add__less,axiom,
% 9.00/5.10 ! [X2: complex,R2: real,Y3: complex,S2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7687_norm__triangle__lt,axiom,
% 9.00/5.10 ! [X2: real,Y3: real,E2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7688_norm__triangle__lt,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex,E2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7689_norm__power__ineq,axiom,
% 9.00/5.10 ! [X2: real,N: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X2 @ N ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X2 ) @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7690_norm__power__ineq,axiom,
% 9.00/5.10 ! [X2: complex,N: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X2 @ N ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X2 ) @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7691_norm__add__leD,axiom,
% 9.00/5.10 ! [A2: real,B3: real,C: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ C )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ C ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7692_norm__add__leD,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex,C: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ C )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ C ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7693_norm__triangle__le,axiom,
% 9.00/5.10 ! [X2: real,Y3: real,E2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7694_norm__triangle__le,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex,E2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7695_norm__triangle__ineq,axiom,
% 9.00/5.10 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7696_norm__triangle__ineq,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7697_norm__triangle__mono,axiom,
% 9.00/5.10 ! [A2: real,R2: real,B3: real,S2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B3 ) @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7698_norm__triangle__mono,axiom,
% 9.00/5.10 ! [A2: complex,R2: real,B3: complex,S2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A2 ) @ R2 )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B3 ) @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ R2 @ S2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7699_norm__diff__triangle__less,axiom,
% 9.00/5.10 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 9.00/5.10 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7700_norm__diff__triangle__less,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 9.00/5.10 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7701_norm__triangle__sub,axiom,
% 9.00/5.10 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y3 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7702_norm__triangle__sub,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y3 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7703_norm__triangle__ineq4,axiom,
% 9.00/5.10 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7704_norm__triangle__ineq4,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7705_norm__diff__triangle__le,axiom,
% 9.00/5.10 ! [X2: real,Y3: real,E1: real,Z: real,E22: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E1 )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z ) ) @ E22 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7706_norm__diff__triangle__le,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex,E1: real,Z: complex,E22: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E1 )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z ) ) @ E22 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7707_norm__triangle__le__diff,axiom,
% 9.00/5.10 ! [X2: real,Y3: real,E2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X2 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7708_norm__triangle__le__diff,axiom,
% 9.00/5.10 ! [X2: complex,Y3: complex,E2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X2 @ Y3 ) ) @ E2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7709_norm__diff__ineq,axiom,
% 9.00/5.10 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7710_norm__diff__ineq,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7711_norm__triangle__ineq2,axiom,
% 9.00/5.10 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7712_norm__triangle__ineq2,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7713_norm__exp,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X2 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7714_norm__exp,axiom,
% 9.00/5.10 ! [X2: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X2 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7715_set__encode__inf,axiom,
% 9.00/5.10 ! [A3: set_nat] :
% 9.00/5.10 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( nat_set_encode @ A3 )
% 9.00/5.10 = zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7716_power__eq__1__iff,axiom,
% 9.00/5.10 ! [W2: real,N: nat] :
% 9.00/5.10 ( ( ( power_power_real @ W2 @ N )
% 9.00/5.10 = one_one_real )
% 9.00/5.10 => ( ( ( real_V7735802525324610683m_real @ W2 )
% 9.00/5.10 = one_one_real )
% 9.00/5.10 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7717_power__eq__1__iff,axiom,
% 9.00/5.10 ! [W2: complex,N: nat] :
% 9.00/5.10 ( ( ( power_power_complex @ W2 @ N )
% 9.00/5.10 = one_one_complex )
% 9.00/5.10 => ( ( ( real_V1022390504157884413omplex @ W2 )
% 9.00/5.10 = one_one_real )
% 9.00/5.10 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7718_norm__diff__triangle__ineq,axiom,
% 9.00/5.10 ! [A2: real,B3: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A2 @ B3 ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B3 @ D ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7719_norm__diff__triangle__ineq,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A2 @ B3 ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B3 @ D ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7720_norm__sgn,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ( X2 = zero_zero_real )
% 9.00/5.10 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 & ( ( X2 != zero_zero_real )
% 9.00/5.10 => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X2 ) )
% 9.00/5.10 = one_one_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7721_norm__sgn,axiom,
% 9.00/5.10 ! [X2: complex] :
% 9.00/5.10 ( ( ( X2 = zero_zero_complex )
% 9.00/5.10 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 & ( ( X2 != zero_zero_complex )
% 9.00/5.10 => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X2 ) )
% 9.00/5.10 = one_one_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7722_norm__triangle__ineq3,axiom,
% 9.00/5.10 ! [A2: real,B3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A2 ) @ ( real_V7735802525324610683m_real @ B3 ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7723_norm__triangle__ineq3,axiom,
% 9.00/5.10 ! [A2: complex,B3: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A2 ) @ ( real_V1022390504157884413omplex @ B3 ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A2 @ B3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7724_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 9.00/5.10 ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X2: nat] :
% 9.00/5.10 ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S2 ) @ X2 )
% 9.00/5.10 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7725_norm__power__diff,axiom,
% 9.00/5.10 ! [Z: real,W2: real,M: nat] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W2 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7726_norm__power__diff,axiom,
% 9.00/5.10 ! [Z: complex,W2: complex,M: nat] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W2 @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W2 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7727_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 9.00/5.10 ! [V: nat,TreeList2: list_VEBT_VEBT,Vd: vEBT_VEBT,X2: nat] :
% 9.00/5.10 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd ) @ X2 )
% 9.00/5.10 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7728_vebt__member_Osimps_I5_J,axiom,
% 9.00/5.10 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 9.00/5.10 ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( ( X2 != Mi )
% 9.00/5.10 => ( ( X2 != Ma )
% 9.00/5.10 => ( ~ ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma @ X2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7729_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 9.00/5.10 ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X2: nat] :
% 9.00/5.10 ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X2 )
% 9.00/5.10 = ( ( X2 = Mi )
% 9.00/5.10 | ( X2 = Ma )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7730_exp__bound__half,axiom,
% 9.00/5.10 ! [Z: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7731_exp__bound__half,axiom,
% 9.00/5.10 ! [Z: complex] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7732_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.00/5.10 => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [S3: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7733_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [S3: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7734_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [S3: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7735_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat] :
% 9.00/5.10 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vc2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vd2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7736_vebt__member_Oelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Summary2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7737_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat] :
% 9.00/5.10 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vc2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vd2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7738_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat] :
% 9.00/5.10 ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vc2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Vd2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7739_vebt__insert_Osimps_I5_J,axiom,
% 9.00/5.10 ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X2: nat] :
% 9.00/5.10 ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( if_VEBT_VEBT
% 9.00/5.10 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 & ~ ( ( X2 = Mi )
% 9.00/5.10 | ( X2 = Ma ) ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ X2 @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( 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 @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X2 @ Mi ) @ Mi @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7740_vebt__member_Oelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.00/5.10 => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.00/5.10 => Y3 )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Summary2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( ~ ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7741_vebt__member_Oelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT] :
% 9.00/5.10 ( ? [Summary2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7742_exp__bound__lemma,axiom,
% 9.00/5.10 ! [Z: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7743_exp__bound__lemma,axiom,
% 9.00/5.10 ! [Z: complex] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7744_vebt__insert_Oelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.10 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ $true @ B ) ) )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ $true ) ) )
% 9.00/5.10 & ( ( Xa2 != one_one_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) )
% 9.00/5.10 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) ) )
% 9.00/5.10 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) ) )
% 9.00/5.10 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( if_VEBT_VEBT
% 9.00/5.10 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 & ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7745_vebt__succ_Osimps_I6_J,axiom,
% 9.00/5.10 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.10 ( ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( some_nat @ Mi ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ none_nat
% 9.00/5.10 @ ( 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 @ X2 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7746_vebt__pred_Osimps_I7_J,axiom,
% 9.00/5.10 ! [Ma: nat,X2: nat,Mi: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.10 ( ( ( ord_less_nat @ Ma @ X2 )
% 9.00/5.10 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( some_nat @ Ma ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma @ X2 )
% 9.00/5.10 => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( 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 @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_option_nat @ ( ord_less_nat @ Mi @ X2 ) @ ( some_nat @ Mi ) @ none_nat )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7747_vebt__delete_Osimps_I7_J,axiom,
% 9.00/5.10 ! [X2: nat,Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.00/5.10 ( ( ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 | ( ord_less_nat @ Ma @ X2 ) )
% 9.00/5.10 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 9.00/5.10 & ( ~ ( ( ord_less_nat @ X2 @ Mi )
% 9.00/5.10 | ( ord_less_nat @ Ma @ X2 ) )
% 9.00/5.10 => ( ( ( ( X2 = Mi )
% 9.00/5.10 & ( X2 = Ma ) )
% 9.00/5.10 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) )
% 9.00/5.10 & ( ~ ( ( X2 = Mi )
% 9.00/5.10 & ( X2 = Ma ) )
% 9.00/5.10 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X2 )
% 9.00/5.10 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.00/5.10 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( X2 = Mi )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.10 = Ma ) )
% 9.00/5.10 & ( ( X2 != Mi )
% 9.00/5.10 => ( X2 = Ma ) ) )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( X2 = Mi ) @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ Mi )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( X2 = Mi )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
% 9.00/5.10 = Ma ) )
% 9.00/5.10 & ( ( X2 != Mi )
% 9.00/5.10 => ( X2 = Ma ) ) )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X2 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ Summary ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7748_vebt__delete_Oelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.10 ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Leaf @ $false @ B ) ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ? [B: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Xa2
% 9.00/5.10 = ( suc @ zero_zero_nat ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Leaf @ A @ $false ) ) ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ? [N3: nat] :
% 9.00/5.10 ( Xa2
% 9.00/5.10 = ( suc @ ( suc @ N3 ) ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Leaf @ A @ B ) ) ) )
% 9.00/5.10 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 9.00/5.10 => ( ( ( ( Xa2 = Mi2 )
% 9.00/5.10 & ( Xa2 = Ma2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 & ( ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 & ( Xa2 = Ma2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( Xa2 = Mi2 )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 9.00/5.10 = Ma2 ) )
% 9.00/5.10 & ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma2 ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va2 ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( Xa2 = Mi2 )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 9.00/5.10 = Ma2 ) )
% 9.00/5.10 & ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma2 ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va2 ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ Summary2 ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7749_vebt__pred_Oelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.00/5.10 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( Y3 != none_nat ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ? [Uw2: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ Uw2 ) )
% 9.00/5.10 => ( ( Xa2
% 9.00/5.10 = ( suc @ zero_zero_nat ) )
% 9.00/5.10 => ~ ( ( A
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ A
% 9.00/5.10 => ( Y3 = none_nat ) ) ) ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ? [Va2: nat] :
% 9.00/5.10 ( Xa2
% 9.00/5.10 = ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.10 => ~ ( ( B
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.10 & ( ~ B
% 9.00/5.10 => ( ( A
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ A
% 9.00/5.10 => ( Y3 = none_nat ) ) ) ) ) ) )
% 9.00/5.10 => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ Ma2 ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7750_vebt__succ_Oelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.00/5.10 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ! [Uu2: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ B ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ~ ( ( B
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.10 & ( ~ B
% 9.00/5.10 => ( Y3 = none_nat ) ) ) ) )
% 9.00/5.10 => ( ( ? [Uv2: $o,Uw2: $o] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ( ? [N3: nat] :
% 9.00/5.10 ( Xa2
% 9.00/5.10 = ( suc @ N3 ) )
% 9.00/5.10 => ( Y3 != none_nat ) ) )
% 9.00/5.10 => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ( ( ? [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 9.00/5.10 ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 9.00/5.10 => ( Y3 != none_nat ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ Mi2 ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ none_nat
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7751_of__int__code__if,axiom,
% 9.00/5.10 ( ring_1_of_int_int
% 9.00/5.10 = ( ^ [K3: int] :
% 9.00/5.10 ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 9.00/5.10 @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.10 @ ( if_int
% 9.00/5.10 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7752_of__int__code__if,axiom,
% 9.00/5.10 ( ring_1_of_int_real
% 9.00/5.10 = ( ^ [K3: int] :
% 9.00/5.10 ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 9.00/5.10 @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.10 @ ( if_real
% 9.00/5.10 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7753_of__int__code__if,axiom,
% 9.00/5.10 ( ring_1_of_int_rat
% 9.00/5.10 = ( ^ [K3: int] :
% 9.00/5.10 ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 9.00/5.10 @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.10 @ ( if_rat
% 9.00/5.10 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7754_of__int__code__if,axiom,
% 9.00/5.10 ( ring_18347121197199848620nteger
% 9.00/5.10 = ( ^ [K3: int] :
% 9.00/5.10 ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.00/5.10 @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.10 @ ( if_Code_integer
% 9.00/5.10 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7755_of__int__code__if,axiom,
% 9.00/5.10 ( ring_17405671764205052669omplex
% 9.00/5.10 = ( ^ [K3: int] :
% 9.00/5.10 ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 9.00/5.10 @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.10 @ ( if_complex
% 9.00/5.10 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7756_vebt__succ_Opelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.00/5.10 ( ( ( vEBT_vebt_succ @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [Uu2: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ B ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( ( ( B
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.10 & ( ~ B
% 9.00/5.10 => ( Y3 = none_nat ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat ) ) ) ) )
% 9.00/5.10 => ( ! [Uv2: $o,Uw2: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ! [N3: nat] :
% 9.00/5.10 ( ( Xa2
% 9.00/5.10 = ( suc @ N3 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N3 ) ) ) ) ) )
% 9.00/5.10 => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ Mi2 ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ none_nat
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7757_vebt__pred_Opelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: option_nat] :
% 9.00/5.10 ( ( ( vEBT_vebt_pred @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
% 9.00/5.10 => ( ! [A: $o,Uw2: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ Uw2 ) )
% 9.00/5.10 => ( ( Xa2
% 9.00/5.10 = ( suc @ zero_zero_nat ) )
% 9.00/5.10 => ( ( ( A
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ A
% 9.00/5.10 => ( Y3 = none_nat ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ! [Va2: nat] :
% 9.00/5.10 ( ( Xa2
% 9.00/5.10 = ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.10 => ( ( ( B
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ one_one_nat ) ) )
% 9.00/5.10 & ( ~ B
% 9.00/5.10 => ( ( A
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ A
% 9.00/5.10 => ( Y3 = none_nat ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) )
% 9.00/5.10 => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va3 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
% 9.00/5.10 => ( ( Y3 = none_nat )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( some_nat @ Ma2 ) ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 != none_nat )
% 9.00/5.10 & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( if_option_nat
% 9.00/5.10 @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
% 9.00/5.10 @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ none_nat ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7758_vebt__delete_Opelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.10 ( ( ( vEBT_vebt_delete @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ $false @ B ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ) ) ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Xa2
% 9.00/5.10 = ( suc @ zero_zero_nat ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ $false ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ! [N3: nat] :
% 9.00/5.10 ( ( Xa2
% 9.00/5.10 = ( suc @ ( suc @ N3 ) ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
% 9.00/5.10 => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 | ( ord_less_nat @ Ma2 @ Xa2 ) )
% 9.00/5.10 => ( ( ( ( Xa2 = Mi2 )
% 9.00/5.10 & ( Xa2 = Ma2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 & ( ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 & ( Xa2 = Ma2 ) )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( Xa2 = Mi2 )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 9.00/5.10 = Ma2 ) )
% 9.00/5.10 & ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 = none_nat )
% 9.00/5.10 @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma2 ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va2 ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ ( vEBT_Node
% 9.00/5.10 @ ( some_P7363390416028606310at_nat
% 9.00/5.10 @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
% 9.00/5.10 @ ( if_nat
% 9.00/5.10 @ ( ( ( Xa2 = Mi2 )
% 9.00/5.10 => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
% 9.00/5.10 = Ma2 ) )
% 9.00/5.10 & ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
% 9.00/5.10 @ Ma2 ) ) )
% 9.00/5.10 @ ( suc @ ( suc @ Va2 ) )
% 9.00/5.10 @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 @ Summary2 ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7759_monoseq__arctan__series,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.10 => ( topolo6980174941875973593q_real
% 9.00/5.10 @ ^ [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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7760_vebt__insert_Opelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.10 ( ( ( vEBT_vebt_insert @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ $true @ B ) ) )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ $true ) ) )
% 9.00/5.10 & ( ( Xa2 != one_one_nat )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S3 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S3 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( if_VEBT_VEBT
% 9.00/5.10 @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 & ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
% 9.00/5.10 @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7761_monoseq__realpow,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.10 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.10 => ( topolo6980174941875973593q_real @ ( power_power_real @ X2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7762_VEBT__internal_Oinsert_H_Opelims,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: vEBT_VEBT] :
% 9.00/5.10 ( ( ( vEBT_VEBT_insert @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
% 9.00/5.10 & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
% 9.00/5.10 => ( Y3
% 9.00/5.10 = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7763_vebt__member_Opelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7764_vebt__member_Opelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 9.00/5.10 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.00/5.10 => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 9.00/5.10 => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
% 9.00/5.10 => ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7765_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 9.00/5.10 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.00/5.10 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7766_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7767_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_V5719532721284313246member @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S3: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7768_vebt__member_Opelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_vebt_member @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [A: $o,B: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ A @ B ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A @ B ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.10 => A )
% 9.00/5.10 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.10 => ( ( ( Xa2 = one_one_nat )
% 9.00/5.10 => B )
% 9.00/5.10 & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.00/5.10 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( Xa2 != Mi2 )
% 9.00/5.10 => ( ( Xa2 != Ma2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.00/5.10 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7769_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ~ ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 9.00/5.10 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.00/5.10 => ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
% 9.00/5.10 => ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
% 9.00/5.10 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7770_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.10 ( ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 = Y3 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.00/5.10 => ( ~ Y3
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ( ( Y3
% 9.00/5.10 = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) )
% 9.00/5.10 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7771_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 9.00/5.10 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.10 ( ( vEBT_VEBT_membermima @ X2 @ Xa2 )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 ) ) ) )
% 9.00/5.10 => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( Xa2 = Mi2 )
% 9.00/5.10 | ( Xa2 = Ma2 )
% 9.00/5.10 | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) )
% 9.00/5.10 => ~ ! [V2: nat,TreeList: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
% 9.00/5.10 ( ( X2
% 9.00/5.10 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) )
% 9.00/5.10 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ Xa2 ) )
% 9.00/5.10 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
% 9.00/5.10 => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.10 & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7772_ln__series,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.10 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.10 => ( ( ln_ln_real @ X2 )
% 9.00/5.10 = ( suminf_real
% 9.00/5.10 @ ^ [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 @ X2 @ one_one_real ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7773_signed__take__bit__rec,axiom,
% 9.00/5.10 ( bit_ri6519982836138164636nteger
% 9.00/5.10 = ( ^ [N2: nat,A4: code_integer] : ( if_Code_integer @ ( N2 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7774_signed__take__bit__rec,axiom,
% 9.00/5.10 ( bit_ri631733984087533419it_int
% 9.00/5.10 = ( ^ [N2: nat,A4: int] : ( if_int @ ( N2 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7775_arctan__series,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.10 => ( ( arctan @ X2 )
% 9.00/5.10 = ( suminf_real
% 9.00/5.10 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7776_signed__take__bit__of__0,axiom,
% 9.00/5.10 ! [N: nat] :
% 9.00/5.10 ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7777_powser__zero,axiom,
% 9.00/5.10 ! [F: nat > complex] :
% 9.00/5.10 ( ( suminf_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) )
% 9.00/5.10 = ( F @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7778_powser__zero,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) )
% 9.00/5.10 = ( F @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7779_signed__take__bit__0,axiom,
% 9.00/5.10 ! [A2: code_integer] :
% 9.00/5.10 ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A2 )
% 9.00/5.10 = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7780_signed__take__bit__0,axiom,
% 9.00/5.10 ! [A2: int] :
% 9.00/5.10 ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A2 )
% 9.00/5.10 = ( uminus_uminus_int @ ( modulo_modulo_int @ A2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7781_signed__take__bit__int__less__exp,axiom,
% 9.00/5.10 ! [N: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7782_signed__take__bit__int__greater__eq__self__iff,axiom,
% 9.00/5.10 ! [K: int,N: nat] :
% 9.00/5.10 ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.00/5.10 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7783_signed__take__bit__int__less__self__iff,axiom,
% 9.00/5.10 ! [N: nat,K: int] :
% 9.00/5.10 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.00/5.10 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7784_signed__take__bit__int__greater__eq__minus__exp,axiom,
% 9.00/5.10 ! [N: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7785_signed__take__bit__int__less__eq__self__iff,axiom,
% 9.00/5.10 ! [N: nat,K: int] :
% 9.00/5.10 ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ K )
% 9.00/5.10 = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7786_signed__take__bit__int__greater__self__iff,axiom,
% 9.00/5.10 ! [K: int,N: nat] :
% 9.00/5.10 ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.00/5.10 = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7787_signed__take__bit__int__less__eq,axiom,
% 9.00/5.10 ! [N: nat,K: int] :
% 9.00/5.10 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.00/5.10 => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7788_signed__take__bit__int__eq__self,axiom,
% 9.00/5.10 ! [N: nat,K: int] :
% 9.00/5.10 ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.00/5.10 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.10 => ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.00/5.10 = K ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7789_signed__take__bit__int__eq__self__iff,axiom,
% 9.00/5.10 ! [N: nat,K: int] :
% 9.00/5.10 ( ( ( bit_ri631733984087533419it_int @ N @ K )
% 9.00/5.10 = K )
% 9.00/5.10 = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K )
% 9.00/5.10 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7790_signed__take__bit__int__greater__eq,axiom,
% 9.00/5.10 ! [K: int,N: nat] :
% 9.00/5.10 ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
% 9.00/5.10 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7791_suminf__geometric,axiom,
% 9.00/5.10 ! [C: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.00/5.10 => ( ( suminf_real @ ( power_power_real @ C ) )
% 9.00/5.10 = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7792_suminf__geometric,axiom,
% 9.00/5.10 ! [C: complex] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.00/5.10 => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 9.00/5.10 = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7793_suminf__zero,axiom,
% 9.00/5.10 ( ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_real )
% 9.00/5.10 = zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7794_suminf__zero,axiom,
% 9.00/5.10 ( ( suminf_nat
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_nat )
% 9.00/5.10 = zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7795_suminf__zero,axiom,
% 9.00/5.10 ( ( suminf_int
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_int )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7796_round__unique,axiom,
% 9.00/5.10 ! [X2: real,Y3: int] :
% 9.00/5.10 ( ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 9.00/5.10 => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 => ( ( archim8280529875227126926d_real @ X2 )
% 9.00/5.10 = Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7797_round__unique,axiom,
% 9.00/5.10 ! [X2: rat,Y3: int] :
% 9.00/5.10 ( ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 9.00/5.10 => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.10 => ( ( archim7778729529865785530nd_rat @ X2 )
% 9.00/5.10 = Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7798_summable__arctan__series,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X2 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7799_tanh__ln__real,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.10 => ( ( tanh_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.10 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7800_round__unique_H,axiom,
% 9.00/5.10 ! [X2: rat,N: int] :
% 9.00/5.10 ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X2 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ( archim7778729529865785530nd_rat @ X2 )
% 9.00/5.10 = N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7801_round__unique_H,axiom,
% 9.00/5.10 ! [X2: real,N: int] :
% 9.00/5.10 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.10 => ( ( archim8280529875227126926d_real @ X2 )
% 9.00/5.10 = N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7802_tanh__0,axiom,
% 9.00/5.10 ( ( tanh_real @ zero_zero_real )
% 9.00/5.10 = zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7803_tanh__real__less__iff,axiom,
% 9.00/5.10 ! [X2: real,Y3: real] :
% 9.00/5.10 ( ( ord_less_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 9.00/5.10 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7804_tanh__real__le__iff,axiom,
% 9.00/5.10 ! [X2: real,Y3: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ ( tanh_real @ Y3 ) )
% 9.00/5.10 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7805_summable__zero,axiom,
% 9.00/5.10 ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7806_summable__zero,axiom,
% 9.00/5.10 ( summable_nat
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7807_summable__zero,axiom,
% 9.00/5.10 ( summable_int
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7808_summable__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > real] :
% 9.00/5.10 ( summable_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7809_summable__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > nat] :
% 9.00/5.10 ( summable_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7810_summable__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > int] :
% 9.00/5.10 ( summable_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7811_round__0,axiom,
% 9.00/5.10 ( ( archim8280529875227126926d_real @ zero_zero_real )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7812_round__0,axiom,
% 9.00/5.10 ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7813_tanh__real__neg__iff,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 9.00/5.10 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7814_tanh__real__pos__iff,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 9.00/5.10 = ( ord_less_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7815_tanh__real__nonneg__iff,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X2 ) )
% 9.00/5.10 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7816_tanh__real__nonpos__iff,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ ( tanh_real @ X2 ) @ zero_zero_real )
% 9.00/5.10 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7817_summable__cmult__iff,axiom,
% 9.00/5.10 ! [C: complex,F: nat > complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.00/5.10 = ( ( C = zero_zero_complex )
% 9.00/5.10 | ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7818_summable__cmult__iff,axiom,
% 9.00/5.10 ! [C: real,F: nat > real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.00/5.10 = ( ( C = zero_zero_real )
% 9.00/5.10 | ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7819_summable__divide__iff,axiom,
% 9.00/5.10 ! [F: nat > real,C: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( divide_divide_real @ ( F @ N2 ) @ C ) )
% 9.00/5.10 = ( ( C = zero_zero_real )
% 9.00/5.10 | ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7820_summable__divide__iff,axiom,
% 9.00/5.10 ! [F: nat > complex,C: complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( divide1717551699836669952omplex @ ( F @ N2 ) @ C ) )
% 9.00/5.10 = ( ( C = zero_zero_complex )
% 9.00/5.10 | ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7821_summable__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7822_summable__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( summable_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7823_summable__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( summable_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7824_summable__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7825_summable__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( summable_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7826_summable__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( summable_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7827_summable__geometric__iff,axiom,
% 9.00/5.10 ! [C: real] :
% 9.00/5.10 ( ( summable_real @ ( power_power_real @ C ) )
% 9.00/5.10 = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7828_summable__geometric__iff,axiom,
% 9.00/5.10 ! [C: complex] :
% 9.00/5.10 ( ( summable_complex @ ( power_power_complex @ C ) )
% 9.00/5.10 = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7829_summable__const__iff,axiom,
% 9.00/5.10 ! [C: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [Uu3: nat] : C )
% 9.00/5.10 = ( C = zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7830_summable__comparison__test_H,axiom,
% 9.00/5.10 ! [G3: nat > real,N6: nat,F: nat > real] :
% 9.00/5.10 ( ( summable_real @ G3 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7831_summable__comparison__test_H,axiom,
% 9.00/5.10 ! [G3: nat > real,N6: nat,F: nat > complex] :
% 9.00/5.10 ( ( summable_real @ G3 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7832_summable__comparison__test,axiom,
% 9.00/5.10 ! [F: nat > real,G3: nat > real] :
% 9.00/5.10 ( ? [N8: nat] :
% 9.00/5.10 ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( ( summable_real @ G3 )
% 9.00/5.10 => ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7833_summable__comparison__test,axiom,
% 9.00/5.10 ! [F: nat > complex,G3: nat > real] :
% 9.00/5.10 ( ? [N8: nat] :
% 9.00/5.10 ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( ( summable_real @ G3 )
% 9.00/5.10 => ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7834_suminf__le,axiom,
% 9.00/5.10 ! [F: nat > real,G3: nat > real] :
% 9.00/5.10 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.10 => ( ( summable_real @ F )
% 9.00/5.10 => ( ( summable_real @ G3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7835_suminf__le,axiom,
% 9.00/5.10 ! [F: nat > nat,G3: nat > nat] :
% 9.00/5.10 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.10 => ( ( summable_nat @ F )
% 9.00/5.10 => ( ( summable_nat @ G3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7836_suminf__le,axiom,
% 9.00/5.10 ! [F: nat > int,G3: nat > int] :
% 9.00/5.10 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.10 => ( ( summable_int @ F )
% 9.00/5.10 => ( ( summable_int @ G3 )
% 9.00/5.10 => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7837_summable__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7838_summable__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_nat ) )
% 9.00/5.10 => ( summable_nat @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7839_summable__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_int ) )
% 9.00/5.10 => ( summable_int @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7840_summable__mult__D,axiom,
% 9.00/5.10 ! [C: complex,F: nat > complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) ) )
% 9.00/5.10 => ( ( C != zero_zero_complex )
% 9.00/5.10 => ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7841_summable__mult__D,axiom,
% 9.00/5.10 ! [C: real,F: nat > real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) ) )
% 9.00/5.10 => ( ( C != zero_zero_real )
% 9.00/5.10 => ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7842_summable__zero__power,axiom,
% 9.00/5.10 summable_int @ ( power_power_int @ zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7843_summable__zero__power,axiom,
% 9.00/5.10 summable_real @ ( power_power_real @ zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7844_summable__zero__power,axiom,
% 9.00/5.10 summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7845_powser__insidea,axiom,
% 9.00/5.10 ! [F: nat > real,X2: real,Z: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7846_powser__insidea,axiom,
% 9.00/5.10 ! [F: nat > complex,X2: complex,Z: complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7847_suminf__nonneg,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7848_suminf__nonneg,axiom,
% 9.00/5.10 ! [F: nat > nat] :
% 9.00/5.10 ( ( summable_nat @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7849_suminf__nonneg,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( ( summable_int @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7850_suminf__eq__zero__iff,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ( suminf_real @ F )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [N2: nat] :
% 9.00/5.10 ( ( F @ N2 )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7851_suminf__eq__zero__iff,axiom,
% 9.00/5.10 ! [F: nat > nat] :
% 9.00/5.10 ( ( summable_nat @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ( suminf_nat @ F )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [N2: nat] :
% 9.00/5.10 ( ( F @ N2 )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7852_suminf__eq__zero__iff,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( ( summable_int @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ( suminf_int @ F )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 = ( ! [N2: nat] :
% 9.00/5.10 ( ( F @ N2 )
% 9.00/5.10 = zero_zero_int ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7853_suminf__pos,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7854_suminf__pos,axiom,
% 9.00/5.10 ! [F: nat > nat] :
% 9.00/5.10 ( ( summable_nat @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7855_suminf__pos,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( ( summable_int @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.10 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7856_summable__zero__power_H,axiom,
% 9.00/5.10 ! [F: nat > complex] :
% 9.00/5.10 ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7857_summable__zero__power_H,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7858_summable__zero__power_H,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( summable_int
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_int @ ( F @ N2 ) @ ( power_power_int @ zero_zero_int @ N2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7859_summable__0__powser,axiom,
% 9.00/5.10 ! [F: nat > complex] :
% 9.00/5.10 ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7860_summable__0__powser,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7861_summable__norm__comparison__test,axiom,
% 9.00/5.10 ! [F: nat > complex,G3: nat > real] :
% 9.00/5.10 ( ? [N8: nat] :
% 9.00/5.10 ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( ( summable_real @ G3 )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7862_summable__rabs__comparison__test,axiom,
% 9.00/5.10 ! [F: nat > real,G3: nat > real] :
% 9.00/5.10 ( ? [N8: nat] :
% 9.00/5.10 ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G3 @ N3 ) ) )
% 9.00/5.10 => ( ( summable_real @ G3 )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7863_summable__rabs,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( abs_abs_real @ ( F @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7864_tanh__real__lt__1,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_real @ ( tanh_real @ X2 ) @ one_one_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7865_suminf__pos__iff,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 9.00/5.10 = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7866_suminf__pos__iff,axiom,
% 9.00/5.10 ! [F: nat > nat] :
% 9.00/5.10 ( ( summable_nat @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 9.00/5.10 = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7867_suminf__pos__iff,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( ( summable_int @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 9.00/5.10 = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7868_suminf__pos2,axiom,
% 9.00/5.10 ! [F: nat > real,I: nat] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.10 => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7869_suminf__pos2,axiom,
% 9.00/5.10 ! [F: nat > nat,I: nat] :
% 9.00/5.10 ( ( summable_nat @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.00/5.10 => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7870_suminf__pos2,axiom,
% 9.00/5.10 ! [F: nat > int,I: nat] :
% 9.00/5.10 ( ( summable_int @ F )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.10 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.00/5.10 => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7871_summable__geometric,axiom,
% 9.00/5.10 ! [C: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.00/5.10 => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7872_summable__geometric,axiom,
% 9.00/5.10 ! [C: complex] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.00/5.10 => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7873_complete__algebra__summable__geometric,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ one_one_real )
% 9.00/5.10 => ( summable_real @ ( power_power_real @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7874_complete__algebra__summable__geometric,axiom,
% 9.00/5.10 ! [X2: complex] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ one_one_real )
% 9.00/5.10 => ( summable_complex @ ( power_power_complex @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7875_suminf__split__head,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real @ F )
% 9.00/5.10 => ( ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) ) )
% 9.00/5.10 = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7876_round__mono,axiom,
% 9.00/5.10 ! [X2: rat,Y3: rat] :
% 9.00/5.10 ( ( ord_less_eq_rat @ X2 @ Y3 )
% 9.00/5.10 => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X2 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7877_summable__norm,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7878_summable__norm,axiom,
% 9.00/5.10 ! [F: nat > complex] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7879_ceiling__ge__round,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X2 ) @ ( archim7802044766580827645g_real @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7880_tanh__real__gt__neg1,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7881_powser__inside,axiom,
% 9.00/5.10 ! [F: nat > real,X2: real,Z: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X2 ) )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7882_powser__inside,axiom,
% 9.00/5.10 ! [F: nat > complex,X2: complex,Z: complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 9.00/5.10 => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X2 ) )
% 9.00/5.10 => ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7883_powser__split__head_I1_J,axiom,
% 9.00/5.10 ! [F: nat > complex,Z: complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 => ( ( suminf_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 9.00/5.10 @ ( times_times_complex
% 9.00/5.10 @ ( suminf_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 @ Z ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7884_powser__split__head_I1_J,axiom,
% 9.00/5.10 ! [F: nat > real,Z: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 => ( ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 = ( plus_plus_real @ ( F @ zero_zero_nat )
% 9.00/5.10 @ ( times_times_real
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 @ Z ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7885_powser__split__head_I2_J,axiom,
% 9.00/5.10 ! [F: nat > complex,Z: complex] :
% 9.00/5.10 ( ( summable_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 => ( ( times_times_complex
% 9.00/5.10 @ ( suminf_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 @ Z )
% 9.00/5.10 = ( minus_minus_complex
% 9.00/5.10 @ ( suminf_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) )
% 9.00/5.10 @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7886_powser__split__head_I2_J,axiom,
% 9.00/5.10 ! [F: nat > real,Z: real] :
% 9.00/5.10 ( ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 => ( ( times_times_real
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 @ Z )
% 9.00/5.10 = ( minus_minus_real
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ Z @ N2 ) ) )
% 9.00/5.10 @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7887_suminf__exist__split,axiom,
% 9.00/5.10 ! [R2: real,F: nat > real] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.00/5.10 => ( ( summable_real @ F )
% 9.00/5.10 => ? [N9: nat] :
% 9.00/5.10 ! [N5: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N9 @ N5 )
% 9.00/5.10 => ( ord_less_real
% 9.00/5.10 @ ( real_V7735802525324610683m_real
% 9.00/5.10 @ ( suminf_real
% 9.00/5.10 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 9.00/5.10 @ R2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7888_suminf__exist__split,axiom,
% 9.00/5.10 ! [R2: real,F: nat > complex] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.00/5.10 => ( ( summable_complex @ F )
% 9.00/5.10 => ? [N9: nat] :
% 9.00/5.10 ! [N5: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N9 @ N5 )
% 9.00/5.10 => ( ord_less_real
% 9.00/5.10 @ ( real_V1022390504157884413omplex
% 9.00/5.10 @ ( suminf_complex
% 9.00/5.10 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N5 ) ) ) )
% 9.00/5.10 @ R2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7889_summable__power__series,axiom,
% 9.00/5.10 ! [F: nat > real,Z: real] :
% 9.00/5.10 ( ! [I2: nat] : ( ord_less_eq_real @ ( F @ I2 ) @ one_one_real )
% 9.00/5.10 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) )
% 9.00/5.10 => ( ( ord_less_eq_real @ zero_zero_real @ Z )
% 9.00/5.10 => ( ( ord_less_real @ Z @ one_one_real )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z @ I4 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7890_Abel__lemma,axiom,
% 9.00/5.10 ! [R2: real,R0: real,A2: nat > complex,M5: real] :
% 9.00/5.10 ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.00/5.10 => ( ( ord_less_real @ R2 @ R0 )
% 9.00/5.10 => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M5 )
% 9.00/5.10 => ( summable_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A2 @ N2 ) ) @ ( power_power_real @ R2 @ N2 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7891_summable__ratio__test,axiom,
% 9.00/5.10 ! [C: real,N6: nat,F: nat > real] :
% 9.00/5.10 ( ( ord_less_real @ C @ one_one_real )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 9.00/5.10 => ( summable_real @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7892_summable__ratio__test,axiom,
% 9.00/5.10 ! [C: real,N6: nat,F: nat > complex] :
% 9.00/5.10 ( ( ord_less_real @ C @ one_one_real )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ( ord_less_eq_nat @ N6 @ N3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 9.00/5.10 => ( summable_complex @ F ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7893_round__diff__minimal,axiom,
% 9.00/5.10 ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7894_round__diff__minimal,axiom,
% 9.00/5.10 ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7895_of__int__round__le,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ ( plus_plus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7896_of__int__round__le,axiom,
% 9.00/5.10 ! [X2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ ( plus_plus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7897_of__int__round__ge,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_eq_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7898_of__int__round__ge,axiom,
% 9.00/5.10 ! [X2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7899_of__int__round__gt,axiom,
% 9.00/5.10 ! [X2: rat] : ( ord_less_rat @ ( minus_minus_rat @ X2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7900_of__int__round__gt,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_real @ ( minus_minus_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7901_of__int__round__abs__le,axiom,
% 9.00/5.10 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X2 ) ) @ X2 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7902_of__int__round__abs__le,axiom,
% 9.00/5.10 ! [X2: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X2 ) ) @ X2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7903_accp__subset,axiom,
% 9.00/5.10 ! [R1: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,R22: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o] :
% 9.00/5.10 ( ( ord_le7862513914298786254_nat_o @ R1 @ R22 )
% 9.00/5.10 => ( ord_le8126618931240741628_nat_o @ ( accp_P8646395344606611882on_nat @ R22 ) @ ( accp_P8646395344606611882on_nat @ R1 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7904_accp__subset,axiom,
% 9.00/5.10 ! [R1: product_prod_nat_nat > product_prod_nat_nat > $o,R22: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.00/5.10 ( ( ord_le5604493270027003598_nat_o @ R1 @ R22 )
% 9.00/5.10 => ( ord_le704812498762024988_nat_o @ ( accp_P4275260045618599050at_nat @ R22 ) @ ( accp_P4275260045618599050at_nat @ R1 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7905_accp__subset,axiom,
% 9.00/5.10 ! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
% 9.00/5.10 ( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
% 9.00/5.10 => ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7906_accp__subset,axiom,
% 9.00/5.10 ! [R1: list_nat > list_nat > $o,R22: list_nat > list_nat > $o] :
% 9.00/5.10 ( ( ord_le6558929396352911974_nat_o @ R1 @ R22 )
% 9.00/5.10 => ( ord_le1520216061033275535_nat_o @ ( accp_list_nat @ R22 ) @ ( accp_list_nat @ R1 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7907_accp__subset,axiom,
% 9.00/5.10 ! [R1: nat > nat > $o,R22: nat > nat > $o] :
% 9.00/5.10 ( ( ord_le2646555220125990790_nat_o @ R1 @ R22 )
% 9.00/5.10 => ( ord_less_eq_nat_o @ ( accp_nat @ R22 ) @ ( accp_nat @ R1 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7908_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_o,X2: $o > complex,Y3: $o > complex] :
% 9.00/5.10 ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7909_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_real,X2: real > complex,Y3: real > complex] :
% 9.00/5.10 ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7910_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_nat,X2: nat > complex,Y3: nat > complex] :
% 9.00/5.10 ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7911_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_int,X2: int > complex,Y3: int > complex] :
% 9.00/5.10 ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7912_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_complex,X2: complex > complex,Y3: complex > complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7913_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_Extended_enat,X2: extended_enat > complex,Y3: extended_enat > complex] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_complex ) ) ) )
% 9.00/5.10 => ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_complex @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_complex ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7914_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_o,X2: $o > real,Y3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7915_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 9.00/5.10 ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7916_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7917_prod_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 9.00/5.10 ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != one_one_real ) ) ) )
% 9.00/5.10 => ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( times_times_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != one_one_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7918_sum__gp,axiom,
% 9.00/5.10 ! [N: nat,M: nat,X2: rat] :
% 9.00/5.10 ( ( ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = zero_zero_rat ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( ( X2 = one_one_rat )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.00/5.10 & ( ( X2 != one_one_rat )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7919_sum__gp,axiom,
% 9.00/5.10 ! [N: nat,M: nat,X2: complex] :
% 9.00/5.10 ( ( ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = zero_zero_complex ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( ( X2 = one_one_complex )
% 9.00/5.10 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.00/5.10 & ( ( X2 != one_one_complex )
% 9.00/5.10 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7920_sum__gp,axiom,
% 9.00/5.10 ! [N: nat,M: nat,X2: real] :
% 9.00/5.10 ( ( ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ N @ M )
% 9.00/5.10 => ( ( ( X2 = one_one_real )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) )
% 9.00/5.10 & ( ( X2 != one_one_real )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.10 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7921_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_o,X2: $o > real,Y3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7922_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_real,X2: real > real,Y3: real > real] :
% 9.00/5.10 ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7923_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_nat,X2: nat > real,Y3: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7924_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_int,X2: int > real,Y3: int > real] :
% 9.00/5.10 ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7925_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_complex,X2: complex > real,Y3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite3207457112153483333omplex
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [I4: complex] :
% 9.00/5.10 ( ( member_complex @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7926_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_Extended_enat,X2: extended_enat > real,Y3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_real ) ) ) )
% 9.00/5.10 => ( finite4001608067531595151d_enat
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [I4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_real @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7927_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_o,X2: $o > rat,Y3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( finite_finite_o
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [I4: $o] :
% 9.00/5.10 ( ( member_o @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7928_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_real,X2: real > rat,Y3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( finite_finite_real
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [I4: real] :
% 9.00/5.10 ( ( member_real @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7929_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_nat,X2: nat > rat,Y3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( finite_finite_nat
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [I4: nat] :
% 9.00/5.10 ( ( member_nat @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7930_sum_Ofinite__Collect__op,axiom,
% 9.00/5.10 ! [I5: set_int,X2: int > rat,Y3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( X2 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( Y3 @ I4 )
% 9.00/5.10 != zero_zero_rat ) ) ) )
% 9.00/5.10 => ( finite_finite_int
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [I4: int] :
% 9.00/5.10 ( ( member_int @ I4 @ I5 )
% 9.00/5.10 & ( ( plus_plus_rat @ ( X2 @ I4 ) @ ( Y3 @ I4 ) )
% 9.00/5.10 != zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7931_geometric__deriv__sums,axiom,
% 9.00/5.10 ! [Z: real] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
% 9.00/5.10 => ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( power_power_real @ Z @ N2 ) )
% 9.00/5.10 @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7932_geometric__deriv__sums,axiom,
% 9.00/5.10 ! [Z: complex] :
% 9.00/5.10 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
% 9.00/5.10 => ( sums_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( power_power_complex @ Z @ N2 ) )
% 9.00/5.10 @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7933_log__base__10__eq1,axiom,
% 9.00/5.10 ! [X2: real] :
% 9.00/5.10 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.10 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.10 = ( 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 @ X2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7934_semiring__norm_I80_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.10 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7935_semiring__norm_I73_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.10 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7936_sum_Oneutral__const,axiom,
% 9.00/5.10 ! [A3: set_nat] :
% 9.00/5.10 ( ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Uu3: nat] : zero_zero_nat
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7937_sum_Oneutral__const,axiom,
% 9.00/5.10 ! [A3: set_complex] :
% 9.00/5.10 ( ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Uu3: complex] : zero_zero_complex
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_complex ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7938_sum_Oneutral__const,axiom,
% 9.00/5.10 ! [A3: set_int] :
% 9.00/5.10 ( ( groups4538972089207619220nt_int
% 9.00/5.10 @ ^ [Uu3: int] : zero_zero_int
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7939_sum_Oneutral__const,axiom,
% 9.00/5.10 ! [A3: set_nat] :
% 9.00/5.10 ( ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [Uu3: nat] : zero_zero_real
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7940_sums__zero,axiom,
% 9.00/5.10 ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_real
% 9.00/5.10 @ zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7941_sums__zero,axiom,
% 9.00/5.10 ( sums_nat
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_nat
% 9.00/5.10 @ zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7942_sums__zero,axiom,
% 9.00/5.10 ( sums_int
% 9.00/5.10 @ ^ [N2: nat] : zero_zero_int
% 9.00/5.10 @ zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7943_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: real > real] :
% 9.00/5.10 ( ( groups8097168146408367636l_real @ G3 @ bot_bot_set_real )
% 9.00/5.10 = zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7944_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: real > rat] :
% 9.00/5.10 ( ( groups1300246762558778688al_rat @ G3 @ bot_bot_set_real )
% 9.00/5.10 = zero_zero_rat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7945_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: real > nat] :
% 9.00/5.10 ( ( groups1935376822645274424al_nat @ G3 @ bot_bot_set_real )
% 9.00/5.10 = zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7946_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: real > int] :
% 9.00/5.10 ( ( groups1932886352136224148al_int @ G3 @ bot_bot_set_real )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7947_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: $o > real] :
% 9.00/5.10 ( ( groups8691415230153176458o_real @ G3 @ bot_bot_set_o )
% 9.00/5.10 = zero_zero_real ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7948_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: $o > rat] :
% 9.00/5.10 ( ( groups7872700643590313910_o_rat @ G3 @ bot_bot_set_o )
% 9.00/5.10 = zero_zero_rat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7949_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: $o > nat] :
% 9.00/5.10 ( ( groups8507830703676809646_o_nat @ G3 @ bot_bot_set_o )
% 9.00/5.10 = zero_zero_nat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7950_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: $o > int] :
% 9.00/5.10 ( ( groups8505340233167759370_o_int @ G3 @ bot_bot_set_o )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7951_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: nat > rat] :
% 9.00/5.10 ( ( groups2906978787729119204at_rat @ G3 @ bot_bot_set_nat )
% 9.00/5.10 = zero_zero_rat ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7952_sum_Oempty,axiom,
% 9.00/5.10 ! [G3: nat > int] :
% 9.00/5.10 ( ( groups3539618377306564664at_int @ G3 @ bot_bot_set_nat )
% 9.00/5.10 = zero_zero_int ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7953_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > real] :
% 9.00/5.10 ( ~ ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7954_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > real] :
% 9.00/5.10 ( ~ ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7955_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.10 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7956_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_nat,G3: nat > rat] :
% 9.00/5.10 ( ~ ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7957_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > rat] :
% 9.00/5.10 ( ~ ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7958_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > rat] :
% 9.00/5.10 ( ~ ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7959_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.10 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7960_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > nat] :
% 9.00/5.10 ( ~ ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7961_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > nat] :
% 9.00/5.10 ( ~ ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7962_sum_Oinfinite,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.10 ( ~ ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7963_sum__eq__0__iff,axiom,
% 9.00/5.10 ! [F2: set_int,F: int > nat] :
% 9.00/5.10 ( ( finite_finite_int @ F2 )
% 9.00/5.10 => ( ( ( groups4541462559716669496nt_nat @ F @ F2 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [X: int] :
% 9.00/5.10 ( ( member_int @ X @ F2 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7964_sum__eq__0__iff,axiom,
% 9.00/5.10 ! [F2: set_complex,F: complex > nat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ F2 )
% 9.00/5.10 => ( ( ( groups5693394587270226106ex_nat @ F @ F2 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ F2 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7965_sum__eq__0__iff,axiom,
% 9.00/5.10 ! [F2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.10 ( ( finite6177210948735845034at_nat @ F2 )
% 9.00/5.10 => ( ( ( groups977919841031483927at_nat @ F @ F2 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [X: product_prod_nat_nat] :
% 9.00/5.10 ( ( member8440522571783428010at_nat @ X @ F2 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7966_sum__eq__0__iff,axiom,
% 9.00/5.10 ! [F2: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ F2 )
% 9.00/5.10 => ( ( ( groups2027974829824023292at_nat @ F @ F2 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [X: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X @ F2 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7967_sum__eq__0__iff,axiom,
% 9.00/5.10 ! [F2: set_nat,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ F2 )
% 9.00/5.10 => ( ( ( groups3542108847815614940at_nat @ F @ F2 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 = ( ! [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ F2 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_nat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7968_semiring__norm_I81_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.00/5.10 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7969_semiring__norm_I72_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.10 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7970_semiring__norm_I77_J,axiom,
% 9.00/5.10 ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7971_semiring__norm_I70_J,axiom,
% 9.00/5.10 ! [M: num] :
% 9.00/5.10 ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7972_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_real,A2: real,B3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ S )
% 9.00/5.10 => ( ( ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [K3: real] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7973_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_o,A2: $o,B3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ S )
% 9.00/5.10 => ( ( ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [K3: $o] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7974_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_int,A2: int,B3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ( ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [K3: int] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7975_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_complex,A2: complex,B3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [K3: complex] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7976_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.10 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.10 => ( ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.10 => ( ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [K3: extended_enat] : ( if_real @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7977_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_real,A2: real,B3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ S )
% 9.00/5.10 => ( ( ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [K3: real] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7978_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_o,A2: $o,B3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ S )
% 9.00/5.10 => ( ( ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [K3: $o] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7979_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_nat,A2: nat,B3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ S )
% 9.00/5.10 => ( ( ( member_nat @ A2 @ S )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_nat @ A2 @ S )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [K3: nat] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7980_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_int,A2: int,B3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ( ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [K3: int] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7981_sum_Odelta_H,axiom,
% 9.00/5.10 ! [S: set_complex,A2: complex,B3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [K3: complex] : ( if_rat @ ( A2 = K3 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7982_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_real,A2: real,B3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ S )
% 9.00/5.10 => ( ( ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7983_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_o,A2: $o,B3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ S )
% 9.00/5.10 => ( ( ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [K3: $o] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7984_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_int,A2: int,B3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ( ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [K3: int] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7985_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_complex,A2: complex,B3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7986_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.10 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.10 => ( ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.10 => ( ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_real )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_real ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7987_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_real,A2: real,B3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ S )
% 9.00/5.10 => ( ( ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7988_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_o,A2: $o,B3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ S )
% 9.00/5.10 => ( ( ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ A2 @ S )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [K3: $o] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7989_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_nat,A2: nat,B3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ S )
% 9.00/5.10 => ( ( ( member_nat @ A2 @ S )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_nat @ A2 @ S )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [K3: nat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7990_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_int,A2: int,B3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ( ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ A2 @ S )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [K3: int] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7991_sum_Odelta,axiom,
% 9.00/5.10 ! [S: set_complex,A2: complex,B3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = ( B3 @ A2 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ zero_zero_rat )
% 9.00/5.10 @ S )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7992_sum__abs,axiom,
% 9.00/5.10 ! [F: int > int,A3: set_int] :
% 9.00/5.10 ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A3 ) )
% 9.00/5.10 @ ( groups4538972089207619220nt_int
% 9.00/5.10 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7993_sum__abs,axiom,
% 9.00/5.10 ! [F: nat > real,A3: set_nat] :
% 9.00/5.10 ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
% 9.00/5.10 @ ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7994_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_real,X2: real,G3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7995_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_o,X2: $o,G3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7996_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_int,X2: int,G3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7997_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7998_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ~ ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_7999_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_real,X2: real,G3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8000_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_o,X2: $o,G3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8001_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_nat,X2: nat,G3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8002_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_int,X2: int,G3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8003_sum_Oinsert,axiom,
% 9.00/5.10 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8004_sum__abs__ge__zero,axiom,
% 9.00/5.10 ! [F: int > int,A3: set_int] :
% 9.00/5.10 ( ord_less_eq_int @ zero_zero_int
% 9.00/5.10 @ ( groups4538972089207619220nt_int
% 9.00/5.10 @ ^ [I4: int] : ( abs_abs_int @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8005_sum__abs__ge__zero,axiom,
% 9.00/5.10 ! [F: nat > real,A3: set_nat] :
% 9.00/5.10 ( ord_less_eq_real @ zero_zero_real
% 9.00/5.10 @ ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( abs_abs_real @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8006_semiring__norm_I74_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
% 9.00/5.10 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8007_semiring__norm_I79_J,axiom,
% 9.00/5.10 ! [M: num,N: num] :
% 9.00/5.10 ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.10 = ( ord_less_eq_num @ M @ N ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8008_powser__sums__zero__iff,axiom,
% 9.00/5.10 ! [A2: nat > complex,X2: complex] :
% 9.00/5.10 ( ( sums_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.00/5.10 @ X2 )
% 9.00/5.10 = ( ( A2 @ zero_zero_nat )
% 9.00/5.10 = X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8009_powser__sums__zero__iff,axiom,
% 9.00/5.10 ! [A2: nat > real,X2: real] :
% 9.00/5.10 ( ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.00/5.10 @ X2 )
% 9.00/5.10 = ( ( A2 @ zero_zero_nat )
% 9.00/5.10 = X2 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8010_sum_Ocl__ivl__Suc,axiom,
% 9.00/5.10 ! [N: nat,M: nat,G3: nat > rat] :
% 9.00/5.10 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = zero_zero_rat ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8011_sum_Ocl__ivl__Suc,axiom,
% 9.00/5.10 ! [N: nat,M: nat,G3: nat > int] :
% 9.00/5.10 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = zero_zero_int ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8012_sum_Ocl__ivl__Suc,axiom,
% 9.00/5.10 ! [N: nat,M: nat,G3: nat > nat] :
% 9.00/5.10 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = zero_zero_nat ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8013_sum_Ocl__ivl__Suc,axiom,
% 9.00/5.10 ! [N: nat,M: nat,G3: nat > real] :
% 9.00/5.10 ( ( ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 & ( ~ ( ord_less_nat @ ( suc @ N ) @ M )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.10 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8014_sum__zero__power,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > complex] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( C @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_complex ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8015_sum__zero__power,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > rat] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( C @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8016_sum__zero__power,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > real] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( C @ zero_zero_nat ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8017_sum__zero__power_H,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > rat,D: nat > rat] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [I4: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ zero_zero_rat @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8018_sum__zero__power_H,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > complex,D: nat > complex] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [I4: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ zero_zero_complex @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_complex ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8019_sum__zero__power_H,axiom,
% 9.00/5.10 ! [A3: set_nat,C: nat > real,D: nat > real] :
% 9.00/5.10 ( ( ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.00/5.10 & ( ~ ( ( finite_finite_nat @ A3 )
% 9.00/5.10 & ( member_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ zero_zero_real @ I4 ) ) @ ( D @ I4 ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8020_sums__le,axiom,
% 9.00/5.10 ! [F: nat > nat,G3: nat > nat,S2: nat,T: nat] :
% 9.00/5.10 ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.10 => ( ( sums_nat @ F @ S2 )
% 9.00/5.10 => ( ( sums_nat @ G3 @ T )
% 9.00/5.10 => ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8021_sums__le,axiom,
% 9.00/5.10 ! [F: nat > int,G3: nat > int,S2: int,T: int] :
% 9.00/5.10 ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.10 => ( ( sums_int @ F @ S2 )
% 9.00/5.10 => ( ( sums_int @ G3 @ T )
% 9.00/5.10 => ( ord_less_eq_int @ S2 @ T ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8022_sums__0,axiom,
% 9.00/5.10 ! [F: nat > real] :
% 9.00/5.10 ( ! [N3: nat] :
% 9.00/5.10 ( ( F @ N3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( sums_real @ F @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8023_sums__0,axiom,
% 9.00/5.10 ! [F: nat > nat] :
% 9.00/5.10 ( ! [N3: nat] :
% 9.00/5.10 ( ( F @ N3 )
% 9.00/5.10 = zero_zero_nat )
% 9.00/5.10 => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8024_sums__0,axiom,
% 9.00/5.10 ! [F: nat > int] :
% 9.00/5.10 ( ! [N3: nat] :
% 9.00/5.10 ( ( F @ N3 )
% 9.00/5.10 = zero_zero_int )
% 9.00/5.10 => ( sums_int @ F @ zero_zero_int ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8025_sum_Oneutral,axiom,
% 9.00/5.10 ! [A3: set_nat,G3: nat > nat] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ( G3 @ X5 )
% 9.00/5.10 = zero_zero_nat ) )
% 9.00/5.10 => ( ( groups3542108847815614940at_nat @ G3 @ A3 )
% 9.00/5.10 = zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8026_sum_Oneutral,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > complex] :
% 9.00/5.10 ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ( G3 @ X5 )
% 9.00/5.10 = zero_zero_complex ) )
% 9.00/5.10 => ( ( groups7754918857620584856omplex @ G3 @ A3 )
% 9.00/5.10 = zero_zero_complex ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8027_sum_Oneutral,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > int] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ( G3 @ X5 )
% 9.00/5.10 = zero_zero_int ) )
% 9.00/5.10 => ( ( groups4538972089207619220nt_int @ G3 @ A3 )
% 9.00/5.10 = zero_zero_int ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8028_sum_Oneutral,axiom,
% 9.00/5.10 ! [A3: set_nat,G3: nat > real] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ( G3 @ X5 )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ G3 @ A3 )
% 9.00/5.10 = zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8029_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: real > real,A3: set_real] :
% 9.00/5.10 ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 9.00/5.10 != zero_zero_real )
% 9.00/5.10 => ~ ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8030_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: $o > real,A3: set_o] :
% 9.00/5.10 ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 9.00/5.10 != zero_zero_real )
% 9.00/5.10 => ~ ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8031_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: int > real,A3: set_int] :
% 9.00/5.10 ( ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 9.00/5.10 != zero_zero_real )
% 9.00/5.10 => ~ ! [A: int] :
% 9.00/5.10 ( ( member_int @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8032_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: real > rat,A3: set_real] :
% 9.00/5.10 ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_rat )
% 9.00/5.10 => ~ ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8033_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: $o > rat,A3: set_o] :
% 9.00/5.10 ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_rat )
% 9.00/5.10 => ~ ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8034_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: nat > rat,A3: set_nat] :
% 9.00/5.10 ( ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_rat )
% 9.00/5.10 => ~ ! [A: nat] :
% 9.00/5.10 ( ( member_nat @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8035_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: int > rat,A3: set_int] :
% 9.00/5.10 ( ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_rat )
% 9.00/5.10 => ~ ! [A: int] :
% 9.00/5.10 ( ( member_int @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_rat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8036_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: real > nat,A3: set_real] :
% 9.00/5.10 ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_nat )
% 9.00/5.10 => ~ ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8037_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: $o > nat,A3: set_o] :
% 9.00/5.10 ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_nat )
% 9.00/5.10 => ~ ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8038_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.00/5.10 ! [G3: int > nat,A3: set_int] :
% 9.00/5.10 ( ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 9.00/5.10 != zero_zero_nat )
% 9.00/5.10 => ~ ! [A: int] :
% 9.00/5.10 ( ( member_int @ A @ A3 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_nat ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8039_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_real,F: real > complex,G3: real > real] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( groups8097168146408367636l_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8040_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_o,F: $o > complex,G3: $o > real] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( groups8691415230153176458o_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8041_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_set_nat,F: set_nat > complex,G3: set_nat > real] :
% 9.00/5.10 ( ! [X5: set_nat] :
% 9.00/5.10 ( ( member_set_nat @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( groups5107569545109728110t_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8042_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_int,F: int > complex,G3: int > real] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8043_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_nat,F: nat > complex,G3: nat > real] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8044_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_complex,F: complex > complex,G3: complex > real] :
% 9.00/5.10 ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8045_sum__norm__le,axiom,
% 9.00/5.10 ! [S: set_nat,F: nat > real,G3: nat > real] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ S )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( groups6591440286371151544t_real @ G3 @ S ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8046_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_real,F: real > rat,G3: real > rat] :
% 9.00/5.10 ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K4 ) @ ( groups1300246762558778688al_rat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8047_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_o,F: $o > rat,G3: $o > rat] :
% 9.00/5.10 ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K4 ) @ ( groups7872700643590313910_o_rat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8048_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_nat,F: nat > rat,G3: nat > rat] :
% 9.00/5.10 ( ! [I2: nat] :
% 9.00/5.10 ( ( member_nat @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K4 ) @ ( groups2906978787729119204at_rat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8049_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_int,F: int > rat,G3: int > rat] :
% 9.00/5.10 ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K4 ) @ ( groups3906332499630173760nt_rat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8050_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_real,F: real > nat,G3: real > nat] :
% 9.00/5.10 ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K4 ) @ ( groups1935376822645274424al_nat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8051_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_o,F: $o > nat,G3: $o > nat] :
% 9.00/5.10 ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K4 ) @ ( groups8507830703676809646_o_nat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8052_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_int,F: int > nat,G3: int > nat] :
% 9.00/5.10 ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K4 ) @ ( groups4541462559716669496nt_nat @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8053_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_real,F: real > int,G3: real > int] :
% 9.00/5.10 ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K4 ) @ ( groups1932886352136224148al_int @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8054_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_o,F: $o > int,G3: $o > int] :
% 9.00/5.10 ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_int @ ( groups8505340233167759370_o_int @ F @ K4 ) @ ( groups8505340233167759370_o_int @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8055_sum__mono,axiom,
% 9.00/5.10 ! [K4: set_nat,F: nat > int,G3: nat > int] :
% 9.00/5.10 ( ! [I2: nat] :
% 9.00/5.10 ( ( member_nat @ I2 @ K4 )
% 9.00/5.10 => ( ord_less_eq_int @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K4 ) @ ( groups3539618377306564664at_int @ G3 @ K4 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8056_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_o,B2: set_nat,G3: $o > nat > nat,R: $o > nat > $o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( groups8507830703676809646_o_nat
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( ( member_nat @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( groups8507830703676809646_o_nat
% 9.00/5.10 @ ^ [X: $o] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8057_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_real,B2: set_nat,G3: real > nat > nat,R: real > nat > $o] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( groups1935376822645274424al_nat
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( ( member_nat @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( groups1935376822645274424al_nat
% 9.00/5.10 @ ^ [X: real] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8058_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_int,B2: set_nat,G3: int > nat > nat,R: int > nat > $o] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( groups4541462559716669496nt_nat
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( ( member_nat @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( groups4541462559716669496nt_nat
% 9.00/5.10 @ ^ [X: int] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8059_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_complex,B2: set_nat,G3: complex > nat > nat,R: complex > nat > $o] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( groups5693394587270226106ex_nat
% 9.00/5.10 @ ^ [X: complex] :
% 9.00/5.10 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( ( member_nat @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( groups5693394587270226106ex_nat
% 9.00/5.10 @ ^ [X: complex] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8060_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,B2: set_nat,G3: extended_enat > nat > nat,R: extended_enat > nat > $o] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( finite_finite_nat @ B2 )
% 9.00/5.10 => ( ( groups2027974829824023292at_nat
% 9.00/5.10 @ ^ [X: extended_enat] :
% 9.00/5.10 ( groups3542108847815614940at_nat @ ( G3 @ X )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( ( member_nat @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat
% 9.00/5.10 @ ^ [Y: nat] :
% 9.00/5.10 ( groups2027974829824023292at_nat
% 9.00/5.10 @ ^ [X: extended_enat] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [X: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8061_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_o,B2: set_complex,G3: $o > complex > complex,R: $o > complex > $o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.10 => ( ( groups5328290441151304332omplex
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( groups7754918857620584856omplex @ ( G3 @ X )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( ( member_complex @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( groups5328290441151304332omplex
% 9.00/5.10 @ ^ [X: $o] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8062_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_real,B2: set_complex,G3: real > complex > complex,R: real > complex > $o] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.10 => ( ( groups5754745047067104278omplex
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( groups7754918857620584856omplex @ ( G3 @ X )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( ( member_complex @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( groups5754745047067104278omplex
% 9.00/5.10 @ ^ [X: real] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8063_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_nat,B2: set_complex,G3: nat > complex > complex,R: nat > complex > $o] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.10 => ( ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( groups7754918857620584856omplex @ ( G3 @ X )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( ( member_complex @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( groups2073611262835488442omplex
% 9.00/5.10 @ ^ [X: nat] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8064_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_int,B2: set_complex,G3: int > complex > complex,R: int > complex > $o] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.10 => ( ( groups3049146728041665814omplex
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( groups7754918857620584856omplex @ ( G3 @ X )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( ( member_complex @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( groups3049146728041665814omplex
% 9.00/5.10 @ ^ [X: int] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8065_sum_Oswap__restrict,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,B2: set_complex,G3: extended_enat > complex > complex,R: extended_enat > complex > $o] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.10 => ( ( groups6818542070133387226omplex
% 9.00/5.10 @ ^ [X: extended_enat] :
% 9.00/5.10 ( groups7754918857620584856omplex @ ( G3 @ X )
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( ( member_complex @ Y @ B2 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ A3 )
% 9.00/5.10 = ( groups7754918857620584856omplex
% 9.00/5.10 @ ^ [Y: complex] :
% 9.00/5.10 ( groups6818542070133387226omplex
% 9.00/5.10 @ ^ [X: extended_enat] : ( G3 @ X @ Y )
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [X: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.10 & ( R @ X @ Y ) ) ) )
% 9.00/5.10 @ B2 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8066_sums__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( sums_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.00/5.10 @ ( groups3539618377306564664at_int @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8067_sums__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( sums_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.00/5.10 @ ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8068_sums__If__finite__set,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( sums_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A3 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.00/5.10 @ ( groups6591440286371151544t_real @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8069_sums__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( sums_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.00/5.10 @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8070_sums__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( sums_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.00/5.10 @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8071_sums__If__finite,axiom,
% 9.00/5.10 ! [P: nat > $o,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.00/5.10 => ( sums_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.00/5.10 @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8072_sums__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > int] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_int ) )
% 9.00/5.10 => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8073_sums__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > nat] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_nat ) )
% 9.00/5.10 => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8074_sums__finite,axiom,
% 9.00/5.10 ! [N6: set_nat,F: nat > real] :
% 9.00/5.10 ( ( finite_finite_nat @ N6 )
% 9.00/5.10 => ( ! [N3: nat] :
% 9.00/5.10 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.10 => ( ( F @ N3 )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8075_norm__sum,axiom,
% 9.00/5.10 ! [F: nat > complex,A3: set_nat] :
% 9.00/5.10 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A3 ) )
% 9.00/5.10 @ ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8076_norm__sum,axiom,
% 9.00/5.10 ! [F: complex > complex,A3: set_complex] :
% 9.00/5.10 ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A3 ) )
% 9.00/5.10 @ ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [I4: complex] : ( real_V1022390504157884413omplex @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8077_norm__sum,axiom,
% 9.00/5.10 ! [F: nat > real,A3: set_nat] :
% 9.00/5.10 ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A3 ) )
% 9.00/5.10 @ ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [I4: nat] : ( real_V7735802525324610683m_real @ ( F @ I4 ) )
% 9.00/5.10 @ A3 ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8078_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > real] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8079_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > real] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8080_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > real] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ zero_zero_real ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8081_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > rat] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8082_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > rat] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8083_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > rat] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8084_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > rat] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ zero_zero_rat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8085_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > nat] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8086_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > nat] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8087_sum__nonpos,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > nat] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
% 9.00/5.10 => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ zero_zero_nat ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8088_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > real] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8089_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > real] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8090_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > real] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8091_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > rat] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8092_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > rat] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8093_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > rat] :
% 9.00/5.10 ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8094_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > rat] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8095_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > nat] :
% 9.00/5.10 ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8096_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > nat] :
% 9.00/5.10 ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8097_sum__nonneg,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > nat] :
% 9.00/5.10 ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8098_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: real > rat,I5: set_real,G3: real > rat,I: real] :
% 9.00/5.10 ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 9.00/5.10 = ( groups1300246762558778688al_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_real @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_real @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8099_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: $o > rat,I5: set_o,G3: $o > rat,I: $o] :
% 9.00/5.10 ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
% 9.00/5.10 = ( groups7872700643590313910_o_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_o @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_o @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8100_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: nat > rat,I5: set_nat,G3: nat > rat,I: nat] :
% 9.00/5.10 ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 9.00/5.10 = ( groups2906978787729119204at_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: nat] :
% 9.00/5.10 ( ( member_nat @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_nat @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_nat @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8101_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: int > rat,I5: set_int,G3: int > rat,I: int] :
% 9.00/5.10 ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 9.00/5.10 = ( groups3906332499630173760nt_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_int @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_int @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8102_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: complex > rat,I5: set_complex,G3: complex > rat,I: complex] :
% 9.00/5.10 ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 9.00/5.10 = ( groups5058264527183730370ex_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_complex @ I @ I5 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8103_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: extended_enat > rat,I5: set_Extended_enat,G3: extended_enat > rat,I: extended_enat] :
% 9.00/5.10 ( ( ( groups1392844769737527556at_rat @ F @ I5 )
% 9.00/5.10 = ( groups1392844769737527556at_rat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_Extended_enat @ I @ I5 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8104_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: real > nat,I5: set_real,G3: real > nat,I: real] :
% 9.00/5.10 ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 9.00/5.10 = ( groups1935376822645274424al_nat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_real @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_real @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8105_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: $o > nat,I5: set_o,G3: $o > nat,I: $o] :
% 9.00/5.10 ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
% 9.00/5.10 = ( groups8507830703676809646_o_nat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_o @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_o @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8106_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: int > nat,I5: set_int,G3: int > nat,I: int] :
% 9.00/5.10 ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 9.00/5.10 = ( groups4541462559716669496nt_nat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_int @ I @ I5 )
% 9.00/5.10 => ( ( finite_finite_int @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8107_sum__mono__inv,axiom,
% 9.00/5.10 ! [F: complex > nat,I5: set_complex,G3: complex > nat,I: complex] :
% 9.00/5.10 ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 9.00/5.10 = ( groups5693394587270226106ex_nat @ G3 @ I5 ) )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ I5 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ I2 ) @ ( G3 @ I2 ) ) )
% 9.00/5.10 => ( ( member_complex @ I @ I5 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = ( G3 @ I ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8108_sum__cong__Suc,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > nat,G3: nat > nat] :
% 9.00/5.10 ( ~ ( member_nat @ zero_zero_nat @ A3 )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ ( suc @ X5 ) @ A3 )
% 9.00/5.10 => ( ( F @ ( suc @ X5 ) )
% 9.00/5.10 = ( G3 @ ( suc @ X5 ) ) ) )
% 9.00/5.10 => ( ( groups3542108847815614940at_nat @ F @ A3 )
% 9.00/5.10 = ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8109_sum__cong__Suc,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > real,G3: nat > real] :
% 9.00/5.10 ( ~ ( member_nat @ zero_zero_nat @ A3 )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ ( suc @ X5 ) @ A3 )
% 9.00/5.10 => ( ( F @ ( suc @ X5 ) )
% 9.00/5.10 = ( G3 @ ( suc @ X5 ) ) ) )
% 9.00/5.10 => ( ( groups6591440286371151544t_real @ F @ A3 )
% 9.00/5.10 = ( groups6591440286371151544t_real @ G3 @ A3 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8110_sums__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > real] :
% 9.00/5.10 ( sums_real
% 9.00/5.10 @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 9.00/5.10 @ ( F @ I ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8111_sums__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > nat] :
% 9.00/5.10 ( sums_nat
% 9.00/5.10 @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.00/5.10 @ ( F @ I ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8112_sums__single,axiom,
% 9.00/5.10 ! [I: nat,F: nat > int] :
% 9.00/5.10 ( sums_int
% 9.00/5.10 @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 9.00/5.10 @ ( F @ I ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8113_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_o,G3: $o > real,P: $o > $o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [X: $o] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8114_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_real,G3: real > real,P: real > $o] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8115_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > real,P: int > $o] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8116_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > real,P: complex > $o] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8117_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > real,P: extended_enat > $o] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3
% 9.00/5.10 @ ( collec4429806609662206161d_enat
% 9.00/5.10 @ ^ [X: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8118_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_o,G3: $o > rat,P: $o > $o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat @ G3
% 9.00/5.10 @ ( collect_o
% 9.00/5.10 @ ^ [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [X: $o] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8119_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_real,G3: real > rat,P: real > $o] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat @ G3
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8120_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_nat,G3: nat > rat,P: nat > $o] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3
% 9.00/5.10 @ ( collect_nat
% 9.00/5.10 @ ^ [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups2906978787729119204at_rat
% 9.00/5.10 @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8121_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > rat,P: int > $o] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8122_sum_Ointer__filter,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > rat,P: complex > $o] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ A3 )
% 9.00/5.10 & ( P @ X ) ) ) )
% 9.00/5.10 = ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8123_sums__If__finite__set_H,axiom,
% 9.00/5.10 ! [G3: nat > real,S: real,A3: set_nat,S5: real,F: nat > real] :
% 9.00/5.10 ( ( sums_real @ G3 @ S )
% 9.00/5.10 => ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( S5
% 9.00/5.10 = ( plus_plus_real @ S
% 9.00/5.10 @ ( groups6591440286371151544t_real
% 9.00/5.10 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 9.00/5.10 @ A3 ) ) )
% 9.00/5.10 => ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : ( if_real @ ( member_nat @ N2 @ A3 ) @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 9.00/5.10 @ S5 ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8124_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_int,T: set_int,G3: int > real,I: int > int,F: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ( finite_finite_int @ T )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: int] :
% 9.00/5.10 ( ( member_int @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8125_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_int,T: set_complex,G3: complex > real,I: complex > int,F: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ T )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: complex] :
% 9.00/5.10 ( ( member_complex @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8126_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_int,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > int,F: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ T )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8127_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_complex,T: set_int,G3: int > real,I: int > complex,F: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ( finite_finite_int @ T )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: int] :
% 9.00/5.10 ( ( member_int @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8128_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_complex,T: set_complex,G3: complex > real,I: complex > complex,F: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ T )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: complex] :
% 9.00/5.10 ( ( member_complex @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8129_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_complex,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > complex,F: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ T )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8130_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_Extended_enat,T: set_int,G3: int > real,I: int > extended_enat,F: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S2 )
% 9.00/5.10 => ( ( finite_finite_int @ T )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: int] :
% 9.00/5.10 ( ( member_int @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups8778361861064173332t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8131_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_Extended_enat,T: set_complex,G3: complex > real,I: complex > extended_enat,F: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S2 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ T )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: complex] :
% 9.00/5.10 ( ( member_complex @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups5808333547571424918x_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8132_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_Extended_enat,T: set_Extended_enat,G3: extended_enat > real,I: extended_enat > extended_enat,F: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S2 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ T )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ S2 ) @ ( groups4148127829035722712t_real @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8133_sum__le__included,axiom,
% 9.00/5.10 ! [S2: set_nat,T: set_nat,G3: nat > rat,I: nat > nat,F: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ S2 )
% 9.00/5.10 => ( ( finite_finite_nat @ T )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ T )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ S2 )
% 9.00/5.10 => ? [Xa: nat] :
% 9.00/5.10 ( ( member_nat @ Xa @ T )
% 9.00/5.10 & ( ( I @ Xa )
% 9.00/5.10 = X5 )
% 9.00/5.10 & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ Xa ) ) ) )
% 9.00/5.10 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S2 ) @ ( groups2906978787729119204at_rat @ G3 @ T ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8134_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups8097168146408367636l_real @ F @ A3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8135_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups8691415230153176458o_real @ F @ A3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8136_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups8778361861064173332t_real @ F @ A3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8137_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups5808333547571424918x_real @ F @ A3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8138_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups4148127829035722712t_real @ F @ A3 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 = ( ! [X: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8139_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ A3 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 = ( ! [X: real] :
% 9.00/5.10 ( ( member_real @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8140_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups7872700643590313910_o_rat @ F @ A3 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 = ( ! [X: $o] :
% 9.00/5.10 ( ( member_o @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8141_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ A3 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 = ( ! [X: nat] :
% 9.00/5.10 ( ( member_nat @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8142_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ A3 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 = ( ! [X: int] :
% 9.00/5.10 ( ( member_int @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8143_sum__nonneg__eq__0__iff,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ A3 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 = ( ! [X: complex] :
% 9.00/5.10 ( ( member_complex @ X @ A3 )
% 9.00/5.10 => ( ( F @ X )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8144_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > real,G3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: int] :
% 9.00/5.10 ( ( member_int @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8145_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > real,G3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: complex] :
% 9.00/5.10 ( ( member_complex @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8146_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_real @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8147_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: nat] :
% 9.00/5.10 ( ( member_nat @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8148_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > rat,G3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: int] :
% 9.00/5.10 ( ( member_int @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8149_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: complex] :
% 9.00/5.10 ( ( member_complex @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8150_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_rat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8151_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > nat,G3: int > nat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: int] :
% 9.00/5.10 ( ( member_int @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8152_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > nat,G3: complex > nat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: complex] :
% 9.00/5.10 ( ( member_complex @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8153_sum__strict__mono__ex1,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > nat,G3: extended_enat > nat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ? [X4: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X4 @ A3 )
% 9.00/5.10 & ( ord_less_nat @ ( F @ X4 ) @ ( G3 @ X4 ) ) )
% 9.00/5.10 => ( ord_less_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8154_sum_Orelated,axiom,
% 9.00/5.10 ! [R: real > real > $o,S: set_int,H2: int > real,G3: int > real] :
% 9.00/5.10 ( ( R @ zero_zero_real @ zero_zero_real )
% 9.00/5.10 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups8778361861064173332t_real @ H2 @ S ) @ ( groups8778361861064173332t_real @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8155_sum_Orelated,axiom,
% 9.00/5.10 ! [R: real > real > $o,S: set_complex,H2: complex > real,G3: complex > real] :
% 9.00/5.10 ( ( R @ zero_zero_real @ zero_zero_real )
% 9.00/5.10 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups5808333547571424918x_real @ H2 @ S ) @ ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8156_sum_Orelated,axiom,
% 9.00/5.10 ! [R: real > real > $o,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.10 ( ( R @ zero_zero_real @ zero_zero_real )
% 9.00/5.10 => ( ! [X1: real,Y1: real,X23: real,Y22: real] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_real @ X1 @ Y1 ) @ ( plus_plus_real @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups4148127829035722712t_real @ H2 @ S ) @ ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8157_sum_Orelated,axiom,
% 9.00/5.10 ! [R: rat > rat > $o,S: set_nat,H2: nat > rat,G3: nat > rat] :
% 9.00/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.00/5.10 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite_finite_nat @ S )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups2906978787729119204at_rat @ H2 @ S ) @ ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8158_sum_Orelated,axiom,
% 9.00/5.10 ! [R: rat > rat > $o,S: set_int,H2: int > rat,G3: int > rat] :
% 9.00/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.00/5.10 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S ) @ ( groups3906332499630173760nt_rat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8159_sum_Orelated,axiom,
% 9.00/5.10 ! [R: rat > rat > $o,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 9.00/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.00/5.10 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S ) @ ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8160_sum_Orelated,axiom,
% 9.00/5.10 ! [R: rat > rat > $o,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.10 ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.00/5.10 => ( ! [X1: rat,Y1: rat,X23: rat,Y22: rat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_rat @ X1 @ Y1 ) @ ( plus_plus_rat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups1392844769737527556at_rat @ H2 @ S ) @ ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8161_sum_Orelated,axiom,
% 9.00/5.10 ! [R: nat > nat > $o,S: set_int,H2: int > nat,G3: int > nat] :
% 9.00/5.10 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.00/5.10 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite_finite_int @ S )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S ) @ ( groups4541462559716669496nt_nat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8162_sum_Orelated,axiom,
% 9.00/5.10 ! [R: nat > nat > $o,S: set_complex,H2: complex > nat,G3: complex > nat] :
% 9.00/5.10 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.00/5.10 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ S )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S ) @ ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8163_sum_Orelated,axiom,
% 9.00/5.10 ! [R: nat > nat > $o,S: set_Extended_enat,H2: extended_enat > nat,G3: extended_enat > nat] :
% 9.00/5.10 ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.00/5.10 => ( ! [X1: nat,Y1: nat,X23: nat,Y22: nat] :
% 9.00/5.10 ( ( ( R @ X1 @ X23 )
% 9.00/5.10 & ( R @ Y1 @ Y22 ) )
% 9.00/5.10 => ( R @ ( plus_plus_nat @ X1 @ Y1 ) @ ( plus_plus_nat @ X23 @ Y22 ) ) )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.10 => ( R @ ( H2 @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( R @ ( groups2027974829824023292at_nat @ H2 @ S ) @ ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8164_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > real,G3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_complex )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8165_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8166_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > real,G3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_real )
% 9.00/5.10 => ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8167_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > real,G3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_o )
% 9.00/5.10 => ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8168_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_int,F: int > real,G3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_int )
% 9.00/5.10 => ( ! [X5: int] :
% 9.00/5.10 ( ( member_int @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_real @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8169_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_complex,F: complex > rat,G3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_complex )
% 9.00/5.10 => ( ! [X5: complex] :
% 9.00/5.10 ( ( member_complex @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8170_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,F: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.10 => ( ! [X5: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8171_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_real,F: real > rat,G3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_real )
% 9.00/5.10 => ( ! [X5: real] :
% 9.00/5.10 ( ( member_real @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8172_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_o,F: $o > rat,G3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_o )
% 9.00/5.10 => ( ! [X5: $o] :
% 9.00/5.10 ( ( member_o @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8173_sum__strict__mono,axiom,
% 9.00/5.10 ! [A3: set_nat,F: nat > rat,G3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( A3 != bot_bot_set_nat )
% 9.00/5.10 => ( ! [X5: nat] :
% 9.00/5.10 ( ( member_nat @ X5 @ A3 )
% 9.00/5.10 => ( ord_less_rat @ ( F @ X5 ) @ ( G3 @ X5 ) ) )
% 9.00/5.10 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8174_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_real,X2: real,G3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( groups8097168146408367636l_real @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8175_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_o,X2: $o,G3: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( groups8691415230153176458o_real @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8691415230153176458o_real @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8176_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_int,X2: int,G3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( groups8778361861064173332t_real @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8177_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( groups5808333547571424918x_real @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8178_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.10 = ( groups4148127829035722712t_real @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8179_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_real,X2: real,G3: real > rat] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_real @ X2 @ A3 )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8180_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_o,X2: $o,G3: $o > rat] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_o @ X2 @ A3 )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat @ G3 @ ( insert_o @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups7872700643590313910_o_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8181_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_nat,X2: nat,G3: nat > rat] :
% 9.00/5.10 ( ( finite_finite_nat @ A3 )
% 9.00/5.10 => ( ( ( member_nat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.10 = ( groups2906978787729119204at_rat @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_nat @ X2 @ A3 )
% 9.00/5.10 => ( ( groups2906978787729119204at_rat @ G3 @ ( insert_nat @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups2906978787729119204at_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8182_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_int,X2: int,G3: int > rat] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_int @ X2 @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3 @ ( insert_int @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8183_sum_Oinsert__if,axiom,
% 9.00/5.10 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) )
% 9.00/5.10 & ( ~ ( member_complex @ X2 @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.10 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8184_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_real,T5: set_real,S: set_real,I: real > real,J: real > real,T2: set_real,G3: real > real,H2: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ S5 )
% 9.00/5.10 => ( ( finite_finite_real @ T5 )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 9.00/5.10 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.10 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8185_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_real,T5: set_o,S: set_real,I: $o > real,J: real > $o,T2: set_o,G3: real > real,H2: $o > real] :
% 9.00/5.10 ( ( finite_finite_real @ S5 )
% 9.00/5.10 => ( ( finite_finite_o @ T5 )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 9.00/5.10 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.10 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8186_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_o,T5: set_real,S: set_o,I: real > $o,J: $o > real,T2: set_real,G3: $o > real,H2: real > real] :
% 9.00/5.10 ( ( finite_finite_o @ S5 )
% 9.00/5.10 => ( ( finite_finite_real @ T5 )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( member_real @ ( J @ A ) @ ( minus_minus_set_real @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ ( minus_minus_set_real @ T2 @ T5 ) )
% 9.00/5.10 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: real] :
% 9.00/5.10 ( ( member_real @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.10 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8187_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_o,T5: set_o,S: set_o,I: $o > $o,J: $o > $o,T2: set_o,G3: $o > real,H2: $o > real] :
% 9.00/5.10 ( ( finite_finite_o @ S5 )
% 9.00/5.10 => ( ( finite_finite_o @ T5 )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( member_o @ ( J @ A ) @ ( minus_minus_set_o @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ ( minus_minus_set_o @ T2 @ T5 ) )
% 9.00/5.10 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: $o] :
% 9.00/5.10 ( ( member_o @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.10 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8188_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_real,T5: set_int,S: set_real,I: int > real,J: real > int,T2: set_int,G3: real > real,H2: int > real] :
% 9.00/5.10 ( ( finite_finite_real @ S5 )
% 9.00/5.10 => ( ( finite_finite_int @ T5 )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 9.00/5.10 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.10 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8189_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_o,T5: set_int,S: set_o,I: int > $o,J: $o > int,T2: set_int,G3: $o > real,H2: int > real] :
% 9.00/5.10 ( ( finite_finite_o @ S5 )
% 9.00/5.10 => ( ( finite_finite_int @ T5 )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( member_int @ ( J @ A ) @ ( minus_minus_set_int @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ ( minus_minus_set_int @ T2 @ T5 ) )
% 9.00/5.10 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: int] :
% 9.00/5.10 ( ( member_int @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.10 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8190_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_real,T5: set_complex,S: set_real,I: complex > real,J: real > complex,T2: set_complex,G3: real > real,H2: complex > real] :
% 9.00/5.10 ( ( finite_finite_real @ S5 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ T5 )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 9.00/5.10 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.10 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8191_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_o,T5: set_complex,S: set_o,I: complex > $o,J: $o > complex,T2: set_complex,G3: $o > real,H2: complex > real] :
% 9.00/5.10 ( ( finite_finite_o @ S5 )
% 9.00/5.10 => ( ( finite3207457112153483333omplex @ T5 )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( member_complex @ ( J @ A ) @ ( minus_811609699411566653omplex @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ T2 @ T5 ) )
% 9.00/5.10 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: complex] :
% 9.00/5.10 ( ( member_complex @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.10 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8192_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_real,T5: set_Extended_enat,S: set_real,I: extended_enat > real,J: real > extended_enat,T2: set_Extended_enat,G3: real > real,H2: extended_enat > real] :
% 9.00/5.10 ( ( finite_finite_real @ S5 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ T5 )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ ( minus_minus_set_real @ S @ S5 ) )
% 9.00/5.10 => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 9.00/5.10 => ( member_real @ ( I @ B ) @ ( minus_minus_set_real @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: real] :
% 9.00/5.10 ( ( member_real @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.10 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8193_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.00/5.10 ! [S5: set_o,T5: set_Extended_enat,S: set_o,I: extended_enat > $o,J: $o > extended_enat,T2: set_Extended_enat,G3: $o > real,H2: extended_enat > real] :
% 9.00/5.10 ( ( finite_finite_o @ S5 )
% 9.00/5.10 => ( ( finite4001608067531595151d_enat @ T5 )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( ( I @ ( J @ A ) )
% 9.00/5.10 = A ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ ( minus_minus_set_o @ S @ S5 ) )
% 9.00/5.10 => ( member_Extended_enat @ ( J @ A ) @ ( minus_925952699566721837d_enat @ T2 @ T5 ) ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 9.00/5.10 => ( ( J @ ( I @ B ) )
% 9.00/5.10 = B ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ T2 @ T5 ) )
% 9.00/5.10 => ( member_o @ ( I @ B ) @ ( minus_minus_set_o @ S @ S5 ) ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S5 )
% 9.00/5.10 => ( ( G3 @ A )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [B: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ B @ T5 )
% 9.00/5.10 => ( ( H2 @ B )
% 9.00/5.10 = zero_zero_real ) )
% 9.00/5.10 => ( ! [A: $o] :
% 9.00/5.10 ( ( member_o @ A @ S )
% 9.00/5.10 => ( ( H2 @ ( J @ A ) )
% 9.00/5.10 = ( G3 @ A ) ) )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.10 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8194_sums__mult2__iff,axiom,
% 9.00/5.10 ! [C: complex,F: nat > complex,D: complex] :
% 9.00/5.10 ( ( C != zero_zero_complex )
% 9.00/5.10 => ( ( sums_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ ( F @ N2 ) @ C )
% 9.00/5.10 @ ( times_times_complex @ D @ C ) )
% 9.00/5.10 = ( sums_complex @ F @ D ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8195_sums__mult2__iff,axiom,
% 9.00/5.10 ! [C: real,F: nat > real,D: real] :
% 9.00/5.10 ( ( C != zero_zero_real )
% 9.00/5.10 => ( ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ C )
% 9.00/5.10 @ ( times_times_real @ D @ C ) )
% 9.00/5.10 = ( sums_real @ F @ D ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8196_sums__mult__iff,axiom,
% 9.00/5.10 ! [C: complex,F: nat > complex,D: complex] :
% 9.00/5.10 ( ( C != zero_zero_complex )
% 9.00/5.10 => ( ( sums_complex
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.00/5.10 @ ( times_times_complex @ C @ D ) )
% 9.00/5.10 = ( sums_complex @ F @ D ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8197_sums__mult__iff,axiom,
% 9.00/5.10 ! [C: real,F: nat > real,D: real] :
% 9.00/5.10 ( ( C != zero_zero_real )
% 9.00/5.10 => ( ( sums_real
% 9.00/5.10 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.00/5.10 @ ( times_times_real @ C @ D ) )
% 9.00/5.10 = ( sums_real @ F @ D ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8198_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_real,F: real > real,B2: real,I: real] :
% 9.00/5.10 ( ( finite_finite_real @ S2 )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8097168146408367636l_real @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_real @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8199_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_o,F: $o > real,B2: real,I: $o] :
% 9.00/5.10 ( ( finite_finite_o @ S2 )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8691415230153176458o_real @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_o @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8200_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_int,F: int > real,B2: real,I: int] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8778361861064173332t_real @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_int @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8201_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_complex,F: complex > real,B2: real,I: complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups5808333547571424918x_real @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_complex @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8202_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_Extended_enat,F: extended_enat > real,B2: real,I: extended_enat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S2 )
% 9.00/5.10 => ( ! [I2: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups4148127829035722712t_real @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_Extended_enat @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8203_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_real,F: real > rat,B2: rat,I: real] :
% 9.00/5.10 ( ( finite_finite_real @ S2 )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_real @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8204_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_o,F: $o > rat,B2: rat,I: $o] :
% 9.00/5.10 ( ( finite_finite_o @ S2 )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_o @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8205_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_nat,F: nat > rat,B2: rat,I: nat] :
% 9.00/5.10 ( ( finite_finite_nat @ S2 )
% 9.00/5.10 => ( ! [I2: nat] :
% 9.00/5.10 ( ( member_nat @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_nat @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8206_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_int,F: int > rat,B2: rat,I: int] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_int @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8207_sum__nonneg__leq__bound,axiom,
% 9.00/5.10 ! [S2: set_complex,F: complex > rat,B2: rat,I: complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
% 9.00/5.10 = B2 )
% 9.00/5.10 => ( ( member_complex @ I @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8208_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_real,F: real > real,I: real] :
% 9.00/5.10 ( ( finite_finite_real @ S2 )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8097168146408367636l_real @ F @ S2 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( ( member_real @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8209_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_o,F: $o > real,I: $o] :
% 9.00/5.10 ( ( finite_finite_o @ S2 )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8691415230153176458o_real @ F @ S2 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( ( member_o @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8210_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_int,F: int > real,I: int] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups8778361861064173332t_real @ F @ S2 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( ( member_int @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8211_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_complex,F: complex > real,I: complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups5808333547571424918x_real @ F @ S2 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( ( member_complex @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8212_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_Extended_enat,F: extended_enat > real,I: extended_enat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ S2 )
% 9.00/5.10 => ( ! [I2: extended_enat] :
% 9.00/5.10 ( ( member_Extended_enat @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups4148127829035722712t_real @ F @ S2 )
% 9.00/5.10 = zero_zero_real )
% 9.00/5.10 => ( ( member_Extended_enat @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8213_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_real,F: real > rat,I: real] :
% 9.00/5.10 ( ( finite_finite_real @ S2 )
% 9.00/5.10 => ( ! [I2: real] :
% 9.00/5.10 ( ( member_real @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups1300246762558778688al_rat @ F @ S2 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 => ( ( member_real @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8214_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_o,F: $o > rat,I: $o] :
% 9.00/5.10 ( ( finite_finite_o @ S2 )
% 9.00/5.10 => ( ! [I2: $o] :
% 9.00/5.10 ( ( member_o @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups7872700643590313910_o_rat @ F @ S2 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 => ( ( member_o @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8215_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_nat,F: nat > rat,I: nat] :
% 9.00/5.10 ( ( finite_finite_nat @ S2 )
% 9.00/5.10 => ( ! [I2: nat] :
% 9.00/5.10 ( ( member_nat @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups2906978787729119204at_rat @ F @ S2 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 => ( ( member_nat @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8216_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_int,F: int > rat,I: int] :
% 9.00/5.10 ( ( finite_finite_int @ S2 )
% 9.00/5.10 => ( ! [I2: int] :
% 9.00/5.10 ( ( member_int @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups3906332499630173760nt_rat @ F @ S2 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 => ( ( member_int @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8217_sum__nonneg__0,axiom,
% 9.00/5.10 ! [S2: set_complex,F: complex > rat,I: complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ S2 )
% 9.00/5.10 => ( ! [I2: complex] :
% 9.00/5.10 ( ( member_complex @ I2 @ S2 )
% 9.00/5.10 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.10 => ( ( ( groups5058264527183730370ex_rat @ F @ S2 )
% 9.00/5.10 = zero_zero_rat )
% 9.00/5.10 => ( ( member_complex @ I @ S2 )
% 9.00/5.10 => ( ( F @ I )
% 9.00/5.10 = zero_zero_rat ) ) ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8218_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_real,G3: real > real,B2: set_real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.10 = ( groups8097168146408367636l_real
% 9.00/5.10 @ ^ [X: real] : ( if_real @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8219_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_o,G3: $o > real,B2: set_o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( groups8691415230153176458o_real @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.10 = ( groups8691415230153176458o_real
% 9.00/5.10 @ ^ [X: $o] : ( if_real @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8220_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > real,B2: set_int] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.10 = ( groups8778361861064173332t_real
% 9.00/5.10 @ ^ [X: int] : ( if_real @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8221_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > real,B2: set_complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.10 = ( groups5808333547571424918x_real
% 9.00/5.10 @ ^ [X: complex] : ( if_real @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8222_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.10 = ( groups4148127829035722712t_real
% 9.00/5.10 @ ^ [X: extended_enat] : ( if_real @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_real )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8223_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_real,G3: real > rat,B2: set_real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ B2 ) )
% 9.00/5.10 = ( groups1300246762558778688al_rat
% 9.00/5.10 @ ^ [X: real] : ( if_rat @ ( member_real @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8224_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_o,G3: $o > rat,B2: set_o] :
% 9.00/5.10 ( ( finite_finite_o @ A3 )
% 9.00/5.10 => ( ( groups7872700643590313910_o_rat @ G3 @ ( inf_inf_set_o @ A3 @ B2 ) )
% 9.00/5.10 = ( groups7872700643590313910_o_rat
% 9.00/5.10 @ ^ [X: $o] : ( if_rat @ ( member_o @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8225_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > rat,B2: set_int] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.10 = ( groups3906332499630173760nt_rat
% 9.00/5.10 @ ^ [X: int] : ( if_rat @ ( member_int @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8226_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.10 = ( groups5058264527183730370ex_rat
% 9.00/5.10 @ ^ [X: complex] : ( if_rat @ ( member_complex @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8227_sum_Ointer__restrict,axiom,
% 9.00/5.10 ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
% 9.00/5.10 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.10 => ( ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.10 = ( groups1392844769737527556at_rat
% 9.00/5.10 @ ^ [X: extended_enat] : ( if_rat @ ( member_Extended_enat @ X @ B2 ) @ ( G3 @ X ) @ zero_zero_rat )
% 9.00/5.10 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8228_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.10 ! [A3: set_real,G3: real > real] :
% 9.00/5.10 ( ( finite_finite_real @ A3 )
% 9.00/5.10 => ( ( groups8097168146408367636l_real @ G3
% 9.00/5.10 @ ( minus_minus_set_real @ A3
% 9.00/5.10 @ ( collect_real
% 9.00/5.10 @ ^ [X: real] :
% 9.00/5.10 ( ( G3 @ X )
% 9.00/5.10 = zero_zero_real ) ) ) )
% 9.00/5.10 = ( groups8097168146408367636l_real @ G3 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8229_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.10 ! [A3: set_int,G3: int > real] :
% 9.00/5.10 ( ( finite_finite_int @ A3 )
% 9.00/5.10 => ( ( groups8778361861064173332t_real @ G3
% 9.00/5.10 @ ( minus_minus_set_int @ A3
% 9.00/5.10 @ ( collect_int
% 9.00/5.10 @ ^ [X: int] :
% 9.00/5.10 ( ( G3 @ X )
% 9.00/5.10 = zero_zero_real ) ) ) )
% 9.00/5.10 = ( groups8778361861064173332t_real @ G3 @ A3 ) ) ) ).
% 9.00/5.10
% 9.00/5.10 thf(fact_8230_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.10 ! [A3: set_complex,G3: complex > real] :
% 9.00/5.10 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.10 => ( ( groups5808333547571424918x_real @ G3
% 9.00/5.10 @ ( minus_811609699411566653omplex @ A3
% 9.00/5.10 @ ( collect_complex
% 9.00/5.10 @ ^ [X: complex] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_real ) ) ) )
% 9.00/5.11 = ( groups5808333547571424918x_real @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8231_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3
% 9.00/5.11 @ ( minus_925952699566721837d_enat @ A3
% 9.00/5.11 @ ( collec4429806609662206161d_enat
% 9.00/5.11 @ ^ [X: extended_enat] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_real ) ) ) )
% 9.00/5.11 = ( groups4148127829035722712t_real @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8232_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_real,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3
% 9.00/5.11 @ ( minus_minus_set_real @ A3
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [X: real] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_rat ) ) ) )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8233_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ G3
% 9.00/5.11 @ ( minus_minus_set_int @ A3
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [X: int] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_rat ) ) ) )
% 9.00/5.11 = ( groups3906332499630173760nt_rat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8234_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3
% 9.00/5.11 @ ( minus_811609699411566653omplex @ A3
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [X: complex] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_rat ) ) ) )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8235_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3
% 9.00/5.11 @ ( minus_925952699566721837d_enat @ A3
% 9.00/5.11 @ ( collec4429806609662206161d_enat
% 9.00/5.11 @ ^ [X: extended_enat] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_rat ) ) ) )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8236_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_real,G3: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ G3
% 9.00/5.11 @ ( minus_minus_set_real @ A3
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [X: real] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_nat ) ) ) )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8237_sum_Osetdiff__irrelevant,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ G3
% 9.00/5.11 @ ( minus_minus_set_int @ A3
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [X: int] :
% 9.00/5.11 ( ( G3 @ X )
% 9.00/5.11 = zero_zero_nat ) ) ) )
% 9.00/5.11 = ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8238_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 9.00/5.11 ! [G3: nat > nat,M: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8239_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 9.00/5.11 ! [G3: nat > real,M: nat,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N ) ) )
% 9.00/5.11 = ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8240_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.00/5.11 ! [G3: nat > nat,M: nat,K: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8241_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.00/5.11 ! [G3: nat > real,M: nat,K: nat,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) ) )
% 9.00/5.11 = ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( plus_plus_nat @ I4 @ K ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8242_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_real,I: real,F: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ I5 )
% 9.00/5.11 => ( ( member_real @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8243_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_o,I: $o,F: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ I5 )
% 9.00/5.11 => ( ( member_o @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8244_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_int,I: int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ I5 )
% 9.00/5.11 => ( ( member_int @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8245_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_complex,I: complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.11 => ( ( member_complex @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8246_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_Extended_enat,I: extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ I5 )
% 9.00/5.11 => ( ( member_Extended_enat @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8247_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_real,I: real,F: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ I5 )
% 9.00/5.11 => ( ( member_real @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8248_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_o,I: $o,F: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ I5 )
% 9.00/5.11 => ( ( member_o @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8249_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_nat,I: nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ I5 )
% 9.00/5.11 => ( ( member_nat @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8250_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_int,I: int,F: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ I5 )
% 9.00/5.11 => ( ( member_int @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8251_sum__pos2,axiom,
% 9.00/5.11 ! [I5: set_complex,I: complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.11 => ( ( member_complex @ I @ I5 )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8252_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_complex )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8253_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups4148127829035722712t_real @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8254_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_real,F: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_real )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8255_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_o,F: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_o )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8256_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_int )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8257_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_complex )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8258_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups1392844769737527556at_rat @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8259_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_real,F: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_real )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8260_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_o,F: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_o )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8261_sum__pos,axiom,
% 9.00/5.11 ! [I5: set_nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ I5 )
% 9.00/5.11 => ( ( I5 != bot_bot_set_nat )
% 9.00/5.11 => ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8262_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_real,K4: real,F: real > real] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8263_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_o,K4: real,F: $o > real] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8264_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_complex,K4: real,F: complex > real] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8265_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_int,K4: real,F: int > real] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8266_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_real,K4: rat,F: real > rat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8267_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_o,K4: rat,F: $o > rat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8268_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_complex,K4: rat,F: complex > rat] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8269_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_nat,K4: rat,F: nat > rat] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) @ ( groups2906978787729119204at_rat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8270_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_int,K4: rat,F: int > rat] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8271_sum__bounded__below,axiom,
% 9.00/5.11 ! [A3: set_real,K4: nat,F: real > nat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ K4 @ ( F @ I2 ) ) )
% 9.00/5.11 => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) @ ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8272_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > real,K4: real] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8273_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > real,K4: real] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8274_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > real,K4: real] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8275_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > real,K4: real] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8276_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8277_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8278_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8279_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8280_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8281_sum__bounded__above,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > nat,K4: nat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8282_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups8097168146408367636l_real @ G3 @ C2 )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8283_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups8691415230153176458o_real @ G3 @ C2 )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8284_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: complex] :
% 9.00/5.11 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups5808333547571424918x_real @ G3 @ C2 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8285_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups4148127829035722712t_real @ G3 @ C2 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8286_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups1300246762558778688al_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8287_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8288_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: complex] :
% 9.00/5.11 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8289_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups1392844769737527556at_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8290_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups1935376822645274424al_nat @ G3 @ C2 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8291_sum_Osame__carrier,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ B2 ) )
% 9.00/5.11 = ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ C2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8292_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > real,H2: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups8097168146408367636l_real @ G3 @ C2 )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8293_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > real,H2: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups8691415230153176458o_real @ G3 @ C2 )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ G3 @ A3 )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8294_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > real,H2: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: complex] :
% 9.00/5.11 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups5808333547571424918x_real @ G3 @ C2 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8295_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( ( groups4148127829035722712t_real @ G3 @ C2 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8296_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > rat,H2: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups1300246762558778688al_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8297_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > rat,H2: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups7872700643590313910_o_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8298_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_complex,A3: set_complex,B2: set_complex,G3: complex > rat,H2: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: complex] :
% 9.00/5.11 ( ( member_complex @ A @ ( minus_811609699411566653omplex @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups5058264527183730370ex_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8299_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_Extended_enat,A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ A @ ( minus_925952699566721837d_enat @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( ( groups1392844769737527556at_rat @ G3 @ C2 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8300_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_real,A3: set_real,B2: set_real,G3: real > nat,H2: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: real] :
% 9.00/5.11 ( ( member_real @ A @ ( minus_minus_set_real @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( ( groups1935376822645274424al_nat @ G3 @ C2 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ G3 @ A3 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8301_sum_Osame__carrierI,axiom,
% 9.00/5.11 ! [C2: set_o,A3: set_o,B2: set_o,G3: $o > nat,H2: $o > nat] :
% 9.00/5.11 ( ( finite_finite_o @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ C2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ B2 @ C2 )
% 9.00/5.11 => ( ! [A: $o] :
% 9.00/5.11 ( ( member_o @ A @ ( minus_minus_set_o @ C2 @ A3 ) )
% 9.00/5.11 => ( ( G3 @ A )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ C2 @ B2 ) )
% 9.00/5.11 => ( ( H2 @ B )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( ( groups8507830703676809646_o_nat @ G3 @ C2 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ C2 ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat @ G3 @ A3 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8302_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 9.00/5.11 = ( groups5808333547571424918x_real @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8303_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 9.00/5.11 = ( groups4148127829035722712t_real @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8304_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8305_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8306_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ S )
% 9.00/5.11 = ( groups5693394587270226106ex_nat @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8307_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ S )
% 9.00/5.11 = ( groups2027974829824023292at_nat @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8308_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ S )
% 9.00/5.11 = ( groups5690904116761175830ex_int @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8309_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ S )
% 9.00/5.11 = ( groups2025484359314973016at_int @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8310_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_nat,S: set_nat,G3: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ S )
% 9.00/5.11 = ( groups2906978787729119204at_rat @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8311_sum_Omono__neutral__left,axiom,
% 9.00/5.11 ! [T2: set_nat,S: set_nat,G3: nat > int] :
% 9.00/5.11 ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ S )
% 9.00/5.11 = ( groups3539618377306564664at_int @ G3 @ T2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8312_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ T2 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8313_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ T2 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8314_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8315_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8316_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ T2 )
% 9.00/5.11 = ( groups5693394587270226106ex_nat @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8317_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ T2 )
% 9.00/5.11 = ( groups2027974829824023292at_nat @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8318_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ T2 )
% 9.00/5.11 = ( groups5690904116761175830ex_int @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8319_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ T2 )
% 9.00/5.11 = ( groups2025484359314973016at_int @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8320_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_nat,S: set_nat,G3: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups2906978787729119204at_rat @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8321_sum_Omono__neutral__right,axiom,
% 9.00/5.11 ! [T2: set_nat,S: set_nat,G3: nat > int] :
% 9.00/5.11 ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ T2 )
% 9.00/5.11 = ( groups3539618377306564664at_int @ G3 @ S ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8322_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8323_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8324_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8325_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8326_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ S )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8327_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ G3 @ S )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8328_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8329_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8330_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,H2: real > nat,G3: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ G3 @ S )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8331_sum_Omono__neutral__cong__left,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,H2: $o > nat,G3: $o > nat] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat @ G3 @ S )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8332_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,G3: real > real,H2: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ T2 )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8333_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,G3: $o > real,H2: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ G3 @ T2 )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8334_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > real,H2: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ T2 )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8335_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > real,H2: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ T2 )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8336_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,G3: real > rat,H2: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8337_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,G3: $o > rat,H2: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8338_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,G3: complex > rat,H2: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ S @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8339_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,G3: extended_enat > rat,H2: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ S @ T2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ T2 )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8340_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,G3: real > nat,H2: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ S @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ G3 @ T2 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8341_sum_Omono__neutral__cong__right,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,G3: $o > nat,H2: $o > nat] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ S @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat @ G3 @ T2 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ H2 @ S ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8342_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8343_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8344_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8345_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8346_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8347_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8348_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8349_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,G3: extended_enat > int] :
% 9.00/5.11 ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8350_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_nat,A3: set_nat,G3: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8351_sum_Osubset__diff,axiom,
% 9.00/5.11 ! [B2: set_nat,A3: set_nat,G3: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( minus_minus_set_nat @ A3 @ B2 ) ) @ ( groups3539618377306564664at_int @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8352_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8353_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8354_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8355_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8356_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8357_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8358_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8359_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat,F: nat > int] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8360_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8361_sum__diff,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8362_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,H2: real > real,G3: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ S )
% 9.00/5.11 = ( groups8097168146408367636l_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8363_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,H2: $o > real,G3: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ G3 @ S )
% 9.00/5.11 = ( groups8691415230153176458o_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8364_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_int,S: set_int,H2: int > real,G3: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ T2 )
% 9.00/5.11 => ( ( finite_finite_int @ S )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ G3 @ S )
% 9.00/5.11 = ( groups8778361861064173332t_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8365_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,H2: complex > real,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ S )
% 9.00/5.11 = ( groups5808333547571424918x_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8366_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ S )
% 9.00/5.11 = ( groups4148127829035722712t_real @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8367_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_real,S: set_real,H2: real > rat,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ ( minus_minus_set_real @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ ( minus_minus_set_real @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( inf_inf_set_real @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ S )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8368_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_o,S: set_o,H2: $o > rat,G3: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ ( minus_minus_set_o @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ ( minus_minus_set_o @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( inf_inf_set_o @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ G3 @ S )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8369_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_int,S: set_int,H2: int > rat,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ T2 )
% 9.00/5.11 => ( ( finite_finite_int @ S )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ ( minus_minus_set_int @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ ( minus_minus_set_int @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( inf_inf_set_int @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ G3 @ S )
% 9.00/5.11 = ( groups3906332499630173760nt_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8370_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_complex,S: set_complex,H2: complex > rat,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ ( minus_811609699411566653omplex @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ S )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8371_sum_Omono__neutral__cong,axiom,
% 9.00/5.11 ! [T2: set_Extended_enat,S: set_Extended_enat,H2: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ T2 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ T2 @ S ) )
% 9.00/5.11 => ( ( H2 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ ( minus_925952699566721837d_enat @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ I2 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ S @ T2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = ( H2 @ X5 ) ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ S )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ H2 @ T2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8372_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8373_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8374_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8375_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8376_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8377_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8378_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8379_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8380_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8381_sum_Ounion__inter,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8382_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > real,B2: set_int] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8383_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > real,B2: set_complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8384_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > real,B2: set_Extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8385_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > rat,B2: set_int] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8386_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > rat,B2: set_complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8387_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > rat,B2: set_Extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8388_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > nat,B2: set_int] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8389_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > nat,B2: set_complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8390_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > nat,B2: set_Extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8391_sum_OInt__Diff,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > int,B2: set_complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8392_sums__mult__D,axiom,
% 9.00/5.11 ! [C: real,F: nat > real,A2: real] :
% 9.00/5.11 ( ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ C @ ( F @ N2 ) )
% 9.00/5.11 @ A2 )
% 9.00/5.11 => ( ( C != zero_zero_real )
% 9.00/5.11 => ( sums_real @ F @ ( divide_divide_real @ A2 @ C ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8393_sums__mult__D,axiom,
% 9.00/5.11 ! [C: complex,F: nat > complex,A2: complex] :
% 9.00/5.11 ( ( sums_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ C @ ( F @ N2 ) )
% 9.00/5.11 @ A2 )
% 9.00/5.11 => ( ( C != zero_zero_complex )
% 9.00/5.11 => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A2 @ C ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8394_sums__Suc__imp,axiom,
% 9.00/5.11 ! [F: nat > real,S2: real] :
% 9.00/5.11 ( ( ( F @ zero_zero_nat )
% 9.00/5.11 = zero_zero_real )
% 9.00/5.11 => ( ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 => ( sums_real @ F @ S2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8395_sums__Suc,axiom,
% 9.00/5.11 ! [F: nat > real,L: real] :
% 9.00/5.11 ( ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.00/5.11 @ L )
% 9.00/5.11 => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8396_sums__Suc,axiom,
% 9.00/5.11 ! [F: nat > nat,L: nat] :
% 9.00/5.11 ( ( sums_nat
% 9.00/5.11 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.00/5.11 @ L )
% 9.00/5.11 => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8397_sums__Suc,axiom,
% 9.00/5.11 ! [F: nat > int,L: int] :
% 9.00/5.11 ( ( sums_int
% 9.00/5.11 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.00/5.11 @ L )
% 9.00/5.11 => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8398_sums__Suc__iff,axiom,
% 9.00/5.11 ! [F: nat > real,S2: real] :
% 9.00/5.11 ( ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( F @ ( suc @ N2 ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( sums_real @ F @ ( plus_plus_real @ S2 @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8399_sums__zero__iff__shift,axiom,
% 9.00/5.11 ! [N: nat,F: nat > real,S2: real] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( ord_less_nat @ I2 @ N )
% 9.00/5.11 => ( ( F @ I2 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( sums_real
% 9.00/5.11 @ ^ [I4: nat] : ( F @ ( plus_plus_nat @ I4 @ N ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( sums_real @ F @ S2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8400_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_real,P: real > $o,H2: real > real,G3: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups8097168146408367636l_real
% 9.00/5.11 @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups8097168146408367636l_real @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups8097168146408367636l_real @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8401_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_int,P: int > $o,H2: int > real,G3: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real
% 9.00/5.11 @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups8778361861064173332t_real @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8402_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_complex,P: complex > $o,H2: complex > real,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real
% 9.00/5.11 @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8403_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > real,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real
% 9.00/5.11 @ ^ [X: extended_enat] : ( if_real @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8404_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_real,P: real > $o,H2: real > rat,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat
% 9.00/5.11 @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1300246762558778688al_rat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8405_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_int,P: int > $o,H2: int > rat,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat
% 9.00/5.11 @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8406_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_complex,P: complex > $o,H2: complex > rat,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat
% 9.00/5.11 @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ H2 @ ( inf_inf_set_complex @ A3 @ ( collect_complex @ P ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8407_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,P: extended_enat > $o,H2: extended_enat > rat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat
% 9.00/5.11 @ ^ [X: extended_enat] : ( if_rat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ H2 @ ( inf_in8357106775501769908d_enat @ A3 @ ( collec4429806609662206161d_enat @ P ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ ( uminus417252749190364093d_enat @ ( collec4429806609662206161d_enat @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8408_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_real,P: real > $o,H2: real > nat,G3: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [X: real] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups1935376822645274424al_nat @ H2 @ ( inf_inf_set_real @ A3 @ ( collect_real @ P ) ) ) @ ( groups1935376822645274424al_nat @ G3 @ ( inf_inf_set_real @ A3 @ ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8409_sum_OIf__cases,axiom,
% 9.00/5.11 ! [A3: set_int,P: int > $o,H2: int > nat,G3: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat
% 9.00/5.11 @ ^ [X: int] : ( if_nat @ ( P @ X ) @ ( H2 @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ H2 @ ( inf_inf_set_int @ A3 @ ( collect_int @ P ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8410_sum__power__add,axiom,
% 9.00/5.11 ! [X2: complex,M: nat,I5: set_nat] :
% 9.00/5.11 ( ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 = ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ I5 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8411_sum__power__add,axiom,
% 9.00/5.11 ! [X2: rat,M: nat,I5: set_nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 = ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ I5 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8412_sum__power__add,axiom,
% 9.00/5.11 ! [X2: int,M: nat,I5: set_nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 = ( times_times_int @ ( power_power_int @ X2 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ I5 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8413_sum__power__add,axiom,
% 9.00/5.11 ! [X2: real,M: nat,I5: set_nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( plus_plus_nat @ M @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 = ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ I5 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8414_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,F: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8415_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_o,A3: set_o,F: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8416_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8417_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8418_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,F: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8419_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_o,A3: set_o,F: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: $o] :
% 9.00/5.11 ( ( member_o @ B @ ( minus_minus_set_o @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8420_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: complex] :
% 9.00/5.11 ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8421_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ B @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8422_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_nat,A3: set_nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: nat] :
% 9.00/5.11 ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8423_sum__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,F: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ! [B: real] :
% 9.00/5.11 ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B ) ) )
% 9.00/5.11 => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8424_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ A3 ) @ ( groups8778361861064173332t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8425_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8426_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8427_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ A3 ) @ ( groups3906332499630173760nt_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8428_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8429_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_rat ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8430_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( inf_inf_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8431_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8432_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8433_sum_Ounion__inter__neutral,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( G3 @ X5 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8434_sum_OatLeastAtMost__rev,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat,M: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8435_sum_OatLeastAtMost__rev,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat,M: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ N @ M ) )
% 9.00/5.11 = ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ N @ M ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8436_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_complex,X2: complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8437_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8438_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_complex,X2: complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8439_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8440_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_complex,X2: complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8441_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8442_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_complex,X2: complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( member_complex @ X2 @ A3 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8443_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,X2: extended_enat,G3: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( member_Extended_enat @ X2 @ A3 )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8444_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_real,X2: real,G3: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( member_real @ X2 @ A3 )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8445_sum_Oremove,axiom,
% 9.00/5.11 ! [A3: set_real,X2: real,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( member_real @ X2 @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ A3 )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8446_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > real,X2: complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8447_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > real,X2: extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8448_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > rat,X2: complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8449_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > rat,X2: extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8450_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > nat,X2: complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8451_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > nat,X2: extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ X2 ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8452_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_complex,G3: complex > int,X2: complex] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ ( insert_complex @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8453_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,G3: extended_enat > int,X2: extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ ( insert_Extended_enat @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ X2 ) @ ( groups2025484359314973016at_int @ G3 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ X2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8454_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_real,G3: real > real,X2: real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ X2 ) @ ( groups8097168146408367636l_real @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8455_sum_Oinsert__remove,axiom,
% 9.00/5.11 ! [A3: set_real,G3: real > rat,X2: real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ ( insert_real @ X2 @ A3 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ X2 ) @ ( groups1300246762558778688al_rat @ G3 @ ( minus_minus_set_real @ A3 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8456_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_complex,A2: complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ A3 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 9.00/5.11 = ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8457_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 = ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8458_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_real,A2: real,F: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8459_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_o,A2: $o,F: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ( ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8460_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_int,A2: int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8461_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_complex,A2: complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ A3 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8462_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,A2: extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8463_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_real,A2: real,F: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8464_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_o,A2: $o,F: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ( ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8465_sum__diff1,axiom,
% 9.00/5.11 ! [A3: set_int,A2: int,F: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8466_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8467_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8468_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8469_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8470_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8471_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8472_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8473_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ A3 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8474_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( plus_plus_int @ ( groups2025484359314973016at_int @ F @ A3 ) @ ( groups2025484359314973016at_int @ F @ B2 ) ) @ ( groups2025484359314973016at_int @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8475_sum__Un,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat,F: nat > int] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A3 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8476_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_complex )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ A3 ) @ ( groups5808333547571424918x_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8477_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 9.00/5.11 = bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ A3 ) @ ( groups4148127829035722712t_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8478_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_complex )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ A3 ) @ ( groups5058264527183730370ex_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8479_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 9.00/5.11 = bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ A3 ) @ ( groups1392844769737527556at_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8480_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_complex )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ A3 ) @ ( groups5693394587270226106ex_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8481_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 9.00/5.11 = bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ A3 ) @ ( groups2027974829824023292at_nat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8482_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_complex @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_complex )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ A3 ) @ ( groups5690904116761175830ex_int @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8483_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ( inf_in8357106775501769908d_enat @ A3 @ B2 )
% 9.00/5.11 = bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ( groups2025484359314973016at_int @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups2025484359314973016at_int @ G3 @ A3 ) @ ( groups2025484359314973016at_int @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8484_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_real,B2: set_real,G3: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_real )
% 9.00/5.11 => ( ( groups8097168146408367636l_real @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups8097168146408367636l_real @ G3 @ A3 ) @ ( groups8097168146408367636l_real @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8485_sum_Ounion__disjoint,axiom,
% 9.00/5.11 ! [A3: set_real,B2: set_real,G3: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ( inf_inf_set_real @ A3 @ B2 )
% 9.00/5.11 = bot_bot_set_real )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat @ G3 @ ( sup_sup_set_real @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups1300246762558778688al_rat @ G3 @ A3 ) @ ( groups1300246762558778688al_rat @ G3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8486_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8487_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8488_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8489_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8490_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8491_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8492_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,G3: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ G3 @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ G3 @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8493_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8494_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,G3: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ G3 @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ G3 @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8495_sum_Ounion__diff2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,G3: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ G3 @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ G3 @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8496_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > real] :
% 9.00/5.11 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups8778361861064173332t_real @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups8778361861064173332t_real @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8497_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8498_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups4148127829035722712t_real @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_real @ ( plus_plus_real @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups4148127829035722712t_real @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups4148127829035722712t_real @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8499_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > rat] :
% 9.00/5.11 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups3906332499630173760nt_rat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups3906332499630173760nt_rat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8500_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8501_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_rat @ ( plus_plus_rat @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups1392844769737527556at_rat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups1392844769737527556at_rat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8502_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ B2 @ A3 ) ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8503_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8504_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ B2 @ A3 ) ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8505_sum__Un2,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) ) @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ B2 @ A3 ) ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8506_suminf__finite,axiom,
% 9.00/5.11 ! [N6: set_nat,F: nat > int] :
% 9.00/5.11 ( ( finite_finite_nat @ N6 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.11 => ( ( F @ N3 )
% 9.00/5.11 = zero_zero_int ) )
% 9.00/5.11 => ( ( suminf_int @ F )
% 9.00/5.11 = ( groups3539618377306564664at_int @ F @ N6 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8507_suminf__finite,axiom,
% 9.00/5.11 ! [N6: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( finite_finite_nat @ N6 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.11 => ( ( F @ N3 )
% 9.00/5.11 = zero_zero_nat ) )
% 9.00/5.11 => ( ( suminf_nat @ F )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ F @ N6 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8508_suminf__finite,axiom,
% 9.00/5.11 ! [N6: set_nat,F: nat > real] :
% 9.00/5.11 ( ( finite_finite_nat @ N6 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ~ ( member_nat @ N3 @ N6 )
% 9.00/5.11 => ( ( F @ N3 )
% 9.00/5.11 = zero_zero_real ) )
% 9.00/5.11 => ( ( suminf_real @ F )
% 9.00/5.11 = ( groups6591440286371151544t_real @ F @ N6 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8509_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_complex,A2: complex,B3: complex > real,C: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5808333547571424918x_real
% 9.00/5.11 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5808333547571424918x_real
% 9.00/5.11 @ ^ [K3: complex] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8510_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > real,C: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups4148127829035722712t_real
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups4148127829035722712t_real
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups4148127829035722712t_real @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8511_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_complex,A2: complex,B3: complex > rat,C: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat
% 9.00/5.11 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5058264527183730370ex_rat
% 9.00/5.11 @ ^ [K3: complex] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8512_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > rat,C: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups1392844769737527556at_rat
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups1392844769737527556at_rat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8513_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_complex,A2: complex,B3: complex > nat,C: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat
% 9.00/5.11 @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat
% 9.00/5.11 @ ^ [K3: complex] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8514_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > nat,C: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_nat @ ( B3 @ A2 ) @ ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_nat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups2027974829824023292at_nat @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8515_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_complex,A2: complex,B3: complex > int,C: complex > int] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ S )
% 9.00/5.11 => ( ( ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int
% 9.00/5.11 @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_complex @ A2 @ S )
% 9.00/5.11 => ( ( groups5690904116761175830ex_int
% 9.00/5.11 @ ^ [K3: complex] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S @ ( insert_complex @ A2 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8516_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_Extended_enat,A2: extended_enat,B3: extended_enat > int,C: extended_enat > int] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ( ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups2025484359314973016at_int
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_int @ ( B3 @ A2 ) @ ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_Extended_enat @ A2 @ S )
% 9.00/5.11 => ( ( groups2025484359314973016at_int
% 9.00/5.11 @ ^ [K3: extended_enat] : ( if_int @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups2025484359314973016at_int @ C @ ( minus_925952699566721837d_enat @ S @ ( insert_Extended_enat @ A2 @ bot_bo7653980558646680370d_enat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8517_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_real,A2: real,B3: real > real,C: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( ( member_real @ A2 @ S )
% 9.00/5.11 => ( ( groups8097168146408367636l_real
% 9.00/5.11 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_real @ ( B3 @ A2 ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.11 => ( ( groups8097168146408367636l_real
% 9.00/5.11 @ ^ [K3: real] : ( if_real @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8518_sum_Odelta__remove,axiom,
% 9.00/5.11 ! [S: set_real,A2: real,B3: real > rat,C: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( ( member_real @ A2 @ S )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat
% 9.00/5.11 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( plus_plus_rat @ ( B3 @ A2 ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) )
% 9.00/5.11 & ( ~ ( member_real @ A2 @ S )
% 9.00/5.11 => ( ( groups1300246762558778688al_rat
% 9.00/5.11 @ ^ [K3: real] : ( if_rat @ ( K3 = A2 ) @ ( B3 @ K3 ) @ ( C @ K3 ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S @ ( insert_real @ A2 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8519_powser__sums__if,axiom,
% 9.00/5.11 ! [M: nat,Z: complex] :
% 9.00/5.11 ( sums_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( if_complex @ ( N2 = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N2 ) )
% 9.00/5.11 @ ( power_power_complex @ Z @ M ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8520_powser__sums__if,axiom,
% 9.00/5.11 ! [M: nat,Z: real] :
% 9.00/5.11 ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( if_real @ ( N2 = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N2 ) )
% 9.00/5.11 @ ( power_power_real @ Z @ M ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8521_powser__sums__if,axiom,
% 9.00/5.11 ! [M: nat,Z: int] :
% 9.00/5.11 ( sums_int
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_int @ ( if_int @ ( N2 = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N2 ) )
% 9.00/5.11 @ ( power_power_int @ Z @ M ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8522_powser__sums__zero,axiom,
% 9.00/5.11 ! [A2: nat > complex] :
% 9.00/5.11 ( sums_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( A2 @ N2 ) @ ( power_power_complex @ zero_zero_complex @ N2 ) )
% 9.00/5.11 @ ( A2 @ zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8523_powser__sums__zero,axiom,
% 9.00/5.11 ! [A2: nat > real] :
% 9.00/5.11 ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( A2 @ N2 ) @ ( power_power_real @ zero_zero_real @ N2 ) )
% 9.00/5.11 @ ( A2 @ zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8524_sum__shift__lb__Suc0__0,axiom,
% 9.00/5.11 ! [F: nat > rat,K: nat] :
% 9.00/5.11 ( ( ( F @ zero_zero_nat )
% 9.00/5.11 = zero_zero_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.00/5.11 = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8525_sum__shift__lb__Suc0__0,axiom,
% 9.00/5.11 ! [F: nat > int,K: nat] :
% 9.00/5.11 ( ( ( F @ zero_zero_nat )
% 9.00/5.11 = zero_zero_int )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.00/5.11 = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8526_sum__shift__lb__Suc0__0,axiom,
% 9.00/5.11 ! [F: nat > nat,K: nat] :
% 9.00/5.11 ( ( ( F @ zero_zero_nat )
% 9.00/5.11 = zero_zero_nat )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8527_sum__shift__lb__Suc0__0,axiom,
% 9.00/5.11 ! [F: nat > real,K: nat] :
% 9.00/5.11 ( ( ( F @ zero_zero_nat )
% 9.00/5.11 = zero_zero_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.00/5.11 = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8528_sum_OatLeast0__atMost__Suc,axiom,
% 9.00/5.11 ! [G3: nat > rat,N: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8529_sum_OatLeast0__atMost__Suc,axiom,
% 9.00/5.11 ! [G3: nat > int,N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8530_sum_OatLeast0__atMost__Suc,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8531_sum_OatLeast0__atMost__Suc,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8532_sum_OatLeast__Suc__atMost,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ M ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8533_sum_OatLeast__Suc__atMost,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ M ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8534_sum_OatLeast__Suc__atMost,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ M ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8535_sum_OatLeast__Suc__atMost,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ M ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8536_sum_Onat__ivl__Suc_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8537_sum_Onat__ivl__Suc_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8538_sum_Onat__ivl__Suc_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8539_sum_Onat__ivl__Suc_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8540_cong__exp__iff__simps_I3_J,axiom,
% 9.00/5.11 ! [N: num,Q3: num] :
% 9.00/5.11 ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 != zero_z3403309356797280102nteger ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8541_cong__exp__iff__simps_I3_J,axiom,
% 9.00/5.11 ! [N: num,Q3: num] :
% 9.00/5.11 ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 != zero_zero_int ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8542_cong__exp__iff__simps_I3_J,axiom,
% 9.00/5.11 ! [N: num,Q3: num] :
% 9.00/5.11 ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 != zero_zero_nat ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8543_numeral__3__eq__3,axiom,
% 9.00/5.11 ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 9.00/5.11 = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8544_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,B3: real,F: real > real] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8545_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_o,A3: set_o,B3: $o,F: $o > real] :
% 9.00/5.11 ( ( finite_finite_o @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.11 => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ ( groups8691415230153176458o_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8546_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > real] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.11 => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8547_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.11 => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ ( groups4148127829035722712t_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8548_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,B3: real,F: real > rat] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8549_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_o,A3: set_o,B3: $o,F: $o > rat] :
% 9.00/5.11 ( ( finite_finite_o @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_o @ A3 @ B2 )
% 9.00/5.11 => ( ( member_o @ B3 @ ( minus_minus_set_o @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( groups7872700643590313910_o_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8550_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,B3: complex,F: complex > rat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ A3 @ B2 )
% 9.00/5.11 => ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8551_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,B3: extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ A3 @ B2 )
% 9.00/5.11 => ( ( member_Extended_enat @ B3 @ ( minus_925952699566721837d_enat @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ ( groups1392844769737527556at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8552_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_nat,A3: set_nat,B3: nat,F: nat > rat] :
% 9.00/5.11 ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ A3 @ B2 )
% 9.00/5.11 => ( ( member_nat @ B3 @ ( minus_minus_set_nat @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8553_sum__strict__mono2,axiom,
% 9.00/5.11 ! [B2: set_real,A3: set_real,B3: real,F: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_real @ A3 @ B2 )
% 9.00/5.11 => ( ( member_real @ B3 @ ( minus_minus_set_real @ B2 @ A3 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B3 ) )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ B2 )
% 9.00/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8554_member__le__sum,axiom,
% 9.00/5.11 ! [I: complex,A3: set_complex,F: complex > real] :
% 9.00/5.11 ( ( member_complex @ I @ A3 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8555_member__le__sum,axiom,
% 9.00/5.11 ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > real] :
% 9.00/5.11 ( ( member_Extended_enat @ I @ A3 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I ) @ ( groups4148127829035722712t_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8556_member__le__sum,axiom,
% 9.00/5.11 ! [I: real,A3: set_real,F: real > real] :
% 9.00/5.11 ( ( member_real @ I @ A3 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8557_member__le__sum,axiom,
% 9.00/5.11 ! [I: $o,A3: set_o,F: $o > real] :
% 9.00/5.11 ( ( member_o @ I @ A3 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8558_member__le__sum,axiom,
% 9.00/5.11 ! [I: int,A3: set_int,F: int > real] :
% 9.00/5.11 ( ( member_int @ I @ A3 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8559_member__le__sum,axiom,
% 9.00/5.11 ! [I: complex,A3: set_complex,F: complex > rat] :
% 9.00/5.11 ( ( member_complex @ I @ A3 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A3 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8560_member__le__sum,axiom,
% 9.00/5.11 ! [I: extended_enat,A3: set_Extended_enat,F: extended_enat > rat] :
% 9.00/5.11 ( ( member_Extended_enat @ I @ A3 )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ ( minus_925952699566721837d_enat @ A3 @ ( insert_Extended_enat @ I @ bot_bo7653980558646680370d_enat ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1392844769737527556at_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8561_member__le__sum,axiom,
% 9.00/5.11 ! [I: real,A3: set_real,F: real > rat] :
% 9.00/5.11 ( ( member_real @ I @ A3 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ ( minus_minus_set_real @ A3 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8562_member__le__sum,axiom,
% 9.00/5.11 ! [I: $o,A3: set_o,F: $o > rat] :
% 9.00/5.11 ( ( member_o @ I @ A3 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ ( minus_minus_set_o @ A3 @ ( insert_o @ I @ bot_bot_set_o ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups7872700643590313910_o_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8563_member__le__sum,axiom,
% 9.00/5.11 ! [I: int,A3: set_int,F: int > rat] :
% 9.00/5.11 ( ( member_int @ I @ A3 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ ( minus_minus_set_int @ A3 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8564_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8565_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8566_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8567_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A3 ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8568_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_rat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 9.00/5.11 => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A3 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8569_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > nat,K4: nat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8570_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > nat,K4: nat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_o @ A3 ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_o @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8571_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > nat,K4: nat] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A3 ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8572_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > nat,K4: nat] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_nat @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A3 ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8573_sum__bounded__above__strict,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > int,K4: int] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_int @ ( F @ I2 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A3 ) )
% 9.00/5.11 => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A3 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A3 ) ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8574_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > real,K4: real] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_complex )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8575_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,F: extended_enat > real,K4: real] :
% 9.00/5.11 ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups4148127829035722712t_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8576_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > real,K4: real] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_real )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8577_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > real,K4: real] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_o @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_o )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8578_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > real,K4: real] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ I2 ) @ ( divide_divide_real @ K4 @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_int )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8579_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: complex] :
% 9.00/5.11 ( ( member_complex @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_complex )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8580_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,F: extended_enat > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite121521170596916366d_enat @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups1392844769737527556at_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8581_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_real,F: real > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_real @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_real )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8582_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_o,F: $o > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_o @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_o @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_o )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8583_sum__bounded__above__divide,axiom,
% 9.00/5.11 ! [A3: set_nat,F: nat > rat,K4: rat] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ A3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ I2 ) @ ( divide_divide_rat @ K4 @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A3 ) ) ) ) )
% 9.00/5.11 => ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( A3 != bot_bot_set_nat )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A3 ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8584_sum_OSuc__reindex__ivl,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ M )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8585_sum_OSuc__reindex__ivl,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ M )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8586_sum_OSuc__reindex__ivl,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ M )
% 9.00/5.11 @ ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8587_sum_OSuc__reindex__ivl,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( G3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ M )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8588_sum__Suc__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8589_sum__Suc__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8590_sum__Suc__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
% 9.00/5.11 => ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8591_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_real,X2: real > code_integer,A2: real > code_integer,B3: code_integer,Delta: code_integer] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups7713935264441627589nteger @ X2 @ I5 )
% 9.00/5.11 = one_one_Code_integer )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_le3102999989581377725nteger
% 9.00/5.11 @ ( abs_abs_Code_integer
% 9.00/5.11 @ ( minus_8373710615458151222nteger
% 9.00/5.11 @ ( groups7713935264441627589nteger
% 9.00/5.11 @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8592_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_o,X2: $o > code_integer,A2: $o > code_integer,B3: code_integer,Delta: code_integer] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups4406642042086082107nteger @ X2 @ I5 )
% 9.00/5.11 = one_one_Code_integer )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_le3102999989581377725nteger
% 9.00/5.11 @ ( abs_abs_Code_integer
% 9.00/5.11 @ ( minus_8373710615458151222nteger
% 9.00/5.11 @ ( groups4406642042086082107nteger
% 9.00/5.11 @ ^ [I4: $o] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8593_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_nat,X2: nat > code_integer,A2: nat > code_integer,B3: code_integer,Delta: code_integer] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups7501900531339628137nteger @ X2 @ I5 )
% 9.00/5.11 = one_one_Code_integer )
% 9.00/5.11 => ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_le3102999989581377725nteger
% 9.00/5.11 @ ( abs_abs_Code_integer
% 9.00/5.11 @ ( minus_8373710615458151222nteger
% 9.00/5.11 @ ( groups7501900531339628137nteger
% 9.00/5.11 @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8594_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_int,X2: int > code_integer,A2: int > code_integer,B3: code_integer,Delta: code_integer] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups7873554091576472773nteger @ X2 @ I5 )
% 9.00/5.11 = one_one_Code_integer )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_le3102999989581377725nteger
% 9.00/5.11 @ ( abs_abs_Code_integer
% 9.00/5.11 @ ( minus_8373710615458151222nteger
% 9.00/5.11 @ ( groups7873554091576472773nteger
% 9.00/5.11 @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8595_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_real,X2: real > real,A2: real > real,B3: real,Delta: real] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups8097168146408367636l_real @ X2 @ I5 )
% 9.00/5.11 = one_one_real )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_real
% 9.00/5.11 @ ( abs_abs_real
% 9.00/5.11 @ ( minus_minus_real
% 9.00/5.11 @ ( groups8097168146408367636l_real
% 9.00/5.11 @ ^ [I4: real] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8596_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_o,X2: $o > real,A2: $o > real,B3: real,Delta: real] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups8691415230153176458o_real @ X2 @ I5 )
% 9.00/5.11 = one_one_real )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_real
% 9.00/5.11 @ ( abs_abs_real
% 9.00/5.11 @ ( minus_minus_real
% 9.00/5.11 @ ( groups8691415230153176458o_real
% 9.00/5.11 @ ^ [I4: $o] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8597_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_int,X2: int > real,A2: int > real,B3: real,Delta: real] :
% 9.00/5.11 ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups8778361861064173332t_real @ X2 @ I5 )
% 9.00/5.11 = one_one_real )
% 9.00/5.11 => ( ! [I2: int] :
% 9.00/5.11 ( ( member_int @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_real
% 9.00/5.11 @ ( abs_abs_real
% 9.00/5.11 @ ( minus_minus_real
% 9.00/5.11 @ ( groups8778361861064173332t_real
% 9.00/5.11 @ ^ [I4: int] : ( times_times_real @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8598_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_real,X2: real > rat,A2: real > rat,B3: rat,Delta: rat] :
% 9.00/5.11 ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups1300246762558778688al_rat @ X2 @ I5 )
% 9.00/5.11 = one_one_rat )
% 9.00/5.11 => ( ! [I2: real] :
% 9.00/5.11 ( ( member_real @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_rat
% 9.00/5.11 @ ( abs_abs_rat
% 9.00/5.11 @ ( minus_minus_rat
% 9.00/5.11 @ ( groups1300246762558778688al_rat
% 9.00/5.11 @ ^ [I4: real] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8599_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_o,X2: $o > rat,A2: $o > rat,B3: rat,Delta: rat] :
% 9.00/5.11 ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups7872700643590313910_o_rat @ X2 @ I5 )
% 9.00/5.11 = one_one_rat )
% 9.00/5.11 => ( ! [I2: $o] :
% 9.00/5.11 ( ( member_o @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_rat
% 9.00/5.11 @ ( abs_abs_rat
% 9.00/5.11 @ ( minus_minus_rat
% 9.00/5.11 @ ( groups7872700643590313910_o_rat
% 9.00/5.11 @ ^ [I4: $o] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8600_convex__sum__bound__le,axiom,
% 9.00/5.11 ! [I5: set_nat,X2: nat > rat,A2: nat > rat,B3: rat,Delta: rat] :
% 9.00/5.11 ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ zero_zero_rat @ ( X2 @ I2 ) ) )
% 9.00/5.11 => ( ( ( groups2906978787729119204at_rat @ X2 @ I5 )
% 9.00/5.11 = one_one_rat )
% 9.00/5.11 => ( ! [I2: nat] :
% 9.00/5.11 ( ( member_nat @ I2 @ I5 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A2 @ I2 ) @ B3 ) ) @ Delta ) )
% 9.00/5.11 => ( ord_less_eq_rat
% 9.00/5.11 @ ( abs_abs_rat
% 9.00/5.11 @ ( minus_minus_rat
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( times_times_rat @ ( A2 @ I4 ) @ ( X2 @ I4 ) )
% 9.00/5.11 @ I5 )
% 9.00/5.11 @ B3 ) )
% 9.00/5.11 @ Delta ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8601_num_Osize_I6_J,axiom,
% 9.00/5.11 ! [X32: num] :
% 9.00/5.11 ( ( size_size_num @ ( bit1 @ X32 ) )
% 9.00/5.11 = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8602_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_real,F: real > complex,K4: real] :
% 9.00/5.11 ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8603_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_o,F: $o > complex,K4: real] :
% 9.00/5.11 ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5328290441151304332omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_o @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8604_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_list_nat,F: list_nat > complex,K4: real] :
% 9.00/5.11 ( ! [X5: list_nat] :
% 9.00/5.11 ( ( member_list_nat @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6529277132148336714omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8605_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_set_nat,F: set_nat > complex,K4: real] :
% 9.00/5.11 ( ! [X5: set_nat] :
% 9.00/5.11 ( ( member_set_nat @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8606_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_nat,F: nat > complex,K4: real] :
% 9.00/5.11 ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8607_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_int,F: int > complex,K4: real] :
% 9.00/5.11 ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8608_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_complex,F: complex > complex,K4: real] :
% 9.00/5.11 ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8609_sum__norm__bound,axiom,
% 9.00/5.11 ! [S: set_nat,F: nat > real,K4: real] :
% 9.00/5.11 ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ S )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ K4 ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S ) ) @ K4 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8610_sum_Oub__add__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > rat,P6: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8611_sum_Oub__add__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > int,P6: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.00/5.11 => ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8612_sum_Oub__add__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > nat,P6: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8613_sum_Oub__add__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat,G3: nat > real,P6: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ one_one_nat ) )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N @ P6 ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ M @ N ) ) @ ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P6 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8614_cong__exp__iff__simps_I7_J,axiom,
% 9.00/5.11 ! [Q3: num,N: num] :
% 9.00/5.11 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.00/5.11 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8615_cong__exp__iff__simps_I7_J,axiom,
% 9.00/5.11 ! [Q3: num,N: num] :
% 9.00/5.11 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
% 9.00/5.11 = zero_zero_int ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8616_cong__exp__iff__simps_I7_J,axiom,
% 9.00/5.11 ! [Q3: num,N: num] :
% 9.00/5.11 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.00/5.11 = zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8617_cong__exp__iff__simps_I11_J,axiom,
% 9.00/5.11 ! [M: num,Q3: num] :
% 9.00/5.11 ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
% 9.00/5.11 = zero_z3403309356797280102nteger ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8618_cong__exp__iff__simps_I11_J,axiom,
% 9.00/5.11 ! [M: num,Q3: num] :
% 9.00/5.11 ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
% 9.00/5.11 = zero_zero_int ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8619_cong__exp__iff__simps_I11_J,axiom,
% 9.00/5.11 ! [M: num,Q3: num] :
% 9.00/5.11 ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
% 9.00/5.11 = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
% 9.00/5.11 = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
% 9.00/5.11 = zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8620_sum__le__suminf,axiom,
% 9.00/5.11 ! [F: nat > int,I5: set_nat] :
% 9.00/5.11 ( ( summable_int @ F )
% 9.00/5.11 => ( ( finite_finite_nat @ I5 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.00/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 9.00/5.11 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8621_sum__le__suminf,axiom,
% 9.00/5.11 ! [F: nat > nat,I5: set_nat] :
% 9.00/5.11 ( ( summable_nat @ F )
% 9.00/5.11 => ( ( finite_finite_nat @ I5 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.00/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 9.00/5.11 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8622_sum__le__suminf,axiom,
% 9.00/5.11 ! [F: nat > real,I5: set_nat] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ( finite_finite_nat @ I5 )
% 9.00/5.11 => ( ! [N3: nat] :
% 9.00/5.11 ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8623_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_Pr1261947904930325089at_nat] :
% 9.00/5.11 ( ( ( finite711546835091564841at_nat @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert8211810215607154385at_nat @ X @ ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ Z2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8624_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_complex] :
% 9.00/5.11 ( ( ( finite_card_complex @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: complex,Y: complex,Z2: complex] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_complex @ X @ ( insert_complex @ Y @ ( insert_complex @ Z2 @ bot_bot_set_complex ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8625_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_list_nat] :
% 9.00/5.11 ( ( ( finite_card_list_nat @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: list_nat,Y: list_nat,Z2: list_nat] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_list_nat @ X @ ( insert_list_nat @ Y @ ( insert_list_nat @ Z2 @ bot_bot_set_list_nat ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8626_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_set_nat] :
% 9.00/5.11 ( ( ( finite_card_set_nat @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: set_nat,Y: set_nat,Z2: set_nat] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_set_nat @ X @ ( insert_set_nat @ Y @ ( insert_set_nat @ Z2 @ bot_bot_set_set_nat ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8627_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_real] :
% 9.00/5.11 ( ( ( finite_card_real @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: real,Y: real,Z2: real] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_real @ X @ ( insert_real @ Y @ ( insert_real @ Z2 @ bot_bot_set_real ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8628_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_o] :
% 9.00/5.11 ( ( ( finite_card_o @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: $o,Y: $o,Z2: $o] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_o @ X @ ( insert_o @ Y @ ( insert_o @ Z2 @ bot_bot_set_o ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8629_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_nat] :
% 9.00/5.11 ( ( ( finite_card_nat @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: nat,Y: nat,Z2: nat] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_nat @ X @ ( insert_nat @ Y @ ( insert_nat @ Z2 @ bot_bot_set_nat ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8630_card__3__iff,axiom,
% 9.00/5.11 ! [S: set_int] :
% 9.00/5.11 ( ( ( finite_card_int @ S )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.11 = ( ? [X: int,Y: int,Z2: int] :
% 9.00/5.11 ( ( S
% 9.00/5.11 = ( insert_int @ X @ ( insert_int @ Y @ ( insert_int @ Z2 @ bot_bot_set_int ) ) ) )
% 9.00/5.11 & ( X != Y )
% 9.00/5.11 & ( Y != Z2 )
% 9.00/5.11 & ( X != Z2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8631_exp__le,axiom,
% 9.00/5.11 ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8632_mod__exhaust__less__4,axiom,
% 9.00/5.11 ! [M: nat] :
% 9.00/5.11 ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.11 = zero_zero_nat )
% 9.00/5.11 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.11 | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.11 = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8633_sum__natinterval__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 9.00/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = zero_zero_rat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8634_sum__natinterval__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > int] :
% 9.00/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = zero_zero_int ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8635_sum__natinterval__diff,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > real] :
% 9.00/5.11 ( ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = zero_zero_real ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8636_sum__telescope_H_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.00/5.11 = ( minus_minus_rat @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8637_sum__telescope_H_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.00/5.11 = ( minus_minus_int @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8638_sum__telescope_H_H,axiom,
% 9.00/5.11 ! [M: nat,N: nat,F: nat > real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N ) )
% 9.00/5.11 = ( minus_minus_real @ ( F @ N ) @ ( F @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8639_summable__partial__sum__bound,axiom,
% 9.00/5.11 ! [F: nat > complex,E2: real] :
% 9.00/5.11 ( ( summable_complex @ F )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.11 => ~ ! [N9: nat] :
% 9.00/5.11 ~ ! [M3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N9 @ M3 )
% 9.00/5.11 => ! [N5: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8640_summable__partial__sum__bound,axiom,
% 9.00/5.11 ! [F: nat > real,E2: real] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.11 => ~ ! [N9: nat] :
% 9.00/5.11 ~ ! [M3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N9 @ M3 )
% 9.00/5.11 => ! [N5: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N5 ) ) ) @ E2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8641_geometric__sums,axiom,
% 9.00/5.11 ! [C: real] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.00/5.11 => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8642_geometric__sums,axiom,
% 9.00/5.11 ! [C: complex] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.00/5.11 => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8643_mask__eq__sum__exp,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int )
% 9.00/5.11 = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8644_mask__eq__sum__exp,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer )
% 9.00/5.11 = ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8645_mask__eq__sum__exp,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8646_sum__gp__multiplied,axiom,
% 9.00/5.11 ! [M: nat,N: nat,X2: complex] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.00/5.11 = ( minus_minus_complex @ ( power_power_complex @ X2 @ M ) @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8647_sum__gp__multiplied,axiom,
% 9.00/5.11 ! [M: nat,N: nat,X2: rat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.00/5.11 = ( minus_minus_rat @ ( power_power_rat @ X2 @ M ) @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8648_sum__gp__multiplied,axiom,
% 9.00/5.11 ! [M: nat,N: nat,X2: int] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.00/5.11 = ( minus_minus_int @ ( power_power_int @ X2 @ M ) @ ( power_power_int @ X2 @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8649_sum__gp__multiplied,axiom,
% 9.00/5.11 ! [M: nat,N: nat,X2: real] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ N ) ) )
% 9.00/5.11 = ( minus_minus_real @ ( power_power_real @ X2 @ M ) @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8650_sum_Oin__pairs,axiom,
% 9.00/5.11 ! [G3: nat > rat,M: nat,N: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.11 = ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_rat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8651_sum_Oin__pairs,axiom,
% 9.00/5.11 ! [G3: nat > int,M: nat,N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.11 = ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_int @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8652_sum_Oin__pairs,axiom,
% 9.00/5.11 ! [G3: nat > nat,M: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8653_sum_Oin__pairs,axiom,
% 9.00/5.11 ! [G3: nat > real,M: nat,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.00/5.11 = ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_real @ ( G3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I4 ) ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8654_accp__subset__induct,axiom,
% 9.00/5.11 ! [D4: produc4953844613479565601on_nat > $o,R: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o,X2: produc4953844613479565601on_nat,P: produc4953844613479565601on_nat > $o] :
% 9.00/5.11 ( ( ord_le8126618931240741628_nat_o @ D4 @ ( accp_P8646395344606611882on_nat @ R ) )
% 9.00/5.11 => ( ! [X5: produc4953844613479565601on_nat,Z4: produc4953844613479565601on_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ( R @ Z4 @ X5 )
% 9.00/5.11 => ( D4 @ Z4 ) ) )
% 9.00/5.11 => ( ( D4 @ X2 )
% 9.00/5.11 => ( ! [X5: produc4953844613479565601on_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ! [Z5: produc4953844613479565601on_nat] :
% 9.00/5.11 ( ( R @ Z5 @ X5 )
% 9.00/5.11 => ( P @ Z5 ) )
% 9.00/5.11 => ( P @ X5 ) ) )
% 9.00/5.11 => ( P @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8655_accp__subset__induct,axiom,
% 9.00/5.11 ! [D4: product_prod_nat_nat > $o,R: product_prod_nat_nat > product_prod_nat_nat > $o,X2: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.00/5.11 ( ( ord_le704812498762024988_nat_o @ D4 @ ( accp_P4275260045618599050at_nat @ R ) )
% 9.00/5.11 => ( ! [X5: product_prod_nat_nat,Z4: product_prod_nat_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ( R @ Z4 @ X5 )
% 9.00/5.11 => ( D4 @ Z4 ) ) )
% 9.00/5.11 => ( ( D4 @ X2 )
% 9.00/5.11 => ( ! [X5: product_prod_nat_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ! [Z5: product_prod_nat_nat] :
% 9.00/5.11 ( ( R @ Z5 @ X5 )
% 9.00/5.11 => ( P @ Z5 ) )
% 9.00/5.11 => ( P @ X5 ) ) )
% 9.00/5.11 => ( P @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8656_accp__subset__induct,axiom,
% 9.00/5.11 ! [D4: product_prod_int_int > $o,R: product_prod_int_int > product_prod_int_int > $o,X2: product_prod_int_int,P: product_prod_int_int > $o] :
% 9.00/5.11 ( ( ord_le8369615600986905444_int_o @ D4 @ ( accp_P1096762738010456898nt_int @ R ) )
% 9.00/5.11 => ( ! [X5: product_prod_int_int,Z4: product_prod_int_int] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ( R @ Z4 @ X5 )
% 9.00/5.11 => ( D4 @ Z4 ) ) )
% 9.00/5.11 => ( ( D4 @ X2 )
% 9.00/5.11 => ( ! [X5: product_prod_int_int] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ! [Z5: product_prod_int_int] :
% 9.00/5.11 ( ( R @ Z5 @ X5 )
% 9.00/5.11 => ( P @ Z5 ) )
% 9.00/5.11 => ( P @ X5 ) ) )
% 9.00/5.11 => ( P @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8657_accp__subset__induct,axiom,
% 9.00/5.11 ! [D4: list_nat > $o,R: list_nat > list_nat > $o,X2: list_nat,P: list_nat > $o] :
% 9.00/5.11 ( ( ord_le1520216061033275535_nat_o @ D4 @ ( accp_list_nat @ R ) )
% 9.00/5.11 => ( ! [X5: list_nat,Z4: list_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ( R @ Z4 @ X5 )
% 9.00/5.11 => ( D4 @ Z4 ) ) )
% 9.00/5.11 => ( ( D4 @ X2 )
% 9.00/5.11 => ( ! [X5: list_nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ! [Z5: list_nat] :
% 9.00/5.11 ( ( R @ Z5 @ X5 )
% 9.00/5.11 => ( P @ Z5 ) )
% 9.00/5.11 => ( P @ X5 ) ) )
% 9.00/5.11 => ( P @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8658_accp__subset__induct,axiom,
% 9.00/5.11 ! [D4: nat > $o,R: nat > nat > $o,X2: nat,P: nat > $o] :
% 9.00/5.11 ( ( ord_less_eq_nat_o @ D4 @ ( accp_nat @ R ) )
% 9.00/5.11 => ( ! [X5: nat,Z4: nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ( R @ Z4 @ X5 )
% 9.00/5.11 => ( D4 @ Z4 ) ) )
% 9.00/5.11 => ( ( D4 @ X2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( D4 @ X5 )
% 9.00/5.11 => ( ! [Z5: nat] :
% 9.00/5.11 ( ( R @ Z5 @ X5 )
% 9.00/5.11 => ( P @ Z5 ) )
% 9.00/5.11 => ( P @ X5 ) ) )
% 9.00/5.11 => ( P @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8659_mask__eq__sum__exp__nat,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [Q6: nat] : ( ord_less_nat @ Q6 @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8660_gauss__sum__nat,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [X: nat] : X
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8661_gbinomial__sum__up__index,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8662_gbinomial__sum__up__index,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8663_gbinomial__sum__up__index,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8664_double__arith__series,axiom,
% 9.00/5.11 ! [A2: complex,D: complex,N: nat] :
% 9.00/5.11 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_complex @ A2 @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8665_double__arith__series,axiom,
% 9.00/5.11 ! [A2: rat,D: rat,N: nat] :
% 9.00/5.11 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_rat @ A2 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8666_double__arith__series,axiom,
% 9.00/5.11 ! [A2: extended_enat,D: extended_enat,N: nat] :
% 9.00/5.11 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups7108830773950497114d_enat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_p3455044024723400733d_enat @ A2 @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A2 ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8667_double__arith__series,axiom,
% 9.00/5.11 ! [A2: code_integer,D: code_integer,N: nat] :
% 9.00/5.11 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups7501900531339628137nteger
% 9.00/5.11 @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8668_double__arith__series,axiom,
% 9.00/5.11 ! [A2: int,D: int,N: nat] :
% 9.00/5.11 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8669_double__arith__series,axiom,
% 9.00/5.11 ! [A2: nat,D: nat,N: nat] :
% 9.00/5.11 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8670_double__arith__series,axiom,
% 9.00/5.11 ! [A2: real,D: real,N: nat] :
% 9.00/5.11 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_real @ A2 @ ( times_times_real @ ( semiri5074537144036343181t_real @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ D ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8671_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8672_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8673_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8674_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8675_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8676_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8677_double__gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.11 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8678_arith__series__nat,axiom,
% 9.00/5.11 ! [A2: nat,D: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ I4 @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A2 ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8679_Sum__Icc__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [X: nat] : X
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ M @ N ) )
% 9.00/5.11 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8680_arith__series,axiom,
% 9.00/5.11 ! [A2: code_integer,D: code_integer,N: nat] :
% 9.00/5.11 ( ( groups7501900531339628137nteger
% 9.00/5.11 @ ^ [I4: nat] : ( plus_p5714425477246183910nteger @ A2 @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A2 ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8681_arith__series,axiom,
% 9.00/5.11 ! [A2: int,D: int,N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_int @ A2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( 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 ) ) @ A2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8682_arith__series,axiom,
% 9.00/5.11 ! [A2: nat,D: nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ A2 @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I4 ) @ D ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( 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 ) ) @ A2 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8683_gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8684_gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8685_gauss__sum,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
% 9.00/5.11 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8686_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8687_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8688_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8689_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8690_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8691_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8692_double__gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
% 9.00/5.11 = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8693_sum__gp__offset,axiom,
% 9.00/5.11 ! [X2: rat,M: nat,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
% 9.00/5.11 & ( ( X2 != one_one_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X2 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8694_sum__gp__offset,axiom,
% 9.00/5.11 ! [X2: complex,M: nat,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_complex )
% 9.00/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
% 9.00/5.11 & ( ( X2 != one_one_complex )
% 9.00/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X2 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8695_sum__gp__offset,axiom,
% 9.00/5.11 ! [X2: real,M: nat,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
% 9.00/5.11 & ( ( X2 != one_one_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N ) ) )
% 9.00/5.11 = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X2 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8696_log__base__10__eq2,axiom,
% 9.00/5.11 ! [X2: real] :
% 9.00/5.11 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.11 => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.11 = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8697_gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.11 = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8698_gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.11 = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8699_gauss__sum__from__Suc__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.11 = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8700_gchoose__row__sum__weighted,axiom,
% 9.00/5.11 ! [R2: rat,M: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.00/5.11 = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8701_gchoose__row__sum__weighted,axiom,
% 9.00/5.11 ! [R2: complex,M: nat] :
% 9.00/5.11 ( ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.00/5.11 = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8702_gchoose__row__sum__weighted,axiom,
% 9.00/5.11 ! [R2: real,M: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 9.00/5.11 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.00/5.11 = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8703_and__int_Opinduct,axiom,
% 9.00/5.11 ! [A0: int,A13: int,P: int > int > $o] :
% 9.00/5.11 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
% 9.00/5.11 => ( ! [K2: int,L4: int] :
% 9.00/5.11 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 9.00/5.11 => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.11 & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.11 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.11 => ( P @ K2 @ L4 ) ) )
% 9.00/5.11 => ( P @ A0 @ A13 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8704_prod__decode__aux_Opelims,axiom,
% 9.00/5.11 ! [X2: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.00/5.11 ( ( ( nat_prod_decode_aux @ X2 @ Xa2 )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 9.00/5.11 => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X2 )
% 9.00/5.11 => ( Y3
% 9.00/5.11 = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X2 @ Xa2 ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_nat @ Xa2 @ X2 )
% 9.00/5.11 => ( Y3
% 9.00/5.11 = ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X2 ) ) ) ) ) )
% 9.00/5.11 => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8705_fold__atLeastAtMost__nat_Opinduct,axiom,
% 9.00/5.11 ! [A0: nat > nat > nat,A13: nat,A24: nat,A33: nat,P: ( nat > nat > nat ) > nat > nat > nat > $o] :
% 9.00/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ A0 @ ( produc487386426758144856at_nat @ A13 @ ( product_Pair_nat_nat @ A24 @ A33 ) ) ) )
% 9.00/5.11 => ( ! [F3: nat > nat > nat,A: nat,B: nat,Acc: nat] :
% 9.00/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A @ ( product_Pair_nat_nat @ B @ Acc ) ) ) )
% 9.00/5.11 => ( ( ~ ( ord_less_nat @ B @ A )
% 9.00/5.11 => ( P @ F3 @ ( plus_plus_nat @ A @ one_one_nat ) @ B @ ( F3 @ A @ Acc ) ) )
% 9.00/5.11 => ( P @ F3 @ A @ B @ Acc ) ) )
% 9.00/5.11 => ( P @ A0 @ A13 @ A24 @ A33 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8706_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.00/5.11 ! [X2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 9.00/5.11 ( ( ( vEBT_V1502963449132264192at_nat @ X2 @ Xa2 @ Xb )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ Xa2 @ Xb ) ) )
% 9.00/5.11 => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.00/5.11 => ( ( Y3 = none_P5556105721700978146at_nat )
% 9.00/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Xb ) ) ) ) )
% 9.00/5.11 => ( ! [V2: product_prod_nat_nat] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.00/5.11 => ( ( Xb = none_P5556105721700978146at_nat )
% 9.00/5.11 => ( ( Y3 = none_P5556105721700978146at_nat )
% 9.00/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) ) ) ) )
% 9.00/5.11 => ~ ! [A: product_prod_nat_nat] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_P7363390416028606310at_nat @ A ) )
% 9.00/5.11 => ! [B: product_prod_nat_nat] :
% 9.00/5.11 ( ( Xb
% 9.00/5.11 = ( some_P7363390416028606310at_nat @ B ) )
% 9.00/5.11 => ( ( Y3
% 9.00/5.11 = ( some_P7363390416028606310at_nat @ ( X2 @ A @ B ) ) )
% 9.00/5.11 => ~ ( accp_P3267385326087170368at_nat @ vEBT_V7235779383477046023at_nat @ ( produc2899441246263362727at_nat @ X2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8707_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.00/5.11 ! [X2: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 9.00/5.11 ( ( ( vEBT_V819420779217536731ft_num @ X2 @ Xa2 @ Xb )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ Xa2 @ Xb ) ) )
% 9.00/5.11 => ( ( ( Xa2 = none_num )
% 9.00/5.11 => ( ( Y3 = none_num )
% 9.00/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ none_num @ Xb ) ) ) ) )
% 9.00/5.11 => ( ! [V2: num] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_num @ V2 ) )
% 9.00/5.11 => ( ( Xb = none_num )
% 9.00/5.11 => ( ( Y3 = none_num )
% 9.00/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) ) ) ) )
% 9.00/5.11 => ~ ! [A: num] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_num @ A ) )
% 9.00/5.11 => ! [B: num] :
% 9.00/5.11 ( ( Xb
% 9.00/5.11 = ( some_num @ B ) )
% 9.00/5.11 => ( ( Y3
% 9.00/5.11 = ( some_num @ ( X2 @ A @ B ) ) )
% 9.00/5.11 => ~ ( accp_P7605991808943153877on_num @ vEBT_V452583751252753300el_num @ ( produc5778274026573060048on_num @ X2 @ ( produc8585076106096196333on_num @ ( some_num @ A ) @ ( some_num @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8708_VEBT__internal_Ooption__shift_Opelims,axiom,
% 9.00/5.11 ! [X2: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 9.00/5.11 ( ( ( vEBT_V4262088993061758097ft_nat @ X2 @ Xa2 @ Xb )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ Xa2 @ Xb ) ) )
% 9.00/5.11 => ( ( ( Xa2 = none_nat )
% 9.00/5.11 => ( ( Y3 = none_nat )
% 9.00/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ none_nat @ Xb ) ) ) ) )
% 9.00/5.11 => ( ! [V2: nat] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_nat @ V2 ) )
% 9.00/5.11 => ( ( Xb = none_nat )
% 9.00/5.11 => ( ( Y3 = none_nat )
% 9.00/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) ) ) ) )
% 9.00/5.11 => ~ ! [A: nat] :
% 9.00/5.11 ( ( Xa2
% 9.00/5.11 = ( some_nat @ A ) )
% 9.00/5.11 => ! [B: nat] :
% 9.00/5.11 ( ( Xb
% 9.00/5.11 = ( some_nat @ B ) )
% 9.00/5.11 => ( ( Y3
% 9.00/5.11 = ( some_nat @ ( X2 @ A @ B ) ) )
% 9.00/5.11 => ~ ( accp_P5496254298877145759on_nat @ vEBT_V3895251965096974666el_nat @ ( produc8929957630744042906on_nat @ X2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A ) @ ( some_nat @ B ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8709_divmod__algorithm__code_I8_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8710_divmod__algorithm__code_I8_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8711_divmod__algorithm__code_I8_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_num @ M @ N )
% 9.00/5.11 => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8712_divmod__algorithm__code_I7_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8713_divmod__algorithm__code_I7_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8714_divmod__algorithm__code_I7_J,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 9.00/5.11 & ( ~ ( ord_less_eq_num @ M @ N )
% 9.00/5.11 => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N ) )
% 9.00/5.11 = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8715_divmod__algorithm__code_I2_J,axiom,
% 9.00/5.11 ! [M: num] :
% 9.00/5.11 ( ( unique5052692396658037445od_int @ M @ one )
% 9.00/5.11 = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8716_divmod__algorithm__code_I2_J,axiom,
% 9.00/5.11 ! [M: num] :
% 9.00/5.11 ( ( unique3479559517661332726nteger @ M @ one )
% 9.00/5.11 = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8717_divmod__algorithm__code_I2_J,axiom,
% 9.00/5.11 ! [M: num] :
% 9.00/5.11 ( ( unique5055182867167087721od_nat @ M @ one )
% 9.00/5.11 = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8718_divmod__algorithm__code_I3_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N ) )
% 9.00/5.11 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8719_divmod__algorithm__code_I3_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
% 9.00/5.11 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8720_divmod__algorithm__code_I3_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N ) )
% 9.00/5.11 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8721_divmod__algorithm__code_I4_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8722_divmod__algorithm__code_I4_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
% 9.00/5.11 = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8723_divmod__algorithm__code_I4_J,axiom,
% 9.00/5.11 ! [N: num] :
% 9.00/5.11 ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N ) )
% 9.00/5.11 = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8724_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_o,T: set_o,R: $o > $o > $o,K: $o > nat] :
% 9.00/5.11 ( ( finite_finite_o @ S2 )
% 9.00/5.11 => ( ( finite_finite_o @ T )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [J3: $o] :
% 9.00/5.11 ( ( member_o @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8725_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_o,T: set_real,R: $o > real > $o,K: real > nat] :
% 9.00/5.11 ( ( finite_finite_o @ S2 )
% 9.00/5.11 => ( ( finite_finite_real @ T )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [J3: real] :
% 9.00/5.11 ( ( member_real @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8726_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_real,T: set_o,R: real > $o > $o,K: $o > nat] :
% 9.00/5.11 ( ( finite_finite_real @ S2 )
% 9.00/5.11 => ( ( finite_finite_o @ T )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [J3: $o] :
% 9.00/5.11 ( ( member_o @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8727_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_real,T: set_real,R: real > real > $o,K: real > nat] :
% 9.00/5.11 ( ( finite_finite_real @ S2 )
% 9.00/5.11 => ( ( finite_finite_real @ T )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [J3: real] :
% 9.00/5.11 ( ( member_real @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8728_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_o,T: set_int,R: $o > int > $o,K: int > nat] :
% 9.00/5.11 ( ( finite_finite_o @ S2 )
% 9.00/5.11 => ( ( finite_finite_int @ T )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_int
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [J3: int] :
% 9.00/5.11 ( ( member_int @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8729_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_real,T: set_int,R: real > int > $o,K: int > nat] :
% 9.00/5.11 ( ( finite_finite_real @ S2 )
% 9.00/5.11 => ( ( finite_finite_int @ T )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_int
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [J3: int] :
% 9.00/5.11 ( ( member_int @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups4541462559716669496nt_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8730_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_o,T: set_complex,R: $o > complex > $o,K: complex > nat] :
% 9.00/5.11 ( ( finite_finite_o @ S2 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ T )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_complex
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [J3: complex] :
% 9.00/5.11 ( ( member_complex @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8731_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_real,T: set_complex,R: real > complex > $o,K: complex > nat] :
% 9.00/5.11 ( ( finite_finite_real @ S2 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ T )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_complex
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [J3: complex] :
% 9.00/5.11 ( ( member_complex @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups5693394587270226106ex_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8732_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_o,T: set_Extended_enat,R: $o > extended_enat > $o,K: extended_enat > nat] :
% 9.00/5.11 ( ( finite_finite_o @ S2 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ T )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite121521170596916366d_enat
% 9.00/5.11 @ ( collec4429806609662206161d_enat
% 9.00/5.11 @ ^ [J3: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8733_sum__multicount__gen,axiom,
% 9.00/5.11 ! [S2: set_real,T: set_Extended_enat,R: real > extended_enat > $o,K: extended_enat > nat] :
% 9.00/5.11 ( ( finite_finite_real @ S2 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ T )
% 9.00/5.11 => ( ! [X5: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X5 @ T )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S2 )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = ( K @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite121521170596916366d_enat
% 9.00/5.11 @ ( collec4429806609662206161d_enat
% 9.00/5.11 @ ^ [J3: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ J3 @ T )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S2 )
% 9.00/5.11 = ( groups2027974829824023292at_nat @ K @ T ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8734_sum__subtractf__nat,axiom,
% 9.00/5.11 ! [A3: set_real,G3: real > nat,F: real > nat] :
% 9.00/5.11 ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( groups1935376822645274424al_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8735_sum__subtractf__nat,axiom,
% 9.00/5.11 ! [A3: set_o,G3: $o > nat,F: $o > nat] :
% 9.00/5.11 ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [X: $o] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( groups8507830703676809646_o_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8736_sum__subtractf__nat,axiom,
% 9.00/5.11 ! [A3: set_set_nat,G3: set_nat > nat,F: set_nat > nat] :
% 9.00/5.11 ( ! [X5: set_nat] :
% 9.00/5.11 ( ( member_set_nat @ X5 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( groups8294997508430121362at_nat
% 9.00/5.11 @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( groups8294997508430121362at_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8737_sum__subtractf__nat,axiom,
% 9.00/5.11 ! [A3: set_int,G3: int > nat,F: int > nat] :
% 9.00/5.11 ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat
% 9.00/5.11 @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8738_sum__subtractf__nat,axiom,
% 9.00/5.11 ! [A3: set_nat,G3: nat > nat,F: nat > nat] :
% 9.00/5.11 ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ A3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( G3 @ X5 ) @ ( F @ X5 ) ) )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G3 @ X ) )
% 9.00/5.11 @ A3 )
% 9.00/5.11 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ G3 @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8739_sum__eq__Suc0__iff,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( ? [X: int] :
% 9.00/5.11 ( ( member_int @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 & ! [Y: int] :
% 9.00/5.11 ( ( member_int @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8740_sum__eq__Suc0__iff,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( ? [X: complex] :
% 9.00/5.11 ( ( member_complex @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 & ! [Y: complex] :
% 9.00/5.11 ( ( member_complex @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8741_sum__eq__Suc0__iff,axiom,
% 9.00/5.11 ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.11 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.11 => ( ( ( groups977919841031483927at_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( ? [X: product_prod_nat_nat] :
% 9.00/5.11 ( ( member8440522571783428010at_nat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 & ! [Y: product_prod_nat_nat] :
% 9.00/5.11 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8742_sum__eq__Suc0__iff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( ? [X: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 & ! [Y: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8743_sum__eq__Suc0__iff,axiom,
% 9.00/5.11 ! [A3: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 = ( ? [X: nat] :
% 9.00/5.11 ( ( member_nat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = ( suc @ zero_zero_nat ) )
% 9.00/5.11 & ! [Y: nat] :
% 9.00/5.11 ( ( member_nat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8744_sum__SucD,axiom,
% 9.00/5.11 ! [F: nat > nat,A3: set_nat,N: nat] :
% 9.00/5.11 ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 9.00/5.11 = ( suc @ N ) )
% 9.00/5.11 => ? [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ A3 )
% 9.00/5.11 & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8745_sum__eq__1__iff,axiom,
% 9.00/5.11 ! [A3: set_int,F: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( ( groups4541462559716669496nt_nat @ F @ A3 )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 = ( ? [X: int] :
% 9.00/5.11 ( ( member_int @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 & ! [Y: int] :
% 9.00/5.11 ( ( member_int @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8746_sum__eq__1__iff,axiom,
% 9.00/5.11 ! [A3: set_complex,F: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( ( groups5693394587270226106ex_nat @ F @ A3 )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 = ( ? [X: complex] :
% 9.00/5.11 ( ( member_complex @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 & ! [Y: complex] :
% 9.00/5.11 ( ( member_complex @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8747_sum__eq__1__iff,axiom,
% 9.00/5.11 ! [A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.11 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.11 => ( ( ( groups977919841031483927at_nat @ F @ A3 )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 = ( ? [X: product_prod_nat_nat] :
% 9.00/5.11 ( ( member8440522571783428010at_nat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 & ! [Y: product_prod_nat_nat] :
% 9.00/5.11 ( ( member8440522571783428010at_nat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8748_sum__eq__1__iff,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( ( groups2027974829824023292at_nat @ F @ A3 )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 = ( ? [X: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 & ! [Y: extended_enat] :
% 9.00/5.11 ( ( member_Extended_enat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8749_sum__eq__1__iff,axiom,
% 9.00/5.11 ! [A3: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( ( groups3542108847815614940at_nat @ F @ A3 )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 = ( ? [X: nat] :
% 9.00/5.11 ( ( member_nat @ X @ A3 )
% 9.00/5.11 & ( ( F @ X )
% 9.00/5.11 = one_one_nat )
% 9.00/5.11 & ! [Y: nat] :
% 9.00/5.11 ( ( member_nat @ Y @ A3 )
% 9.00/5.11 => ( ( X != Y )
% 9.00/5.11 => ( ( F @ Y )
% 9.00/5.11 = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8750_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_o,T2: set_o,R: $o > $o > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [J3: $o] :
% 9.00/5.11 ( ( member_o @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8751_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_o,T2: set_real,R: $o > real > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [J3: real] :
% 9.00/5.11 ( ( member_real @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8752_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_real,T2: set_o,R: real > $o > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( finite_finite_o @ T2 )
% 9.00/5.11 => ( ! [X5: $o] :
% 9.00/5.11 ( ( member_o @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [J3: $o] :
% 9.00/5.11 ( ( member_o @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_o @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8753_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_real,T2: set_real,R: real > real > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( finite_finite_real @ T2 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( member_real @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [J3: real] :
% 9.00/5.11 ( ( member_real @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_real @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8754_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_o,T2: set_nat,R: $o > nat > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_nat
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [J3: nat] :
% 9.00/5.11 ( ( member_nat @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8755_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_real,T2: set_nat,R: real > nat > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( finite_finite_nat @ T2 )
% 9.00/5.11 => ( ! [X5: nat] :
% 9.00/5.11 ( ( member_nat @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_nat
% 9.00/5.11 @ ( collect_nat
% 9.00/5.11 @ ^ [J3: nat] :
% 9.00/5.11 ( ( member_nat @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_nat @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8756_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_o,T2: set_int,R: $o > int > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ( finite_finite_int @ T2 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_int
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [J3: int] :
% 9.00/5.11 ( ( member_int @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8757_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_real,T2: set_int,R: real > int > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( finite_finite_int @ T2 )
% 9.00/5.11 => ( ! [X5: int] :
% 9.00/5.11 ( ( member_int @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_int
% 9.00/5.11 @ ( collect_int
% 9.00/5.11 @ ^ [J3: int] :
% 9.00/5.11 ( ( member_int @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_int @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8758_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_o,T2: set_complex,R: $o > complex > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_o @ S )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_o
% 9.00/5.11 @ ( collect_o
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( ( member_o @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat
% 9.00/5.11 @ ^ [I4: $o] :
% 9.00/5.11 ( finite_card_complex
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [J3: complex] :
% 9.00/5.11 ( ( member_complex @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8759_sum__multicount,axiom,
% 9.00/5.11 ! [S: set_real,T2: set_complex,R: real > complex > $o,K: nat] :
% 9.00/5.11 ( ( finite_finite_real @ S )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ T2 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( member_complex @ X5 @ T2 )
% 9.00/5.11 => ( ( finite_card_real
% 9.00/5.11 @ ( collect_real
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( ( member_real @ I4 @ S )
% 9.00/5.11 & ( R @ I4 @ X5 ) ) ) )
% 9.00/5.11 = K ) )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [I4: real] :
% 9.00/5.11 ( finite_card_complex
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [J3: complex] :
% 9.00/5.11 ( ( member_complex @ J3 @ T2 )
% 9.00/5.11 & ( R @ I4 @ J3 ) ) ) )
% 9.00/5.11 @ S )
% 9.00/5.11 = ( times_times_nat @ K @ ( finite_card_complex @ T2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8760_sum__diff__nat,axiom,
% 9.00/5.11 ! [B2: set_complex,A3: set_complex,F: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( ord_le211207098394363844omplex @ B2 @ A3 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8761_sum__diff__nat,axiom,
% 9.00/5.11 ! [B2: set_Pr1261947904930325089at_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.11 ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.11 => ( ( ord_le3146513528884898305at_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8762_sum__diff__nat,axiom,
% 9.00/5.11 ! [B2: set_Extended_enat,A3: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( ord_le7203529160286727270d_enat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ F @ ( minus_925952699566721837d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8763_sum__diff__nat,axiom,
% 9.00/5.11 ! [B2: set_int,A3: set_int,F: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_int @ B2 @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8764_sum__diff__nat,axiom,
% 9.00/5.11 ! [B2: set_nat,A3: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( ord_less_eq_set_nat @ B2 @ A3 )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8765_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: product_prod_nat_nat,A3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.11 ( ( ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member8440522571783428010at_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A3 @ ( insert8211810215607154385at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.00/5.11 = ( groups977919841031483927at_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8766_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: set_nat,A3: set_set_nat,F: set_nat > nat] :
% 9.00/5.11 ( ( ( member_set_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_set_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A3 @ ( insert_set_nat @ A2 @ bot_bot_set_set_nat ) ) )
% 9.00/5.11 = ( groups8294997508430121362at_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8767_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: real,A3: set_real,F: real > nat] :
% 9.00/5.11 ( ( ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_real @ A2 @ A3 )
% 9.00/5.11 => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A3 @ ( insert_real @ A2 @ bot_bot_set_real ) ) )
% 9.00/5.11 = ( groups1935376822645274424al_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8768_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: $o,A3: set_o,F: $o > nat] :
% 9.00/5.11 ( ( ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_o @ A2 @ A3 )
% 9.00/5.11 => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A3 @ ( insert_o @ A2 @ bot_bot_set_o ) ) )
% 9.00/5.11 = ( groups8507830703676809646_o_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8769_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: int,A3: set_int,F: int > nat] :
% 9.00/5.11 ( ( ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_int @ A2 @ A3 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A3 @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
% 9.00/5.11 = ( groups4541462559716669496nt_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8770_sum__diff1__nat,axiom,
% 9.00/5.11 ! [A2: nat,A3: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( ( member_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.11 = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( F @ A2 ) ) ) )
% 9.00/5.11 & ( ~ ( member_nat @ A2 @ A3 )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A3 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ F @ A3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8771_sum__nth__roots,axiom,
% 9.00/5.11 ! [N: nat,C: complex] :
% 9.00/5.11 ( ( ord_less_nat @ one_one_nat @ N )
% 9.00/5.11 => ( ( groups7754918857620584856omplex
% 9.00/5.11 @ ^ [X: complex] : X
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [Z2: complex] :
% 9.00/5.11 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.11 = C ) ) )
% 9.00/5.11 = zero_zero_complex ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8772_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int,F: int > nat] :
% 9.00/5.11 ( ( finite_finite_int @ A3 )
% 9.00/5.11 => ( ( finite_finite_int @ B2 )
% 9.00/5.11 => ( ( groups4541462559716669496nt_nat @ F @ ( sup_sup_set_int @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F @ A3 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) @ ( groups4541462559716669496nt_nat @ F @ ( inf_inf_set_int @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8773_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex,F: complex > nat] :
% 9.00/5.11 ( ( finite3207457112153483333omplex @ A3 )
% 9.00/5.11 => ( ( finite3207457112153483333omplex @ B2 )
% 9.00/5.11 => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A3 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8774_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat,F: extended_enat > nat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.11 => ( ( finite4001608067531595151d_enat @ B2 )
% 9.00/5.11 => ( ( groups2027974829824023292at_nat @ F @ ( sup_su4489774667511045786d_enat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups2027974829824023292at_nat @ F @ A3 ) @ ( groups2027974829824023292at_nat @ F @ B2 ) ) @ ( groups2027974829824023292at_nat @ F @ ( inf_in8357106775501769908d_enat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8775_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.00/5.11 ( ( finite6177210948735845034at_nat @ A3 )
% 9.00/5.11 => ( ( finite6177210948735845034at_nat @ B2 )
% 9.00/5.11 => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A3 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8776_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > nat] :
% 9.00/5.11 ( ( finite4392333629123659920at_nat @ A3 )
% 9.00/5.11 => ( ( finite4392333629123659920at_nat @ B2 )
% 9.00/5.11 => ( ( groups1900718384385340925at_nat @ F @ ( sup_su718114333110466843at_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups1900718384385340925at_nat @ F @ A3 ) @ ( groups1900718384385340925at_nat @ F @ B2 ) ) @ ( groups1900718384385340925at_nat @ F @ ( inf_in4302113700860409141at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8777_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 9.00/5.11 ( ( finite4343798906461161616at_nat @ A3 )
% 9.00/5.11 => ( ( finite4343798906461161616at_nat @ B2 )
% 9.00/5.11 => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A3 ) @ ( groups3860910324918113789at_nat @ F @ B2 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8778_sum__Un__nat,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat,F: nat > nat] :
% 9.00/5.11 ( ( finite_finite_nat @ A3 )
% 9.00/5.11 => ( ( finite_finite_nat @ B2 )
% 9.00/5.11 => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A3 @ B2 ) )
% 9.00/5.11 = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A3 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A3 @ B2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8779_sum__roots__unity,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( ord_less_nat @ one_one_nat @ N )
% 9.00/5.11 => ( ( groups7754918857620584856omplex
% 9.00/5.11 @ ^ [X: complex] : X
% 9.00/5.11 @ ( collect_complex
% 9.00/5.11 @ ^ [Z2: complex] :
% 9.00/5.11 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.11 = one_one_complex ) ) )
% 9.00/5.11 = zero_zero_complex ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8780_divmod__divmod__step,axiom,
% 9.00/5.11 ( unique5055182867167087721od_nat
% 9.00/5.11 = ( ^ [M2: num,N2: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N2 @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8781_divmod__divmod__step,axiom,
% 9.00/5.11 ( unique5052692396658037445od_int
% 9.00/5.11 = ( ^ [M2: num,N2: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N2 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N2 @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8782_divmod__divmod__step,axiom,
% 9.00/5.11 ( unique3479559517661332726nteger
% 9.00/5.11 = ( ^ [M2: num,N2: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N2 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N2 @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8783_Sum__Icc__int,axiom,
% 9.00/5.11 ! [M: int,N: int] :
% 9.00/5.11 ( ( ord_less_eq_int @ M @ N )
% 9.00/5.11 => ( ( groups4538972089207619220nt_int
% 9.00/5.11 @ ^ [X: int] : X
% 9.00/5.11 @ ( set_or1266510415728281911st_int @ M @ N ) )
% 9.00/5.11 = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8784_fold__atLeastAtMost__nat_Opsimps,axiom,
% 9.00/5.11 ! [F: nat > nat > nat,A2: nat,B3: nat,Acc2: nat] :
% 9.00/5.11 ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A2 @ ( product_Pair_nat_nat @ B3 @ Acc2 ) ) ) )
% 9.00/5.11 => ( ( ( ord_less_nat @ B3 @ A2 )
% 9.00/5.11 => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
% 9.00/5.11 = Acc2 ) )
% 9.00/5.11 & ( ~ ( ord_less_nat @ B3 @ A2 )
% 9.00/5.11 => ( ( set_fo2584398358068434914at_nat @ F @ A2 @ B3 @ Acc2 )
% 9.00/5.11 = ( set_fo2584398358068434914at_nat @ F @ ( plus_plus_nat @ A2 @ one_one_nat ) @ B3 @ ( F @ A2 @ Acc2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8785_fold__atLeastAtMost__nat_Opelims,axiom,
% 9.00/5.11 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 9.00/5.11 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) )
% 9.00/5.11 => ~ ( ( ( ( ord_less_nat @ Xb @ Xa2 )
% 9.00/5.11 => ( Y3 = Xc ) )
% 9.00/5.11 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 9.00/5.11 => ( Y3
% 9.00/5.11 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) )
% 9.00/5.11 => ~ ( accp_P6019419558468335806at_nat @ set_fo3699595496184130361el_nat @ ( produc3209952032786966637at_nat @ X2 @ ( produc487386426758144856at_nat @ Xa2 @ ( product_Pair_nat_nat @ Xb @ Xc ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8786_upto_Opinduct,axiom,
% 9.00/5.11 ! [A0: int,A13: int,P: int > int > $o] :
% 9.00/5.11 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A13 ) )
% 9.00/5.11 => ( ! [I2: int,J2: int] :
% 9.00/5.11 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
% 9.00/5.11 => ( ( ( ord_less_eq_int @ I2 @ J2 )
% 9.00/5.11 => ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
% 9.00/5.11 => ( P @ I2 @ J2 ) ) )
% 9.00/5.11 => ( P @ A0 @ A13 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8787_lemma__termdiff2,axiom,
% 9.00/5.11 ! [H2: rat,Z: rat,N: nat] :
% 9.00/5.11 ( ( H2 != zero_zero_rat )
% 9.00/5.11 => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ N ) @ ( power_power_rat @ Z @ N ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.00/5.11 = ( times_times_rat @ H2
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [P5: nat] :
% 9.00/5.11 ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [Q6: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H2 ) @ Q6 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8788_lemma__termdiff2,axiom,
% 9.00/5.11 ! [H2: complex,Z: complex,N: nat] :
% 9.00/5.11 ( ( H2 != zero_zero_complex )
% 9.00/5.11 => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ N ) @ ( power_power_complex @ Z @ N ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.00/5.11 = ( times_times_complex @ H2
% 9.00/5.11 @ ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [P5: nat] :
% 9.00/5.11 ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [Q6: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H2 ) @ Q6 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8789_lemma__termdiff2,axiom,
% 9.00/5.11 ! [H2: real,Z: real,N: nat] :
% 9.00/5.11 ( ( H2 != zero_zero_real )
% 9.00/5.11 => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ N ) @ ( power_power_real @ Z @ N ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.00/5.11 = ( times_times_real @ H2
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [P5: nat] :
% 9.00/5.11 ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [Q6: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H2 ) @ Q6 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q6 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8790_diffs__equiv,axiom,
% 9.00/5.11 ! [C: nat > complex,X2: complex] :
% 9.00/5.11 ( ( summable_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) )
% 9.00/5.11 => ( sums_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( C @ N2 ) ) @ ( power_power_complex @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.00/5.11 @ ( suminf_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8791_diffs__equiv,axiom,
% 9.00/5.11 ! [C: nat > real,X2: real] :
% 9.00/5.11 ( ( summable_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) )
% 9.00/5.11 => ( sums_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( C @ N2 ) ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.00/5.11 @ ( suminf_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8792_lessThan__eq__iff,axiom,
% 9.00/5.11 ! [X2: nat,Y3: nat] :
% 9.00/5.11 ( ( ( set_ord_lessThan_nat @ X2 )
% 9.00/5.11 = ( set_ord_lessThan_nat @ Y3 ) )
% 9.00/5.11 = ( X2 = Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8793_lessThan__eq__iff,axiom,
% 9.00/5.11 ! [X2: int,Y3: int] :
% 9.00/5.11 ( ( ( set_ord_lessThan_int @ X2 )
% 9.00/5.11 = ( set_ord_lessThan_int @ Y3 ) )
% 9.00/5.11 = ( X2 = Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8794_lessThan__eq__iff,axiom,
% 9.00/5.11 ! [X2: real,Y3: real] :
% 9.00/5.11 ( ( ( set_or5984915006950818249n_real @ X2 )
% 9.00/5.11 = ( set_or5984915006950818249n_real @ Y3 ) )
% 9.00/5.11 = ( X2 = Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8795_lessThan__iff,axiom,
% 9.00/5.11 ! [I: $o,K: $o] :
% 9.00/5.11 ( ( member_o @ I @ ( set_ord_lessThan_o @ K ) )
% 9.00/5.11 = ( ord_less_o @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8796_lessThan__iff,axiom,
% 9.00/5.11 ! [I: set_nat,K: set_nat] :
% 9.00/5.11 ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 9.00/5.11 = ( ord_less_set_nat @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8797_lessThan__iff,axiom,
% 9.00/5.11 ! [I: rat,K: rat] :
% 9.00/5.11 ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 9.00/5.11 = ( ord_less_rat @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8798_lessThan__iff,axiom,
% 9.00/5.11 ! [I: num,K: num] :
% 9.00/5.11 ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 9.00/5.11 = ( ord_less_num @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8799_lessThan__iff,axiom,
% 9.00/5.11 ! [I: nat,K: nat] :
% 9.00/5.11 ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 9.00/5.11 = ( ord_less_nat @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8800_lessThan__iff,axiom,
% 9.00/5.11 ! [I: int,K: int] :
% 9.00/5.11 ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 9.00/5.11 = ( ord_less_int @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8801_lessThan__iff,axiom,
% 9.00/5.11 ! [I: real,K: real] :
% 9.00/5.11 ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 9.00/5.11 = ( ord_less_real @ I @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8802_finite__lessThan,axiom,
% 9.00/5.11 ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8803_card__lessThan,axiom,
% 9.00/5.11 ! [U: nat] :
% 9.00/5.11 ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
% 9.00/5.11 = U ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8804_lessThan__subset__iff,axiom,
% 9.00/5.11 ! [X2: rat,Y3: rat] :
% 9.00/5.11 ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X2 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 9.00/5.11 = ( ord_less_eq_rat @ X2 @ Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8805_lessThan__subset__iff,axiom,
% 9.00/5.11 ! [X2: num,Y3: num] :
% 9.00/5.11 ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X2 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 9.00/5.11 = ( ord_less_eq_num @ X2 @ Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8806_lessThan__subset__iff,axiom,
% 9.00/5.11 ! [X2: nat,Y3: nat] :
% 9.00/5.11 ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X2 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 9.00/5.11 = ( ord_less_eq_nat @ X2 @ Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8807_lessThan__subset__iff,axiom,
% 9.00/5.11 ! [X2: int,Y3: int] :
% 9.00/5.11 ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X2 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 9.00/5.11 = ( ord_less_eq_int @ X2 @ Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8808_lessThan__subset__iff,axiom,
% 9.00/5.11 ! [X2: real,Y3: real] :
% 9.00/5.11 ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X2 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 9.00/5.11 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8809_lessThan__0,axiom,
% 9.00/5.11 ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 9.00/5.11 = bot_bot_set_nat ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8810_sum_OlessThan__Suc,axiom,
% 9.00/5.11 ! [G3: nat > rat,N: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8811_sum_OlessThan__Suc,axiom,
% 9.00/5.11 ! [G3: nat > int,N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8812_sum_OlessThan__Suc,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8813_sum_OlessThan__Suc,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) @ ( G3 @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8814_single__Diff__lessThan,axiom,
% 9.00/5.11 ! [K: $o] :
% 9.00/5.11 ( ( minus_minus_set_o @ ( insert_o @ K @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K ) )
% 9.00/5.11 = ( insert_o @ K @ bot_bot_set_o ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8815_single__Diff__lessThan,axiom,
% 9.00/5.11 ! [K: nat] :
% 9.00/5.11 ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 9.00/5.11 = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8816_single__Diff__lessThan,axiom,
% 9.00/5.11 ! [K: int] :
% 9.00/5.11 ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 9.00/5.11 = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8817_single__Diff__lessThan,axiom,
% 9.00/5.11 ! [K: real] :
% 9.00/5.11 ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 9.00/5.11 = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8818_lessThan__non__empty,axiom,
% 9.00/5.11 ! [X2: int] :
% 9.00/5.11 ( ( set_ord_lessThan_int @ X2 )
% 9.00/5.11 != bot_bot_set_int ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8819_lessThan__non__empty,axiom,
% 9.00/5.11 ! [X2: real] :
% 9.00/5.11 ( ( set_or5984915006950818249n_real @ X2 )
% 9.00/5.11 != bot_bot_set_real ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8820_infinite__Iio,axiom,
% 9.00/5.11 ! [A2: int] :
% 9.00/5.11 ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A2 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8821_infinite__Iio,axiom,
% 9.00/5.11 ! [A2: real] :
% 9.00/5.11 ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A2 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8822_lessThan__def,axiom,
% 9.00/5.11 ( set_or890127255671739683et_nat
% 9.00/5.11 = ( ^ [U2: set_nat] :
% 9.00/5.11 ( collect_set_nat
% 9.00/5.11 @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8823_lessThan__def,axiom,
% 9.00/5.11 ( set_ord_lessThan_rat
% 9.00/5.11 = ( ^ [U2: rat] :
% 9.00/5.11 ( collect_rat
% 9.00/5.11 @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8824_lessThan__def,axiom,
% 9.00/5.11 ( set_ord_lessThan_num
% 9.00/5.11 = ( ^ [U2: num] :
% 9.00/5.11 ( collect_num
% 9.00/5.11 @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8825_lessThan__def,axiom,
% 9.00/5.11 ( set_ord_lessThan_nat
% 9.00/5.11 = ( ^ [U2: nat] :
% 9.00/5.11 ( collect_nat
% 9.00/5.11 @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8826_lessThan__def,axiom,
% 9.00/5.11 ( set_ord_lessThan_int
% 9.00/5.11 = ( ^ [U2: int] :
% 9.00/5.11 ( collect_int
% 9.00/5.11 @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8827_lessThan__def,axiom,
% 9.00/5.11 ( set_or5984915006950818249n_real
% 9.00/5.11 = ( ^ [U2: real] :
% 9.00/5.11 ( collect_real
% 9.00/5.11 @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8828_Iio__eq__empty__iff,axiom,
% 9.00/5.11 ! [N: $o] :
% 9.00/5.11 ( ( ( set_ord_lessThan_o @ N )
% 9.00/5.11 = bot_bot_set_o )
% 9.00/5.11 = ( N = bot_bot_o ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8829_Iio__eq__empty__iff,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( ( set_ord_lessThan_nat @ N )
% 9.00/5.11 = bot_bot_set_nat )
% 9.00/5.11 = ( N = bot_bot_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8830_lessThan__strict__subset__iff,axiom,
% 9.00/5.11 ! [M: rat,N: rat] :
% 9.00/5.11 ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N ) )
% 9.00/5.11 = ( ord_less_rat @ M @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8831_lessThan__strict__subset__iff,axiom,
% 9.00/5.11 ! [M: num,N: num] :
% 9.00/5.11 ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N ) )
% 9.00/5.11 = ( ord_less_num @ M @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8832_lessThan__strict__subset__iff,axiom,
% 9.00/5.11 ! [M: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8833_lessThan__strict__subset__iff,axiom,
% 9.00/5.11 ! [M: int,N: int] :
% 9.00/5.11 ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N ) )
% 9.00/5.11 = ( ord_less_int @ M @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8834_lessThan__strict__subset__iff,axiom,
% 9.00/5.11 ! [M: real,N: real] :
% 9.00/5.11 ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N ) )
% 9.00/5.11 = ( ord_less_real @ M @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8835_lessThan__Suc,axiom,
% 9.00/5.11 ! [K: nat] :
% 9.00/5.11 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.00/5.11 = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8836_lessThan__empty__iff,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( ( set_ord_lessThan_nat @ N )
% 9.00/5.11 = bot_bot_set_nat )
% 9.00/5.11 = ( N = zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8837_finite__nat__bounded,axiom,
% 9.00/5.11 ! [S: set_nat] :
% 9.00/5.11 ( ( finite_finite_nat @ S )
% 9.00/5.11 => ? [K2: nat] : ( ord_less_eq_set_nat @ S @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8838_finite__nat__iff__bounded,axiom,
% 9.00/5.11 ( finite_finite_nat
% 9.00/5.11 = ( ^ [S6: set_nat] :
% 9.00/5.11 ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8839_ivl__disj__int__one_I4_J,axiom,
% 9.00/5.11 ! [L: $o,U: $o] :
% 9.00/5.11 ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
% 9.00/5.11 = bot_bot_set_o ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8840_ivl__disj__int__one_I4_J,axiom,
% 9.00/5.11 ! [L: nat,U: nat] :
% 9.00/5.11 ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.00/5.11 = bot_bot_set_nat ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8841_ivl__disj__int__one_I4_J,axiom,
% 9.00/5.11 ! [L: int,U: int] :
% 9.00/5.11 ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.00/5.11 = bot_bot_set_int ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8842_ivl__disj__int__one_I4_J,axiom,
% 9.00/5.11 ! [L: real,U: real] :
% 9.00/5.11 ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ L ) @ ( set_or1222579329274155063t_real @ L @ U ) )
% 9.00/5.11 = bot_bot_set_real ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8843_sum_Onat__diff__reindex,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8844_sum_Onat__diff__reindex,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8845_sum__diff__distrib,axiom,
% 9.00/5.11 ! [Q: int > nat,P: int > nat,N: int] :
% 9.00/5.11 ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.00/5.11 => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N ) ) )
% 9.00/5.11 = ( groups4541462559716669496nt_nat
% 9.00/5.11 @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.00/5.11 @ ( set_ord_lessThan_int @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8846_sum__diff__distrib,axiom,
% 9.00/5.11 ! [Q: real > nat,P: real > nat,N: real] :
% 9.00/5.11 ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.00/5.11 => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N ) ) )
% 9.00/5.11 = ( groups1935376822645274424al_nat
% 9.00/5.11 @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.00/5.11 @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8847_sum__diff__distrib,axiom,
% 9.00/5.11 ! [Q: nat > nat,P: nat > nat,N: nat] :
% 9.00/5.11 ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
% 9.00/5.11 => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8848_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: $o,K: $o] :
% 9.00/5.11 ( ( ( ord_less_o @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.11 = ( insert_o @ X2 @ bot_bot_set_o ) ) )
% 9.00/5.11 & ( ~ ( ord_less_o @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K ) @ ( insert_o @ X2 @ bot_bot_set_o ) )
% 9.00/5.11 = bot_bot_set_o ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8849_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: rat,K: rat] :
% 9.00/5.11 ( ( ( ord_less_rat @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
% 9.00/5.11 = ( insert_rat @ X2 @ bot_bot_set_rat ) ) )
% 9.00/5.11 & ( ~ ( ord_less_rat @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K ) @ ( insert_rat @ X2 @ bot_bot_set_rat ) )
% 9.00/5.11 = bot_bot_set_rat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8850_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: num,K: num] :
% 9.00/5.11 ( ( ( ord_less_num @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
% 9.00/5.11 = ( insert_num @ X2 @ bot_bot_set_num ) ) )
% 9.00/5.11 & ( ~ ( ord_less_num @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K ) @ ( insert_num @ X2 @ bot_bot_set_num ) )
% 9.00/5.11 = bot_bot_set_num ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8851_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: nat,K: nat] :
% 9.00/5.11 ( ( ( ord_less_nat @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.11 = ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.00/5.11 & ( ~ ( ord_less_nat @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K ) @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
% 9.00/5.11 = bot_bot_set_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8852_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: int,K: int] :
% 9.00/5.11 ( ( ( ord_less_int @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.11 = ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.00/5.11 & ( ~ ( ord_less_int @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K ) @ ( insert_int @ X2 @ bot_bot_set_int ) )
% 9.00/5.11 = bot_bot_set_int ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8853_Iio__Int__singleton,axiom,
% 9.00/5.11 ! [X2: real,K: real] :
% 9.00/5.11 ( ( ( ord_less_real @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.11 = ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.00/5.11 & ( ~ ( ord_less_real @ X2 @ K )
% 9.00/5.11 => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K ) @ ( insert_real @ X2 @ bot_bot_set_real ) )
% 9.00/5.11 = bot_bot_set_real ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8854_suminf__le__const,axiom,
% 9.00/5.11 ! [F: nat > int,X2: int] :
% 9.00/5.11 ( ( summable_int @ F )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( ord_less_eq_int @ ( suminf_int @ F ) @ X2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8855_suminf__le__const,axiom,
% 9.00/5.11 ! [F: nat > nat,X2: nat] :
% 9.00/5.11 ( ( summable_nat @ F )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8856_suminf__le__const,axiom,
% 9.00/5.11 ! [F: nat > real,X2: real] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( ord_less_eq_real @ ( suminf_real @ F ) @ X2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8857_sum_OlessThan__Suc__shift,axiom,
% 9.00/5.11 ! [G3: nat > rat,N: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_rat @ ( G3 @ zero_zero_nat )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8858_sum_OlessThan__Suc__shift,axiom,
% 9.00/5.11 ! [G3: nat > int,N: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_int @ ( G3 @ zero_zero_nat )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8859_sum_OlessThan__Suc__shift,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_nat @ ( G3 @ zero_zero_nat )
% 9.00/5.11 @ ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8860_sum_OlessThan__Suc__shift,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
% 9.00/5.11 = ( plus_plus_real @ ( G3 @ zero_zero_nat )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8861_sum__lessThan__telescope_H,axiom,
% 9.00/5.11 ! [F: nat > rat,M: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8862_sum__lessThan__telescope_H,axiom,
% 9.00/5.11 ! [F: nat > int,M: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8863_sum__lessThan__telescope_H,axiom,
% 9.00/5.11 ! [F: nat > real,M: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8864_sum__lessThan__telescope,axiom,
% 9.00/5.11 ! [F: nat > rat,M: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8865_sum__lessThan__telescope,axiom,
% 9.00/5.11 ! [F: nat > int,M: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8866_sum__lessThan__telescope,axiom,
% 9.00/5.11 ! [F: nat > real,M: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ M ) )
% 9.00/5.11 = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8867_summableI__nonneg__bounded,axiom,
% 9.00/5.11 ! [F: nat > int,X2: int] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( summable_int @ F ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8868_summableI__nonneg__bounded,axiom,
% 9.00/5.11 ! [F: nat > nat,X2: nat] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( summable_nat @ F ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8869_summableI__nonneg__bounded,axiom,
% 9.00/5.11 ! [F: nat > real,X2: real] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.00/5.11 => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X2 )
% 9.00/5.11 => ( summable_real @ F ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8870_sum_OatLeast1__atMost__eq,axiom,
% 9.00/5.11 ! [G3: nat > nat,N: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.11 = ( groups3542108847815614940at_nat
% 9.00/5.11 @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8871_sum_OatLeast1__atMost__eq,axiom,
% 9.00/5.11 ! [G3: nat > real,N: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ G3 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.11 = ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8872_fold__atLeastAtMost__nat_Osimps,axiom,
% 9.00/5.11 ( set_fo2584398358068434914at_nat
% 9.00/5.11 = ( ^ [F5: nat > nat > nat,A4: nat,B4: nat,Acc3: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A4 ) @ Acc3 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A4 @ one_one_nat ) @ B4 @ ( F5 @ A4 @ Acc3 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8873_fold__atLeastAtMost__nat_Oelims,axiom,
% 9.00/5.11 ! [X2: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y3: nat] :
% 9.00/5.11 ( ( ( set_fo2584398358068434914at_nat @ X2 @ Xa2 @ Xb @ Xc )
% 9.00/5.11 = Y3 )
% 9.00/5.11 => ( ( ( ord_less_nat @ Xb @ Xa2 )
% 9.00/5.11 => ( Y3 = Xc ) )
% 9.00/5.11 & ( ~ ( ord_less_nat @ Xb @ Xa2 )
% 9.00/5.11 => ( Y3
% 9.00/5.11 = ( set_fo2584398358068434914at_nat @ X2 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X2 @ Xa2 @ Xc ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8874_finite__enumerate__initial__segment,axiom,
% 9.00/5.11 ! [S: set_Extended_enat,N: nat,S2: extended_enat] :
% 9.00/5.11 ( ( finite4001608067531595151d_enat @ S )
% 9.00/5.11 => ( ( ord_less_nat @ N @ ( finite121521170596916366d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) ) )
% 9.00/5.11 => ( ( infini7641415182203889163d_enat @ ( inf_in8357106775501769908d_enat @ S @ ( set_or8419480210114673929d_enat @ S2 ) ) @ N )
% 9.00/5.11 = ( infini7641415182203889163d_enat @ S @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8875_finite__enumerate__initial__segment,axiom,
% 9.00/5.11 ! [S: set_nat,N: nat,S2: nat] :
% 9.00/5.11 ( ( finite_finite_nat @ S )
% 9.00/5.11 => ( ( ord_less_nat @ N @ ( finite_card_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) ) )
% 9.00/5.11 => ( ( infini8530281810654367211te_nat @ ( inf_inf_set_nat @ S @ ( set_ord_lessThan_nat @ S2 ) ) @ N )
% 9.00/5.11 = ( infini8530281810654367211te_nat @ S @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8876_power__diff__1__eq,axiom,
% 9.00/5.11 ! [X2: complex,N: nat] :
% 9.00/5.11 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex )
% 9.00/5.11 = ( times_times_complex @ ( minus_minus_complex @ X2 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8877_power__diff__1__eq,axiom,
% 9.00/5.11 ! [X2: rat,N: nat] :
% 9.00/5.11 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat )
% 9.00/5.11 = ( times_times_rat @ ( minus_minus_rat @ X2 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8878_power__diff__1__eq,axiom,
% 9.00/5.11 ! [X2: int,N: nat] :
% 9.00/5.11 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ one_one_int )
% 9.00/5.11 = ( times_times_int @ ( minus_minus_int @ X2 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8879_power__diff__1__eq,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real )
% 9.00/5.11 = ( times_times_real @ ( minus_minus_real @ X2 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8880_one__diff__power__eq,axiom,
% 9.00/5.11 ! [X2: complex,N: nat] :
% 9.00/5.11 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 9.00/5.11 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8881_one__diff__power__eq,axiom,
% 9.00/5.11 ! [X2: rat,N: nat] :
% 9.00/5.11 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 9.00/5.11 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8882_one__diff__power__eq,axiom,
% 9.00/5.11 ! [X2: int,N: nat] :
% 9.00/5.11 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 9.00/5.11 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8883_one__diff__power__eq,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 9.00/5.11 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8884_geometric__sum,axiom,
% 9.00/5.11 ! [X2: rat,N: nat] :
% 9.00/5.11 ( ( X2 != one_one_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X2 @ one_one_rat ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8885_geometric__sum,axiom,
% 9.00/5.11 ! [X2: complex,N: nat] :
% 9.00/5.11 ( ( X2 != one_one_complex )
% 9.00/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X2 @ one_one_complex ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8886_geometric__sum,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( X2 != one_one_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ one_one_real ) @ ( minus_minus_real @ X2 @ one_one_real ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8887_sum__less__suminf,axiom,
% 9.00/5.11 ! [F: nat > int,N: nat] :
% 9.00/5.11 ( ( summable_int @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_int @ zero_zero_int @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8888_sum__less__suminf,axiom,
% 9.00/5.11 ! [F: nat > nat,N: nat] :
% 9.00/5.11 ( ( summable_nat @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8889_sum__less__suminf,axiom,
% 9.00/5.11 ! [F: nat > real,N: nat] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_real @ zero_zero_real @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8890_sum__gp__strict,axiom,
% 9.00/5.11 ! [X2: rat,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( semiri681578069525770553at_rat @ N ) ) )
% 9.00/5.11 & ( ( X2 != one_one_rat )
% 9.00/5.11 => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8891_sum__gp__strict,axiom,
% 9.00/5.11 ! [X2: complex,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_complex )
% 9.00/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( semiri8010041392384452111omplex @ N ) ) )
% 9.00/5.11 & ( ( X2 != one_one_complex )
% 9.00/5.11 => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8892_sum__gp__strict,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( ( X2 = one_one_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( semiri5074537144036343181t_real @ N ) ) )
% 9.00/5.11 & ( ( X2 != one_one_real )
% 9.00/5.11 => ( ( groups6591440286371151544t_real @ ( power_power_real @ X2 ) @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) ) @ ( minus_minus_real @ one_one_real @ X2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8893_diff__power__eq__sum,axiom,
% 9.00/5.11 ! [X2: complex,N: nat,Y3: complex] :
% 9.00/5.11 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ ( suc @ N ) ) @ ( power_power_complex @ Y3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 9.00/5.11 @ ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X2 @ P5 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8894_diff__power__eq__sum,axiom,
% 9.00/5.11 ! [X2: rat,N: nat,Y3: rat] :
% 9.00/5.11 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ ( suc @ N ) ) @ ( power_power_rat @ Y3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X2 @ P5 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8895_diff__power__eq__sum,axiom,
% 9.00/5.11 ! [X2: int,N: nat,Y3: int] :
% 9.00/5.11 ( ( minus_minus_int @ ( power_power_int @ X2 @ ( suc @ N ) ) @ ( power_power_int @ Y3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X2 @ P5 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8896_diff__power__eq__sum,axiom,
% 9.00/5.11 ! [X2: real,N: nat,Y3: real] :
% 9.00/5.11 ( ( minus_minus_real @ ( power_power_real @ X2 @ ( suc @ N ) ) @ ( power_power_real @ Y3 @ ( suc @ N ) ) )
% 9.00/5.11 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X2 @ P5 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ P5 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8897_power__diff__sumr2,axiom,
% 9.00/5.11 ! [X2: complex,N: nat,Y3: complex] :
% 9.00/5.11 ( ( minus_minus_complex @ ( power_power_complex @ X2 @ N ) @ ( power_power_complex @ Y3 @ N ) )
% 9.00/5.11 = ( times_times_complex @ ( minus_minus_complex @ X2 @ Y3 )
% 9.00/5.11 @ ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X2 @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8898_power__diff__sumr2,axiom,
% 9.00/5.11 ! [X2: rat,N: nat,Y3: rat] :
% 9.00/5.11 ( ( minus_minus_rat @ ( power_power_rat @ X2 @ N ) @ ( power_power_rat @ Y3 @ N ) )
% 9.00/5.11 = ( times_times_rat @ ( minus_minus_rat @ X2 @ Y3 )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X2 @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8899_power__diff__sumr2,axiom,
% 9.00/5.11 ! [X2: int,N: nat,Y3: int] :
% 9.00/5.11 ( ( minus_minus_int @ ( power_power_int @ X2 @ N ) @ ( power_power_int @ Y3 @ N ) )
% 9.00/5.11 = ( times_times_int @ ( minus_minus_int @ X2 @ Y3 )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X2 @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8900_power__diff__sumr2,axiom,
% 9.00/5.11 ! [X2: real,N: nat,Y3: real] :
% 9.00/5.11 ( ( minus_minus_real @ ( power_power_real @ X2 @ N ) @ ( power_power_real @ Y3 @ N ) )
% 9.00/5.11 = ( times_times_real @ ( minus_minus_real @ X2 @ Y3 )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X2 @ I4 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8901_real__sum__nat__ivl__bounded2,axiom,
% 9.00/5.11 ! [N: nat,F: nat > rat,K4: rat,K: nat] :
% 9.00/5.11 ( ! [P7: nat] :
% 9.00/5.11 ( ( ord_less_nat @ P7 @ N )
% 9.00/5.11 => ( ord_less_eq_rat @ ( F @ P7 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_eq_rat @ zero_zero_rat @ K4 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8902_real__sum__nat__ivl__bounded2,axiom,
% 9.00/5.11 ! [N: nat,F: nat > int,K4: int,K: nat] :
% 9.00/5.11 ( ! [P7: nat] :
% 9.00/5.11 ( ( ord_less_nat @ P7 @ N )
% 9.00/5.11 => ( ord_less_eq_int @ ( F @ P7 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_eq_int @ zero_zero_int @ K4 )
% 9.00/5.11 => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8903_real__sum__nat__ivl__bounded2,axiom,
% 9.00/5.11 ! [N: nat,F: nat > nat,K4: nat,K: nat] :
% 9.00/5.11 ( ! [P7: nat] :
% 9.00/5.11 ( ( ord_less_nat @ P7 @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( F @ P7 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_eq_nat @ zero_zero_nat @ K4 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8904_real__sum__nat__ivl__bounded2,axiom,
% 9.00/5.11 ! [N: nat,F: nat > real,K4: real,K: nat] :
% 9.00/5.11 ( ! [P7: nat] :
% 9.00/5.11 ( ( ord_less_nat @ P7 @ N )
% 9.00/5.11 => ( ord_less_eq_real @ ( F @ P7 ) @ K4 ) )
% 9.00/5.11 => ( ( ord_less_eq_real @ zero_zero_real @ K4 )
% 9.00/5.11 => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K4 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8905_sum__less__suminf2,axiom,
% 9.00/5.11 ! [F: nat > int,N: nat,I: nat] :
% 9.00/5.11 ( ( summable_int @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_eq_int @ zero_zero_int @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ( ord_less_eq_nat @ N @ I )
% 9.00/5.11 => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.00/5.11 => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8906_sum__less__suminf2,axiom,
% 9.00/5.11 ! [F: nat > nat,N: nat,I: nat] :
% 9.00/5.11 ( ( summable_nat @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ( ord_less_eq_nat @ N @ I )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.00/5.11 => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8907_sum__less__suminf2,axiom,
% 9.00/5.11 ! [F: nat > real,N: nat,I: nat] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ! [M4: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ N @ M4 )
% 9.00/5.11 => ( ord_less_eq_real @ zero_zero_real @ ( F @ M4 ) ) )
% 9.00/5.11 => ( ( ord_less_eq_nat @ N @ I )
% 9.00/5.11 => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.00/5.11 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8908_sum__atLeastAtMost__code,axiom,
% 9.00/5.11 ! [F: nat > rat,A2: nat,B3: nat] :
% 9.00/5.11 ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.11 = ( set_fo1949268297981939178at_rat
% 9.00/5.11 @ ^ [A4: nat] : ( plus_plus_rat @ ( F @ A4 ) )
% 9.00/5.11 @ A2
% 9.00/5.11 @ B3
% 9.00/5.11 @ zero_zero_rat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8909_sum__atLeastAtMost__code,axiom,
% 9.00/5.11 ! [F: nat > int,A2: nat,B3: nat] :
% 9.00/5.11 ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.11 = ( set_fo2581907887559384638at_int
% 9.00/5.11 @ ^ [A4: nat] : ( plus_plus_int @ ( F @ A4 ) )
% 9.00/5.11 @ A2
% 9.00/5.11 @ B3
% 9.00/5.11 @ zero_zero_int ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8910_sum__atLeastAtMost__code,axiom,
% 9.00/5.11 ! [F: nat > nat,A2: nat,B3: nat] :
% 9.00/5.11 ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.11 = ( set_fo2584398358068434914at_nat
% 9.00/5.11 @ ^ [A4: nat] : ( plus_plus_nat @ ( F @ A4 ) )
% 9.00/5.11 @ A2
% 9.00/5.11 @ B3
% 9.00/5.11 @ zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8911_sum__atLeastAtMost__code,axiom,
% 9.00/5.11 ! [F: nat > real,A2: nat,B3: nat] :
% 9.00/5.11 ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.11 = ( set_fo3111899725591712190t_real
% 9.00/5.11 @ ^ [A4: nat] : ( plus_plus_real @ ( F @ A4 ) )
% 9.00/5.11 @ A2
% 9.00/5.11 @ B3
% 9.00/5.11 @ zero_zero_real ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8912_one__diff__power__eq_H,axiom,
% 9.00/5.11 ! [X2: complex,N: nat] :
% 9.00/5.11 ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X2 @ N ) )
% 9.00/5.11 = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X2 )
% 9.00/5.11 @ ( groups2073611262835488442omplex
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_complex @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8913_one__diff__power__eq_H,axiom,
% 9.00/5.11 ! [X2: rat,N: nat] :
% 9.00/5.11 ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X2 @ N ) )
% 9.00/5.11 = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X2 )
% 9.00/5.11 @ ( groups2906978787729119204at_rat
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_rat @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8914_one__diff__power__eq_H,axiom,
% 9.00/5.11 ! [X2: int,N: nat] :
% 9.00/5.11 ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X2 @ N ) )
% 9.00/5.11 = ( times_times_int @ ( minus_minus_int @ one_one_int @ X2 )
% 9.00/5.11 @ ( groups3539618377306564664at_int
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_int @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8915_one__diff__power__eq_H,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ N ) )
% 9.00/5.11 = ( times_times_real @ ( minus_minus_real @ one_one_real @ X2 )
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [I4: nat] : ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8916_termdiff__converges,axiom,
% 9.00/5.11 ! [X2: real,K4: real,C: nat > real] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X2 ) @ K4 )
% 9.00/5.11 => ( ! [X5: real] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K4 )
% 9.00/5.11 => ( summable_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( C @ N2 ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 9.00/5.11 => ( summable_real
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_real @ ( diffs_real @ C @ N2 ) @ ( power_power_real @ X2 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8917_termdiff__converges,axiom,
% 9.00/5.11 ! [X2: complex,K4: real,C: nat > complex] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X2 ) @ K4 )
% 9.00/5.11 => ( ! [X5: complex] :
% 9.00/5.11 ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K4 )
% 9.00/5.11 => ( summable_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( C @ N2 ) @ ( power_power_complex @ X5 @ N2 ) ) ) )
% 9.00/5.11 => ( summable_complex
% 9.00/5.11 @ ^ [N2: nat] : ( times_times_complex @ ( diffs_complex @ C @ N2 ) @ ( power_power_complex @ X2 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8918_sum__pos__lt__pair,axiom,
% 9.00/5.11 ! [F: nat > real,K: nat] :
% 9.00/5.11 ( ( summable_real @ F )
% 9.00/5.11 => ( ! [D6: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D6 ) @ one_one_nat ) ) ) ) )
% 9.00/5.11 => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8919_in__measure,axiom,
% 9.00/5.11 ! [X2: option_nat,Y3: option_nat,F: option_nat > nat] :
% 9.00/5.11 ( ( member4117937158525611210on_nat @ ( produc5098337634421038937on_nat @ X2 @ Y3 ) @ ( measure_option_nat @ F ) )
% 9.00/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8920_in__measure,axiom,
% 9.00/5.11 ! [X2: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat,F: set_Pr4329608150637261639at_nat > nat] :
% 9.00/5.11 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ X2 @ Y3 ) @ ( measur4922264856574889999at_nat @ F ) )
% 9.00/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8921_in__measure,axiom,
% 9.00/5.11 ! [X2: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,F: set_Pr1261947904930325089at_nat > nat] :
% 9.00/5.11 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X2 @ Y3 ) @ ( measur1827424007717751593at_nat @ F ) )
% 9.00/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8922_in__measure,axiom,
% 9.00/5.11 ! [X2: nat,Y3: nat,F: nat > nat] :
% 9.00/5.11 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measure_nat @ F ) )
% 9.00/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8923_in__measure,axiom,
% 9.00/5.11 ! [X2: int,Y3: int,F: int > nat] :
% 9.00/5.11 ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ ( measure_int @ F ) )
% 9.00/5.11 = ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8924_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_nat,B2: set_nat] :
% 9.00/5.11 ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A3 @ B2 ) @ finite_psubset_nat )
% 9.00/5.11 = ( ( ord_less_set_nat @ A3 @ B2 )
% 9.00/5.11 & ( finite_finite_nat @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8925_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_int,B2: set_int] :
% 9.00/5.11 ( ( member2572552093476627150et_int @ ( produc6363374080413544029et_int @ A3 @ B2 ) @ finite_psubset_int )
% 9.00/5.11 = ( ( ord_less_set_int @ A3 @ B2 )
% 9.00/5.11 & ( finite_finite_int @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8926_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_complex,B2: set_complex] :
% 9.00/5.11 ( ( member351165363924911826omplex @ ( produc3790773574474814305omplex @ A3 @ B2 ) @ finite8643634255014194347omplex )
% 9.00/5.11 = ( ( ord_less_set_complex @ A3 @ B2 )
% 9.00/5.11 & ( finite3207457112153483333omplex @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8927_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_Extended_enat,B2: set_Extended_enat] :
% 9.00/5.11 ( ( member4453595087596390480d_enat @ ( produc6639060556116774935d_enat @ A3 @ B2 ) @ finite4251489430341359785d_enat )
% 9.00/5.11 = ( ( ord_le2529575680413868914d_enat @ A3 @ B2 )
% 9.00/5.11 & ( finite4001608067531595151d_enat @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8928_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.00/5.11 ( ( member1466754251312161552at_nat @ ( produc9060074326276436823at_nat @ A3 @ B2 ) @ finite4695646753290404266at_nat )
% 9.00/5.11 = ( ( ord_le2604355607129572851at_nat @ A3 @ B2 )
% 9.00/5.11 & ( finite4343798906461161616at_nat @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8929_in__finite__psubset,axiom,
% 9.00/5.11 ! [A3: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.00/5.11 ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A3 @ B2 ) @ finite469560695537375940at_nat )
% 9.00/5.11 = ( ( ord_le7866589430770878221at_nat @ A3 @ B2 )
% 9.00/5.11 & ( finite6177210948735845034at_nat @ B2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8930_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > real] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8931_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > set_int] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_set_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8932_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > rat] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8933_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > num] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_num @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8934_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > nat] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8935_monoI1,axiom,
% 9.00/5.11 ! [X6: nat > int] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_int @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.11 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8936_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > real] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8937_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > set_int] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8938_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > rat] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8939_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > num] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8940_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > nat] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8941_monoI2,axiom,
% 9.00/5.11 ! [X6: nat > int] :
% 9.00/5.11 ( ! [M4: nat,N3: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.11 => ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ M4 ) ) )
% 9.00/5.11 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8942_monoseq__def,axiom,
% 9.00/5.11 ( topolo6980174941875973593q_real
% 9.00/5.11 = ( ^ [X8: nat > real] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8943_monoseq__def,axiom,
% 9.00/5.11 ( topolo3100542954746470799et_int
% 9.00/5.11 = ( ^ [X8: nat > set_int] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_set_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8944_monoseq__def,axiom,
% 9.00/5.11 ( topolo4267028734544971653eq_rat
% 9.00/5.11 = ( ^ [X8: nat > rat] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8945_monoseq__def,axiom,
% 9.00/5.11 ( topolo1459490580787246023eq_num
% 9.00/5.11 = ( ^ [X8: nat > num] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_num @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8946_monoseq__def,axiom,
% 9.00/5.11 ( topolo4902158794631467389eq_nat
% 9.00/5.11 = ( ^ [X8: nat > nat] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8947_monoseq__def,axiom,
% 9.00/5.11 ( topolo4899668324122417113eq_int
% 9.00/5.11 = ( ^ [X8: nat > int] :
% 9.00/5.11 ( ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_int @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) )
% 9.00/5.11 | ! [M2: nat,N2: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.11 => ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ M2 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8948_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo6980174941875973593q_real
% 9.00/5.11 = ( ^ [X8: nat > real] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8949_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo3100542954746470799et_int
% 9.00/5.11 = ( ^ [X8: nat > set_int] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8950_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo4267028734544971653eq_rat
% 9.00/5.11 = ( ^ [X8: nat > rat] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8951_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo1459490580787246023eq_num
% 9.00/5.11 = ( ^ [X8: nat > num] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8952_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo4902158794631467389eq_nat
% 9.00/5.11 = ( ^ [X8: nat > nat] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8953_monoseq__Suc,axiom,
% 9.00/5.11 ( topolo4899668324122417113eq_int
% 9.00/5.11 = ( ^ [X8: nat > int] :
% 9.00/5.11 ( ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
% 9.00/5.11 | ! [N2: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8954_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > real] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8955_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > set_int] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8956_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > rat] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8957_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > num] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8958_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > nat] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8959_mono__SucI2,axiom,
% 9.00/5.11 ! [X6: nat > int] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N3 ) ) @ ( X6 @ N3 ) )
% 9.00/5.11 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8960_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > real] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_real @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo6980174941875973593q_real @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8961_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > set_int] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo3100542954746470799et_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8962_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > rat] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo4267028734544971653eq_rat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8963_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > num] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_num @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo1459490580787246023eq_num @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8964_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > nat] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_nat @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo4902158794631467389eq_nat @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8965_mono__SucI1,axiom,
% 9.00/5.11 ! [X6: nat > int] :
% 9.00/5.11 ( ! [N3: nat] : ( ord_less_eq_int @ ( X6 @ N3 ) @ ( X6 @ ( suc @ N3 ) ) )
% 9.00/5.11 => ( topolo4899668324122417113eq_int @ X6 ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8966_Maclaurin__exp__lt,axiom,
% 9.00/5.11 ! [X2: real,N: nat] :
% 9.00/5.11 ( ( X2 != zero_zero_real )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.11 => ? [T6: real] :
% 9.00/5.11 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.00/5.11 & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.11 & ( ( exp_real @ X2 )
% 9.00/5.11 = ( plus_plus_real
% 9.00/5.11 @ ( groups6591440286371151544t_real
% 9.00/5.11 @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 9.00/5.11 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.11 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8967_floor__log__nat__eq__powr__iff,axiom,
% 9.00/5.11 ! [B3: nat,K: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.11 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.11 => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.00/5.11 = ( semiri1314217659103216013at_int @ N ) )
% 9.00/5.11 = ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 9.00/5.11 & ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8968_of__nat__code,axiom,
% 9.00/5.11 ( semiri681578069525770553at_rat
% 9.00/5.11 = ( ^ [N2: nat] :
% 9.00/5.11 ( semiri7787848453975740701ux_rat
% 9.00/5.11 @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
% 9.00/5.11 @ N2
% 9.00/5.11 @ zero_zero_rat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8969_of__nat__code,axiom,
% 9.00/5.11 ( semiri1316708129612266289at_nat
% 9.00/5.11 = ( ^ [N2: nat] :
% 9.00/5.11 ( semiri8422978514062236437ux_nat
% 9.00/5.11 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
% 9.00/5.11 @ N2
% 9.00/5.11 @ zero_zero_nat ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8970_of__nat__code,axiom,
% 9.00/5.11 ( semiri1314217659103216013at_int
% 9.00/5.11 = ( ^ [N2: nat] :
% 9.00/5.11 ( semiri8420488043553186161ux_int
% 9.00/5.11 @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
% 9.00/5.11 @ N2
% 9.00/5.11 @ zero_zero_int ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8971_of__nat__code,axiom,
% 9.00/5.11 ( semiri5074537144036343181t_real
% 9.00/5.11 = ( ^ [N2: nat] :
% 9.00/5.11 ( semiri7260567687927622513x_real
% 9.00/5.11 @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
% 9.00/5.11 @ N2
% 9.00/5.11 @ zero_zero_real ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8972_central__binomial__lower__bound,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.11 => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8973_binomial__0__Suc,axiom,
% 9.00/5.11 ! [K: nat] :
% 9.00/5.11 ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 9.00/5.11 = zero_zero_nat ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8974_binomial__1,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.11 = N ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8975_binomial__eq__0__iff,axiom,
% 9.00/5.11 ! [N: nat,K: nat] :
% 9.00/5.11 ( ( ( binomial @ N @ K )
% 9.00/5.11 = zero_zero_nat )
% 9.00/5.11 = ( ord_less_nat @ N @ K ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8976_binomial__n__0,axiom,
% 9.00/5.11 ! [N: nat] :
% 9.00/5.11 ( ( binomial @ N @ zero_zero_nat )
% 9.00/5.11 = one_one_nat ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8977_zero__less__binomial__iff,axiom,
% 9.00/5.11 ! [N: nat,K: nat] :
% 9.00/5.11 ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
% 9.00/5.11 = ( ord_less_eq_nat @ K @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8978_fact__mono__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8979_fact__ge__self,axiom,
% 9.00/5.11 ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8980_binomial__fact__lemma,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.11 => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
% 9.00/5.11 = ( semiri1408675320244567234ct_nat @ N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8981_binomial__altdef__nat,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.11 => ( ( binomial @ N @ K )
% 9.00/5.11 = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8982_binomial__eq__0,axiom,
% 9.00/5.11 ! [N: nat,K: nat] :
% 9.00/5.11 ( ( ord_less_nat @ N @ K )
% 9.00/5.11 => ( ( binomial @ N @ K )
% 9.00/5.11 = zero_zero_nat ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8983_fact__less__mono__nat,axiom,
% 9.00/5.11 ! [M: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.11 => ( ( ord_less_nat @ M @ N )
% 9.00/5.11 => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8984_binomial__symmetric,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.11 => ( ( binomial @ N @ K )
% 9.00/5.11 = ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8985_binomial__le__pow,axiom,
% 9.00/5.11 ! [R2: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ R2 @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8986_zero__less__binomial,axiom,
% 9.00/5.11 ! [K: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ N )
% 9.00/5.11 => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8987_fact__ge__Suc__0__nat,axiom,
% 9.00/5.11 ! [N: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8988_choose__mult,axiom,
% 9.00/5.11 ! [K: nat,M: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ M )
% 9.00/5.11 => ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.11 => ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
% 9.00/5.11 = ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8989_fact__diff__Suc,axiom,
% 9.00/5.11 ! [N: nat,M: nat] :
% 9.00/5.11 ( ( ord_less_nat @ N @ ( suc @ M ) )
% 9.00/5.11 => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) )
% 9.00/5.11 = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8990_fact__div__fact__le__pow,axiom,
% 9.00/5.11 ! [R2: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ R2 @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8991_real__of__int__floor__add__one__gt,axiom,
% 9.00/5.11 ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8992_floor__eq,axiom,
% 9.00/5.11 ! [N: int,X2: real] :
% 9.00/5.11 ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 9.00/5.11 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.00/5.11 => ( ( archim6058952711729229775r_real @ X2 )
% 9.00/5.11 = N ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8993_real__of__int__floor__add__one__ge,axiom,
% 9.00/5.11 ! [R2: real] : ( ord_less_eq_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8994_real__of__int__floor__gt__diff__one,axiom,
% 9.00/5.11 ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8995_real__of__int__floor__ge__diff__one,axiom,
% 9.00/5.11 ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8996_binomial__code,axiom,
% 9.00/5.11 ( binomial
% 9.00/5.11 = ( ^ [N2: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N2 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N2 @ K3 ) @ one_one_nat ) @ N2 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8997_binomial__maximum_H,axiom,
% 9.00/5.11 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8998_binomial__mono,axiom,
% 9.00/5.11 ! [K: nat,K5: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ K5 )
% 9.00/5.11 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_8999_binomial__maximum,axiom,
% 9.00/5.11 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_9000_binomial__antimono,axiom,
% 9.00/5.11 ! [K: nat,K5: nat,N: nat] :
% 9.00/5.11 ( ( ord_less_eq_nat @ K @ K5 )
% 9.00/5.11 => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 9.00/5.11 => ( ( ord_less_eq_nat @ K5 @ N )
% 9.00/5.11 => ( ord_less_eq_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.00/5.11
% 9.00/5.11 thf(fact_9001_binomial__le__pow2,axiom,
% 9.00/5.12 ! [N: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9002_choose__reduce__nat,axiom,
% 9.00/5.12 ! [N: nat,K: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.12 => ( ( binomial @ N @ K )
% 9.00/5.12 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9003_times__binomial__minus1__eq,axiom,
% 9.00/5.12 ! [K: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.12 => ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
% 9.00/5.12 = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9004_floor__eq2,axiom,
% 9.00/5.12 ! [N: int,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
% 9.00/5.12 => ( ( archim6058952711729229775r_real @ X2 )
% 9.00/5.12 = N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9005_floor__divide__real__eq__div,axiom,
% 9.00/5.12 ! [B3: int,A2: real] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A2 @ ( ring_1_of_int_real @ B3 ) ) )
% 9.00/5.12 = ( divide_divide_int @ ( archim6058952711729229775r_real @ A2 ) @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9006_binomial__less__binomial__Suc,axiom,
% 9.00/5.12 ! [K: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ K @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9007_binomial__strict__antimono,axiom,
% 9.00/5.12 ! [K: nat,K5: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ K @ K5 )
% 9.00/5.12 => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 9.00/5.12 => ( ( ord_less_eq_nat @ K5 @ N )
% 9.00/5.12 => ( ord_less_nat @ ( binomial @ N @ K5 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9008_binomial__strict__mono,axiom,
% 9.00/5.12 ! [K: nat,K5: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ K @ K5 )
% 9.00/5.12 => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K5 ) @ N )
% 9.00/5.12 => ( ord_less_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K5 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9009_binomial__addition__formula,axiom,
% 9.00/5.12 ! [N: nat,K: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( binomial @ N @ ( suc @ K ) )
% 9.00/5.12 = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9010_square__fact__le__2__fact,axiom,
% 9.00/5.12 ! [N: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9011_Maclaurin__lemma,axiom,
% 9.00/5.12 ! [H2: real,F: real > real,J: nat > real,N: nat] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.00/5.12 => ? [B8: real] :
% 9.00/5.12 ( ( F @ H2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H2 @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9012_floor__log__eq__powr__iff,axiom,
% 9.00/5.12 ! [X2: real,B3: real,K: int] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.12 => ( ( ( archim6058952711729229775r_real @ ( log @ B3 @ X2 ) )
% 9.00/5.12 = K )
% 9.00/5.12 = ( ( ord_less_eq_real @ ( powr_real @ B3 @ ( ring_1_of_int_real @ K ) ) @ X2 )
% 9.00/5.12 & ( ord_less_real @ X2 @ ( powr_real @ B3 @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9013_Maclaurin__exp__le,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( exp_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X2 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9014_floor__log2__div2,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.00/5.12 = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9015_floor__log__nat__eq__if,axiom,
% 9.00/5.12 ! [B3: nat,N: nat,K: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( power_power_nat @ B3 @ N ) @ K )
% 9.00/5.12 => ( ( ord_less_nat @ K @ ( power_power_nat @ B3 @ ( plus_plus_nat @ N @ one_one_nat ) ) )
% 9.00/5.12 => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B3 )
% 9.00/5.12 => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B3 ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.00/5.12 = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9016_powr__int,axiom,
% 9.00/5.12 ! [X2: real,I: int] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
% 9.00/5.12 = ( power_power_real @ X2 @ ( nat2 @ I ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ I ) )
% 9.00/5.12 = ( divide_divide_real @ one_one_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9017_nat__dvd__1__iff__1,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ M @ one_one_nat )
% 9.00/5.12 = ( M = one_one_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9018_dvd__1__left,axiom,
% 9.00/5.12 ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9019_dvd__1__iff__1,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( M
% 9.00/5.12 = ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9020_nat__mult__dvd__cancel__disj,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.12 = ( ( K = zero_zero_nat )
% 9.00/5.12 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9021_finite__atMost,axiom,
% 9.00/5.12 ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9022_card__atMost,axiom,
% 9.00/5.12 ! [U: nat] :
% 9.00/5.12 ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 9.00/5.12 = ( suc @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9023_nat__1,axiom,
% 9.00/5.12 ( ( nat2 @ one_one_int )
% 9.00/5.12 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9024_nat__0__iff,axiom,
% 9.00/5.12 ! [I: int] :
% 9.00/5.12 ( ( ( nat2 @ I )
% 9.00/5.12 = zero_zero_nat )
% 9.00/5.12 = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9025_nat__le__0,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.00/5.12 => ( ( nat2 @ Z )
% 9.00/5.12 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9026_zless__nat__conj,axiom,
% 9.00/5.12 ! [W2: int,Z: int] :
% 9.00/5.12 ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.12 & ( ord_less_int @ W2 @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9027_nat__neg__numeral,axiom,
% 9.00/5.12 ! [K: num] :
% 9.00/5.12 ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9028_nat__zminus__int,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9029_int__nat__eq,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.00/5.12 = Z ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.00/5.12 = zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9030_atMost__0,axiom,
% 9.00/5.12 ( ( set_ord_atMost_nat @ zero_zero_nat )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9031_zero__less__nat__eq,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_int @ zero_zero_int @ Z ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9032_card__atLeastAtMost__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.00/5.12 = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9033_nat__ceiling__le__eq,axiom,
% 9.00/5.12 ! [X2: real,A2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) @ A2 )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ A2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9034_odd__Suc__minus__one,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
% 9.00/5.12 = N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9035_even__diff__nat,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.12 = ( ( ord_less_nat @ M @ N )
% 9.00/5.12 | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9036_one__less__nat__eq,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_int @ one_one_int @ Z ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9037_nat__less__numeral__power__cancel__iff,axiom,
% 9.00/5.12 ! [A2: int,X2: num,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 9.00/5.12 = ( ord_less_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9038_numeral__power__less__nat__cancel__iff,axiom,
% 9.00/5.12 ! [X2: num,N: nat,A2: int] :
% 9.00/5.12 ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
% 9.00/5.12 = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9039_nat__le__numeral__power__cancel__iff,axiom,
% 9.00/5.12 ! [A2: int,X2: num,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( nat2 @ A2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) )
% 9.00/5.12 = ( ord_less_eq_int @ A2 @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9040_numeral__power__le__nat__cancel__iff,axiom,
% 9.00/5.12 ! [X2: num,N: nat,A2: int] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
% 9.00/5.12 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X2 ) @ N ) @ A2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9041_gcd__nat_Oextremum,axiom,
% 9.00/5.12 ! [A2: nat] : ( dvd_dvd_nat @ A2 @ zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9042_gcd__nat_Oextremum__strict,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 9.00/5.12 & ( zero_zero_nat != A2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9043_gcd__nat_Oextremum__unique,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 9.00/5.12 = ( A2 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9044_gcd__nat_Onot__eq__extremum,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ( ( A2 != zero_zero_nat )
% 9.00/5.12 = ( ( dvd_dvd_nat @ A2 @ zero_zero_nat )
% 9.00/5.12 & ( A2 != zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9045_gcd__nat_Oextremum__uniqueI,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ zero_zero_nat @ A2 )
% 9.00/5.12 => ( A2 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9046_dvd__diff__nat,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ K @ M )
% 9.00/5.12 => ( ( dvd_dvd_nat @ K @ N )
% 9.00/5.12 => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9047_nat__dvd__iff,axiom,
% 9.00/5.12 ! [Z: int,M: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
% 9.00/5.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( M = zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9048_even__nat__iff,axiom,
% 9.00/5.12 ! [K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
% 9.00/5.12 = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9049_nat__zero__as__int,axiom,
% 9.00/5.12 ( zero_zero_nat
% 9.00/5.12 = ( nat2 @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9050_nat__mono,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ X2 @ Y3 )
% 9.00/5.12 => ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9051_atMost__atLeast0,axiom,
% 9.00/5.12 ( set_ord_atMost_nat
% 9.00/5.12 = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9052_eq__nat__nat__iff,axiom,
% 9.00/5.12 ! [Z: int,Z7: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.00/5.12 => ( ( ( nat2 @ Z )
% 9.00/5.12 = ( nat2 @ Z7 ) )
% 9.00/5.12 = ( Z = Z7 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9053_all__nat,axiom,
% 9.00/5.12 ( ( ^ [P2: nat > $o] :
% 9.00/5.12 ! [X3: nat] : ( P2 @ X3 ) )
% 9.00/5.12 = ( ^ [P3: nat > $o] :
% 9.00/5.12 ! [X: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.00/5.12 => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9054_ex__nat,axiom,
% 9.00/5.12 ( ( ^ [P2: nat > $o] :
% 9.00/5.12 ? [X3: nat] : ( P2 @ X3 ) )
% 9.00/5.12 = ( ^ [P3: nat > $o] :
% 9.00/5.12 ? [X: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.00/5.12 & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9055_lessThan__Suc__atMost,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.00/5.12 = ( set_ord_atMost_nat @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9056_dvd__pos__nat,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( dvd_dvd_nat @ M @ N )
% 9.00/5.12 => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9057_nat__dvd__not__less,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.12 => ( ( ord_less_nat @ M @ N )
% 9.00/5.12 => ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9058_dvd__minus__self,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) )
% 9.00/5.12 = ( ( ord_less_nat @ N @ M )
% 9.00/5.12 | ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9059_atMost__Suc,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 9.00/5.12 = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9060_zdvd__antisym__nonneg,axiom,
% 9.00/5.12 ! [M: int,N: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.00/5.12 => ( ( dvd_dvd_int @ M @ N )
% 9.00/5.12 => ( ( dvd_dvd_int @ N @ M )
% 9.00/5.12 => ( M = N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9061_less__eq__dvd__minus,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( dvd_dvd_nat @ M @ N )
% 9.00/5.12 = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9062_dvd__diffD1,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.12 => ( ( dvd_dvd_nat @ K @ M )
% 9.00/5.12 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( dvd_dvd_nat @ K @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9063_dvd__diffD,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N ) )
% 9.00/5.12 => ( ( dvd_dvd_nat @ K @ N )
% 9.00/5.12 => ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9064_zdvd__not__zless,axiom,
% 9.00/5.12 ! [M: int,N: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ M )
% 9.00/5.12 => ( ( ord_less_int @ M @ N )
% 9.00/5.12 => ~ ( dvd_dvd_int @ N @ M ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9065_dvd__fact,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.00/5.12 => ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9066_finite__nat__iff__bounded__le,axiom,
% 9.00/5.12 ( finite_finite_nat
% 9.00/5.12 = ( ^ [S6: set_nat] :
% 9.00/5.12 ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9067_nat__mono__iff,axiom,
% 9.00/5.12 ! [Z: int,W2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_int @ W2 @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9068_zless__nat__eq__int__zless,axiom,
% 9.00/5.12 ! [M: nat,Z: int] :
% 9.00/5.12 ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9069_nat__le__iff,axiom,
% 9.00/5.12 ! [X2: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N )
% 9.00/5.12 = ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9070_nat__0__le,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
% 9.00/5.12 = Z ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9071_int__eq__iff,axiom,
% 9.00/5.12 ! [M: nat,Z: int] :
% 9.00/5.12 ( ( ( semiri1314217659103216013at_int @ M )
% 9.00/5.12 = Z )
% 9.00/5.12 = ( ( M
% 9.00/5.12 = ( nat2 @ Z ) )
% 9.00/5.12 & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9072_dvd__imp__le,axiom,
% 9.00/5.12 ! [K: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ K @ N )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ord_less_eq_nat @ K @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9073_nat__mult__dvd__cancel1,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.12 = ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9074_dvd__mult__cancel,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.00/5.12 => ( dvd_dvd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9075_bezout__add__strong__nat,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( A2 != zero_zero_nat )
% 9.00/5.12 => ? [D6: nat,X5: nat,Y4: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ D6 @ A2 )
% 9.00/5.12 & ( dvd_dvd_nat @ D6 @ B3 )
% 9.00/5.12 & ( ( times_times_nat @ A2 @ X5 )
% 9.00/5.12 = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ D6 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9076_zdvd__imp__le,axiom,
% 9.00/5.12 ! [Z: int,N: int] :
% 9.00/5.12 ( ( dvd_dvd_int @ Z @ N )
% 9.00/5.12 => ( ( ord_less_int @ zero_zero_int @ N )
% 9.00/5.12 => ( ord_less_eq_int @ Z @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9077_mod__greater__zero__iff__not__dvd,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N ) )
% 9.00/5.12 = ( ~ ( dvd_dvd_nat @ N @ M ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9078_dvd__imp__le__int,axiom,
% 9.00/5.12 ! [I: int,D: int] :
% 9.00/5.12 ( ( I != zero_zero_int )
% 9.00/5.12 => ( ( dvd_dvd_int @ D @ I )
% 9.00/5.12 => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9079_mod__eq__dvd__iff__nat,axiom,
% 9.00/5.12 ! [N: nat,M: nat,Q3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( ( ( modulo_modulo_nat @ M @ Q3 )
% 9.00/5.12 = ( modulo_modulo_nat @ N @ Q3 ) )
% 9.00/5.12 = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9080_real__nat__ceiling__ge,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ X2 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9081_finite__divisors__nat,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.12 => ( finite_finite_nat
% 9.00/5.12 @ ( collect_nat
% 9.00/5.12 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9082_nat__less__eq__zless,axiom,
% 9.00/5.12 ! [W2: int,Z: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_int @ W2 @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9083_nat__le__eq__zle,axiom,
% 9.00/5.12 ! [W2: int,Z: int] :
% 9.00/5.12 ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 9.00/5.12 | ( ord_less_eq_int @ zero_zero_int @ Z ) )
% 9.00/5.12 => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
% 9.00/5.12 = ( ord_less_eq_int @ W2 @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9084_nat__eq__iff2,axiom,
% 9.00/5.12 ! [M: nat,W2: int] :
% 9.00/5.12 ( ( M
% 9.00/5.12 = ( nat2 @ W2 ) )
% 9.00/5.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( W2
% 9.00/5.12 = ( semiri1314217659103216013at_int @ M ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( M = zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9085_nat__eq__iff,axiom,
% 9.00/5.12 ! [W2: int,M: nat] :
% 9.00/5.12 ( ( ( nat2 @ W2 )
% 9.00/5.12 = M )
% 9.00/5.12 = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( W2
% 9.00/5.12 = ( semiri1314217659103216013at_int @ M ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( M = zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9086_split__nat,axiom,
% 9.00/5.12 ! [P: nat > $o,I: int] :
% 9.00/5.12 ( ( P @ ( nat2 @ I ) )
% 9.00/5.12 = ( ! [N2: nat] :
% 9.00/5.12 ( ( I
% 9.00/5.12 = ( semiri1314217659103216013at_int @ N2 ) )
% 9.00/5.12 => ( P @ N2 ) )
% 9.00/5.12 & ( ( ord_less_int @ I @ zero_zero_int )
% 9.00/5.12 => ( P @ zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9087_le__nat__iff,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
% 9.00/5.12 = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9088_nat__add__distrib,axiom,
% 9.00/5.12 ! [Z: int,Z7: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.00/5.12 => ( ( nat2 @ ( plus_plus_int @ Z @ Z7 ) )
% 9.00/5.12 = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9089_nat__mult__distrib,axiom,
% 9.00/5.12 ! [Z: int,Z7: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 9.00/5.12 = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9090_nat__diff__distrib_H,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( nat2 @ ( minus_minus_int @ X2 @ Y3 ) )
% 9.00/5.12 = ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9091_nat__diff__distrib,axiom,
% 9.00/5.12 ! [Z7: int,Z: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z7 )
% 9.00/5.12 => ( ( ord_less_eq_int @ Z7 @ Z )
% 9.00/5.12 => ( ( nat2 @ ( minus_minus_int @ Z @ Z7 ) )
% 9.00/5.12 = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9092_nat__abs__triangle__ineq,axiom,
% 9.00/5.12 ! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9093_nat__div__distrib,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 9.00/5.12 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9094_nat__div__distrib_H,axiom,
% 9.00/5.12 ! [Y3: int,X2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( nat2 @ ( divide_divide_int @ X2 @ Y3 ) )
% 9.00/5.12 = ( divide_divide_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9095_nat__power__eq,axiom,
% 9.00/5.12 ! [Z: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( nat2 @ ( power_power_int @ Z @ N ) )
% 9.00/5.12 = ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9096_nat__floor__neg,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 9.00/5.12 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9097_nat__mod__distrib,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( nat2 @ ( modulo_modulo_int @ X2 @ Y3 ) )
% 9.00/5.12 = ( modulo_modulo_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9098_floor__eq3,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.00/5.12 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 9.00/5.12 = N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9099_dvd__mult__cancel2,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M ) @ M )
% 9.00/5.12 = ( N = one_one_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9100_dvd__mult__cancel1,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N ) @ M )
% 9.00/5.12 = ( N = one_one_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9101_le__nat__floor,axiom,
% 9.00/5.12 ! [X2: nat,A2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X2 ) @ A2 )
% 9.00/5.12 => ( ord_less_eq_nat @ X2 @ ( nat2 @ ( archim6058952711729229775r_real @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9102_dvd__minus__add,axiom,
% 9.00/5.12 ! [Q3: nat,N: nat,R2: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Q3 @ N )
% 9.00/5.12 => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M ) )
% 9.00/5.12 => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N @ Q3 ) )
% 9.00/5.12 = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9103_power__dvd__imp__le,axiom,
% 9.00/5.12 ! [I: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N ) )
% 9.00/5.12 => ( ( ord_less_nat @ one_one_nat @ I )
% 9.00/5.12 => ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9104_mod__nat__eqI,axiom,
% 9.00/5.12 ! [R2: nat,N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_nat @ R2 @ N )
% 9.00/5.12 => ( ( ord_less_eq_nat @ R2 @ M )
% 9.00/5.12 => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M @ R2 ) )
% 9.00/5.12 => ( ( modulo_modulo_nat @ M @ N )
% 9.00/5.12 = R2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9105_mod__int__pos__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 9.00/5.12 = ( ( dvd_dvd_int @ L @ K )
% 9.00/5.12 | ( ( L = zero_zero_int )
% 9.00/5.12 & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 9.00/5.12 | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9106_nat__2,axiom,
% 9.00/5.12 ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.12 = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9107_Suc__nat__eq__nat__zadd1,axiom,
% 9.00/5.12 ! [Z: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Z )
% 9.00/5.12 => ( ( suc @ ( nat2 @ Z ) )
% 9.00/5.12 = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9108_nat__less__iff,axiom,
% 9.00/5.12 ! [W2: int,M: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.00/5.12 => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 9.00/5.12 = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9109_nat__mult__distrib__neg,axiom,
% 9.00/5.12 ! [Z: int,Z7: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ Z @ zero_zero_int )
% 9.00/5.12 => ( ( nat2 @ ( times_times_int @ Z @ Z7 ) )
% 9.00/5.12 = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z7 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9110_odd__pos,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9111_nat__abs__int__diff,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.12 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
% 9.00/5.12 = ( minus_minus_nat @ B3 @ A2 ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.12 => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) )
% 9.00/5.12 = ( minus_minus_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9112_floor__eq4,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.00/5.12 => ( ( nat2 @ ( archim6058952711729229775r_real @ X2 ) )
% 9.00/5.12 = N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9113_dvd__power__iff__le,axiom,
% 9.00/5.12 ! [K: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N ) )
% 9.00/5.12 = ( ord_less_eq_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9114_diff__nat__eq__if,axiom,
% 9.00/5.12 ! [Z7: int,Z: int] :
% 9.00/5.12 ( ( ( ord_less_int @ Z7 @ zero_zero_int )
% 9.00/5.12 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 9.00/5.12 = ( nat2 @ Z ) ) )
% 9.00/5.12 & ( ~ ( ord_less_int @ Z7 @ zero_zero_int )
% 9.00/5.12 => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
% 9.00/5.12 = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z7 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z7 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9115_sum__choose__diagonal,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 9.00/5.12 @ ( set_ord_atMost_nat @ M ) )
% 9.00/5.12 = ( binomial @ ( suc @ N ) @ M ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9116_even__set__encode__iff,axiom,
% 9.00/5.12 ! [A3: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ A3 )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A3 ) )
% 9.00/5.12 = ( ~ ( member_nat @ zero_zero_nat @ A3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9117_atLeast1__atMost__eq__remove0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9118_polynomial__product__nat,axiom,
% 9.00/5.12 ! [M: nat,A2: nat > nat,N: nat,B3: nat > nat,X2: nat] :
% 9.00/5.12 ( ! [I2: nat] :
% 9.00/5.12 ( ( ord_less_nat @ M @ I2 )
% 9.00/5.12 => ( ( A2 @ I2 )
% 9.00/5.12 = zero_zero_nat ) )
% 9.00/5.12 => ( ! [J2: nat] :
% 9.00/5.12 ( ( ord_less_nat @ N @ J2 )
% 9.00/5.12 => ( ( B3 @ J2 )
% 9.00/5.12 = zero_zero_nat ) )
% 9.00/5.12 => ( ( times_times_nat
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( power_power_nat @ X2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_atMost_nat @ M ) )
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [J3: nat] : ( times_times_nat @ ( B3 @ J3 ) @ ( power_power_nat @ X2 @ J3 ) )
% 9.00/5.12 @ ( set_ord_atMost_nat @ N ) ) )
% 9.00/5.12 = ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [R5: nat] :
% 9.00/5.12 ( times_times_nat
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [K3: nat] : ( times_times_nat @ ( A2 @ K3 ) @ ( B3 @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.00/5.12 @ ( set_ord_atMost_nat @ R5 ) )
% 9.00/5.12 @ ( power_power_nat @ X2 @ R5 ) )
% 9.00/5.12 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9119_even__mod__4__div__2,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = ( suc @ zero_zero_nat ) )
% 9.00/5.12 => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9120_divmod__step__nat__def,axiom,
% 9.00/5.12 ( unique5026877609467782581ep_nat
% 9.00/5.12 = ( ^ [L2: num] :
% 9.00/5.12 ( produc2626176000494625587at_nat
% 9.00/5.12 @ ^ [Q6: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9121_divmod__step__int__def,axiom,
% 9.00/5.12 ( unique5024387138958732305ep_int
% 9.00/5.12 = ( ^ [L2: num] :
% 9.00/5.12 ( produc4245557441103728435nt_int
% 9.00/5.12 @ ^ [Q6: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9122_odd__mod__4__div__2,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.00/5.12 => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9123_Bernoulli__inequality__even,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X2 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X2 ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9124_sin__coeff__def,axiom,
% 9.00/5.12 ( sin_coeff
% 9.00/5.12 = ( ^ [N2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9125_vebt__buildup_Oelims,axiom,
% 9.00/5.12 ! [X2: nat,Y3: vEBT_VEBT] :
% 9.00/5.12 ( ( ( vEBT_vebt_buildup @ X2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ( X2 = zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.00/5.12 => ( ( ( X2
% 9.00/5.12 = ( suc @ zero_zero_nat ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.00/5.12 => ~ ! [Va2: nat] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9126_divmod__nat__if,axiom,
% 9.00/5.12 ( divmod_nat
% 9.00/5.12 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.12 ( if_Pro6206227464963214023at_nat
% 9.00/5.12 @ ( ( N2 = zero_zero_nat )
% 9.00/5.12 | ( ord_less_nat @ M2 @ N2 ) )
% 9.00/5.12 @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 9.00/5.12 @ ( produc2626176000494625587at_nat
% 9.00/5.12 @ ^ [Q6: nat] : ( product_Pair_nat_nat @ ( suc @ Q6 ) )
% 9.00/5.12 @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9127_sin__coeff__0,axiom,
% 9.00/5.12 ( ( sin_coeff @ zero_zero_nat )
% 9.00/5.12 = zero_zero_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9128_dvd__antisym,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ M @ N )
% 9.00/5.12 => ( ( dvd_dvd_nat @ N @ M )
% 9.00/5.12 => ( M = N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9129_sin__x__le__x,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9130_sin__le__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( sin_real @ X2 ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9131_cos__le__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( cos_real @ X2 ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9132_abs__sin__x__le__abs__x,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ ( abs_abs_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9133_sin__cos__le1,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9134_sin__x__ge__neg__x,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ ( uminus_uminus_real @ X2 ) @ ( sin_real @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9135_sin__ge__minus__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9136_cos__ge__minus__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9137_abs__sin__le__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X2 ) ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9138_abs__cos__le__one,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X2 ) ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9139_sin__gt__zero__02,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9140_cos__two__less__zero,axiom,
% 9.00/5.12 ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9141_cos__is__zero,axiom,
% 9.00/5.12 ? [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 & ( ( cos_real @ X5 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 & ! [Y5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.00/5.12 & ( ord_less_eq_real @ Y5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 & ( ( cos_real @ Y5 )
% 9.00/5.12 = zero_zero_real ) )
% 9.00/5.12 => ( Y5 = X5 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9142_cos__two__le__zero,axiom,
% 9.00/5.12 ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9143_cos__double__less__one,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X2 ) ) @ one_one_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9144_vebt__buildup_Osimps_I3_J,axiom,
% 9.00/5.12 ! [Va: nat] :
% 9.00/5.12 ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.12 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.12 => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9145_vebt__buildup_Opelims,axiom,
% 9.00/5.12 ! [X2: nat,Y3: vEBT_VEBT] :
% 9.00/5.12 ( ( ( vEBT_vebt_buildup @ X2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X2 )
% 9.00/5.12 => ( ( ( X2 = zero_zero_nat )
% 9.00/5.12 => ( ( Y3
% 9.00/5.12 = ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.12 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 9.00/5.12 => ( ( ( X2
% 9.00/5.12 = ( suc @ zero_zero_nat ) )
% 9.00/5.12 => ( ( Y3
% 9.00/5.12 = ( vEBT_Leaf @ $false @ $false ) )
% 9.00/5.12 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 9.00/5.12 => ~ ! [Va2: nat] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va2 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.00/5.12 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9146_Maclaurin__sin__expansion3,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ X2 )
% 9.00/5.12 & ( ( sin_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9147_Maclaurin__sin__expansion4,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ X2 )
% 9.00/5.12 & ( ( sin_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9148_Maclaurin__sin__expansion2,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( sin_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9149_pi__gt__zero,axiom,
% 9.00/5.12 ord_less_real @ zero_zero_real @ pi ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9150_pi__not__less__zero,axiom,
% 9.00/5.12 ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9151_sin__gt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ pi )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9152_pi__ge__zero,axiom,
% 9.00/5.12 ord_less_eq_real @ zero_zero_real @ pi ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9153_cos__inj__pi,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.00/5.12 => ( ( ( cos_real @ X2 )
% 9.00/5.12 = ( cos_real @ Y3 ) )
% 9.00/5.12 => ( X2 = Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9154_cos__mono__le__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9155_cos__monotone__0__pi__le,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ord_less_eq_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9156_sin__ge__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9157_cos__monotone__0__pi,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9158_cos__mono__less__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.00/5.12 => ( ( ord_less_real @ ( cos_real @ X2 ) @ ( cos_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9159_sin__eq__0__pi,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ pi )
% 9.00/5.12 => ( ( ( sin_real @ X2 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 => ( X2 = zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9160_cos__monotone__minus__pi__0_H,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9161_sincos__principal__value,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ? [Y4: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y4 )
% 9.00/5.12 & ( ord_less_eq_real @ Y4 @ pi )
% 9.00/5.12 & ( ( sin_real @ Y4 )
% 9.00/5.12 = ( sin_real @ X2 ) )
% 9.00/5.12 & ( ( cos_real @ Y4 )
% 9.00/5.12 = ( cos_real @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9162_sin__zero__pi__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ pi )
% 9.00/5.12 => ( ( ( sin_real @ X2 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 = ( X2 = zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9163_pi__less__4,axiom,
% 9.00/5.12 ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9164_pi__ge__two,axiom,
% 9.00/5.12 ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9165_cos__monotone__minus__pi__0,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9166_cos__total,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ pi )
% 9.00/5.12 & ( ( cos_real @ X5 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 & ! [Y5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ zero_zero_real @ Y5 )
% 9.00/5.12 & ( ord_less_eq_real @ Y5 @ pi )
% 9.00/5.12 & ( ( cos_real @ Y5 )
% 9.00/5.12 = Y3 ) )
% 9.00/5.12 => ( Y5 = X5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9167_pi__half__less__two,axiom,
% 9.00/5.12 ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9168_pi__half__le__two,axiom,
% 9.00/5.12 ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9169_sin__pi__divide__n__ge__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( N != zero_zero_nat )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9170_pi__half__gt__zero,axiom,
% 9.00/5.12 ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9171_cos__gt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9172_sin__gt__zero2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9173_sin__lt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ pi @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.00/5.12 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9174_pi__half__ge__zero,axiom,
% 9.00/5.12 ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9175_m2pi__less__pi,axiom,
% 9.00/5.12 ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9176_sin__inj__pi,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ( sin_real @ X2 )
% 9.00/5.12 = ( sin_real @ Y3 ) )
% 9.00/5.12 => ( X2 = Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9177_sin__mono__le__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9178_sin__monotone__2pi__le,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9179_arctan__ubound,axiom,
% 9.00/5.12 ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9180_sin__le__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ pi @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9181_minus__pi__half__less__zero,axiom,
% 9.00/5.12 ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9182_cos__gt__zero__pi,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9183_sin__less__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ord_less_real @ ( sin_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9184_cos__ge__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9185_sin__mono__less__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( sin_real @ X2 ) @ ( sin_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9186_sin__monotone__2pi,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9187_sin__total,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( sin_real @ X5 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 & ! [Y5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.00/5.12 & ( ord_less_eq_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( sin_real @ Y5 )
% 9.00/5.12 = Y3 ) )
% 9.00/5.12 => ( Y5 = X5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9188_arctan__lbound,axiom,
% 9.00/5.12 ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9189_arctan__bounded,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.00/5.12 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9190_sin__pi__divide__n__gt__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9191_sincos__total__pi,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = one_one_real )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ pi )
% 9.00/5.12 & ( X2
% 9.00/5.12 = ( cos_real @ T6 ) )
% 9.00/5.12 & ( Y3
% 9.00/5.12 = ( sin_real @ T6 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9192_sincos__total__pi__half,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = one_one_real )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( X2
% 9.00/5.12 = ( cos_real @ T6 ) )
% 9.00/5.12 & ( Y3
% 9.00/5.12 = ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9193_sincos__total__2pi__le,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = one_one_real )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.00/5.12 & ( X2
% 9.00/5.12 = ( cos_real @ T6 ) )
% 9.00/5.12 & ( Y3
% 9.00/5.12 = ( sin_real @ T6 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9194_cos__zero__lemma,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( cos_real @ X2 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 => ? [N3: nat] :
% 9.00/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.00/5.12 & ( X2
% 9.00/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9195_sin__zero__lemma,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( sin_real @ X2 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 => ? [N3: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
% 9.00/5.12 & ( X2
% 9.00/5.12 = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9196_sincos__total__2pi,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = one_one_real )
% 9.00/5.12 => ~ ! [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.00/5.12 => ( ( X2
% 9.00/5.12 = ( cos_real @ T6 ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 != ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9197_Maclaurin__cos__expansion2,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ X2 )
% 9.00/5.12 & ( ( cos_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9198_Maclaurin__minus__cos__expansion,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ X2 @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ zero_zero_real )
% 9.00/5.12 & ( ( cos_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9199_Maclaurin__cos__expansion,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( cos_real @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( 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 @ X2 @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9200_complex__unimodular__polar,axiom,
% 9.00/5.12 ! [Z: complex] :
% 9.00/5.12 ( ( ( real_V1022390504157884413omplex @ Z )
% 9.00/5.12 = one_one_real )
% 9.00/5.12 => ~ ! [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.00/5.12 => ( Z
% 9.00/5.12 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9201_cos__coeff__0,axiom,
% 9.00/5.12 ( ( cos_coeff @ zero_zero_nat )
% 9.00/5.12 = one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9202_lemma__tan__total,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ord_less_real @ Y3 @ ( tan_real @ X5 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9203_tan__gt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9204_lemma__tan__total1,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ? [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ X5 )
% 9.00/5.12 = Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9205_tan__mono__lt__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9206_tan__monotone_H,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.12 = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9207_tan__monotone,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9208_tan__total,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ? [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ X5 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 & ! [Y5: real] :
% 9.00/5.12 ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y5 )
% 9.00/5.12 & ( ord_less_real @ Y5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ Y5 )
% 9.00/5.12 = Y3 ) )
% 9.00/5.12 => ( Y5 = X5 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9209_tan__pos__pi2__le,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9210_tan__total__pos,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ X5 )
% 9.00/5.12 = Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9211_tan__less__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ord_less_real @ ( tan_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9212_tan__mono__le__eq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9213_tan__mono__le,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( tan_real @ X2 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9214_tan__bound__pi2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X2 ) ) @ one_one_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9215_arctan,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.00/5.12 & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ ( arctan @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9216_arctan__tan,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( arctan @ ( tan_real @ X2 ) )
% 9.00/5.12 = X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9217_arctan__unique,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ( tan_real @ X2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( arctan @ Y3 )
% 9.00/5.12 = X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9218_tan__total__pi4,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ? [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z4 )
% 9.00/5.12 & ( ord_less_real @ Z4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( tan_real @ Z4 )
% 9.00/5.12 = X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9219_sin__tan,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( sin_real @ X2 )
% 9.00/5.12 = ( divide_divide_real @ ( tan_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9220_cos__tan,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( cos_real @ X2 )
% 9.00/5.12 = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9221_Maclaurin__sin__bound,axiom,
% 9.00/5.12 ! [X2: real,N: nat] :
% 9.00/5.12 ( ord_less_eq_real
% 9.00/5.12 @ ( abs_abs_real
% 9.00/5.12 @ ( minus_minus_real @ ( sin_real @ X2 )
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) ) ) )
% 9.00/5.12 @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X2 ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9222_cot__less__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ord_less_real @ ( cot_real @ X2 ) @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9223_real__sqrt__less__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9224_real__sqrt__le__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9225_real__sqrt__gt__0__iff,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9226_real__sqrt__lt__0__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 9.00/5.12 = ( ord_less_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9227_real__sqrt__ge__0__iff,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9228_real__sqrt__le__0__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ zero_zero_real )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ zero_zero_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9229_real__sqrt__lt__1__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( sqrt @ X2 ) @ one_one_real )
% 9.00/5.12 = ( ord_less_real @ X2 @ one_one_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9230_real__sqrt__gt__1__iff,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9231_real__sqrt__le__1__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ one_one_real )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ one_one_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9232_real__sqrt__ge__1__iff,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9233_real__sqrt__pow2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.12 = X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9234_real__sqrt__pow2__iff,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ( power_power_real @ ( sqrt @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.12 = X2 )
% 9.00/5.12 = ( ord_less_eq_real @ zero_zero_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9235_real__sqrt__le__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9236_real__sqrt__less__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.12 => ( ord_less_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9237_sqrt__divide__self__eq,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( divide_divide_real @ ( sqrt @ X2 ) @ X2 )
% 9.00/5.12 = ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9238_real__sqrt__gt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9239_real__sqrt__ge__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9240_real__sqrt__eq__zero__cancel,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( sqrt @ X2 )
% 9.00/5.12 = zero_zero_real )
% 9.00/5.12 => ( X2 = zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9241_real__sqrt__ge__one,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9242_real__inv__sqrt__pow2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.12 = ( inverse_inverse_real @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9243_real__div__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( divide_divide_real @ X2 @ ( sqrt @ X2 ) )
% 9.00/5.12 = ( sqrt @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9244_sqrt__add__le__add__sqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X2 @ Y3 ) ) @ ( plus_plus_real @ ( sqrt @ X2 ) @ ( sqrt @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9245_le__real__sqrt__sumsq,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X2 @ X2 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9246_inverse__powr,axiom,
% 9.00/5.12 ! [Y3: real,A2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( powr_real @ ( inverse_inverse_real @ Y3 ) @ A2 )
% 9.00/5.12 = ( inverse_inverse_real @ ( powr_real @ Y3 @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9247_sqrt2__less__2,axiom,
% 9.00/5.12 ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9248_forall__pos__mono__1,axiom,
% 9.00/5.12 ! [P: real > $o,E2: real] :
% 9.00/5.12 ( ! [D6: real,E: real] :
% 9.00/5.12 ( ( ord_less_real @ D6 @ E )
% 9.00/5.12 => ( ( P @ D6 )
% 9.00/5.12 => ( P @ E ) ) )
% 9.00/5.12 => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.12 => ( P @ E2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9249_forall__pos__mono,axiom,
% 9.00/5.12 ! [P: real > $o,E2: real] :
% 9.00/5.12 ( ! [D6: real,E: real] :
% 9.00/5.12 ( ( ord_less_real @ D6 @ E )
% 9.00/5.12 => ( ( P @ D6 )
% 9.00/5.12 => ( P @ E ) ) )
% 9.00/5.12 => ( ! [N3: nat] :
% 9.00/5.12 ( ( N3 != zero_zero_nat )
% 9.00/5.12 => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.12 => ( P @ E2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9250_real__arch__inverse,axiom,
% 9.00/5.12 ! [E2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.00/5.12 = ( ? [N2: nat] :
% 9.00/5.12 ( ( N2 != zero_zero_nat )
% 9.00/5.12 & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.00/5.12 & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ E2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9251_ln__inverse,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ln_ln_real @ ( inverse_inverse_real @ X2 ) )
% 9.00/5.12 = ( uminus_uminus_real @ ( ln_ln_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9252_real__less__rsqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 9.00/5.12 => ( ord_less_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9253_real__le__rsqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ X2 @ ( sqrt @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9254_sqrt__le__D,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9255_log__inverse,axiom,
% 9.00/5.12 ! [A2: real,X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ A2 )
% 9.00/5.12 => ( ( A2 != one_one_real )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( log @ A2 @ ( inverse_inverse_real @ X2 ) )
% 9.00/5.12 = ( uminus_uminus_real @ ( log @ A2 @ X2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9256_real__le__lsqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9257_real__sqrt__unique,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] :
% 9.00/5.12 ( ( ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.12 = X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( sqrt @ X2 )
% 9.00/5.12 = Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9258_lemma__real__divide__sqrt__less,axiom,
% 9.00/5.12 ! [U: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ U )
% 9.00/5.12 => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9259_real__sqrt__sum__squares__ge1,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9260_real__sqrt__sum__squares__ge2,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] : ( ord_less_eq_real @ Y3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9261_real__sqrt__sum__squares__triangle__ineq,axiom,
% 9.00/5.12 ! [A2: real,C: real,B3: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A2 @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B3 @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9262_sqrt__ge__absD,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ Y3 ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9263_exp__plus__inverse__exp,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9264_real__less__lsqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( sqrt @ X2 ) @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9265_sqrt__sum__squares__le__sum,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9266_real__sqrt__ge__abs1,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9267_real__sqrt__ge__abs2,axiom,
% 9.00/5.12 ! [Y3: real,X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9268_sqrt__sum__squares__le__sum__abs,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9269_ln__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ln_ln_real @ ( sqrt @ X2 ) )
% 9.00/5.12 = ( divide_divide_real @ ( ln_ln_real @ X2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9270_plus__inverse__ge__2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9271_arsinh__real__aux,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9272_real__sqrt__power__even,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( power_power_real @ ( sqrt @ X2 ) @ N )
% 9.00/5.12 = ( power_power_real @ X2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9273_real__sqrt__sum__squares__mult__ge__zero,axiom,
% 9.00/5.12 ! [X2: real,Y3: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9274_arith__geo__mean__sqrt,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X2 @ Y3 ) ) @ ( divide_divide_real @ ( plus_plus_real @ X2 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9275_powr__half__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 = ( sqrt @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9276_real__le__x__sinh,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ord_less_eq_real @ X2 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9277_real__le__abs__sinh,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X2 ) @ ( inverse_inverse_real @ ( exp_real @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9278_cos__x__y__le__one,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X2 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9279_real__sqrt__sum__squares__less,axiom,
% 9.00/5.12 ! [X2: real,U: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9280_arcosh__real__def,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ one_one_real @ X2 )
% 9.00/5.12 => ( ( arcosh_real @ X2 )
% 9.00/5.12 = ( ln_ln_real @ ( plus_plus_real @ X2 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9281_powr__real__of__int,axiom,
% 9.00/5.12 ! [X2: real,N: int] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ zero_zero_int @ N )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 9.00/5.12 = ( power_power_real @ X2 @ ( nat2 @ N ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ zero_zero_int @ N )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 9.00/5.12 = ( inverse_inverse_real @ ( power_power_real @ X2 @ ( nat2 @ ( uminus_uminus_int @ N ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9282_sinh__ln__real,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( sinh_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.12 = ( divide_divide_real @ ( minus_minus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9283_cot__gt__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( cot_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9284_sqrt__sum__squares__half__less,axiom,
% 9.00/5.12 ! [X2: real,U: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ X2 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9285_sin__cos__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X2 ) )
% 9.00/5.12 => ( ( sin_real @ X2 )
% 9.00/5.12 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9286_cos__arcsin,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( cos_real @ ( arcsin @ X2 ) )
% 9.00/5.12 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9287_sin__arccos__abs,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( sin_real @ ( arccos @ Y3 ) )
% 9.00/5.12 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9288_sin__arccos,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( sin_real @ ( arccos @ X2 ) )
% 9.00/5.12 = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9289_le__arcsin__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( arcsin @ X2 ) )
% 9.00/5.12 = ( ord_less_eq_real @ ( sin_real @ Y3 ) @ X2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9290_cos__arccos,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( cos_real @ ( arccos @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9291_sin__arcsin,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9292_cosh__real__pos,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9293_arcosh__cosh__real,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( arcosh_real @ ( cosh_real @ X2 ) )
% 9.00/5.12 = X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9294_cosh__real__nonpos__le__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9295_cosh__real__nonneg__le__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9296_cosh__real__nonneg,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9297_cosh__real__ge__1,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9298_sinh__less__cosh__real,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9299_sinh__le__cosh__real,axiom,
% 9.00/5.12 ! [X2: real] : ( ord_less_eq_real @ ( sinh_real @ X2 ) @ ( cosh_real @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9300_cosh__real__strict__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.12 => ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9301_cosh__real__nonneg__less__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9302_cosh__real__nonpos__less__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_real @ ( cosh_real @ X2 ) @ ( cosh_real @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9303_arccos__le__arccos,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9304_arccos__eq__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 & ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real ) )
% 9.00/5.12 => ( ( ( arccos @ X2 )
% 9.00/5.12 = ( arccos @ Y3 ) )
% 9.00/5.12 = ( X2 = Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9305_arccos__le__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9306_arcsin__minus,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( arcsin @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.12 = ( uminus_uminus_real @ ( arcsin @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9307_arcsin__le__arcsin,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9308_arcsin__eq__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( ( arcsin @ X2 )
% 9.00/5.12 = ( arcsin @ Y3 ) )
% 9.00/5.12 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9309_arcsin__le__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9310_arccos__lbound,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9311_arccos__less__arccos,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9312_arccos__less__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_real @ ( arccos @ X2 ) @ ( arccos @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ Y3 @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9313_arccos__ubound,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9314_arccos__cos,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ( arccos @ ( cos_real @ X2 ) )
% 9.00/5.12 = X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9315_arcsin__less__arcsin,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9316_arcsin__less__mono,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( ord_less_real @ ( arcsin @ X2 ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 = ( ord_less_real @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9317_cos__arccos__abs,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.00/5.12 => ( ( cos_real @ ( arccos @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9318_arccos__cos__eq__abs,axiom,
% 9.00/5.12 ! [Theta: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
% 9.00/5.12 => ( ( arccos @ ( cos_real @ Theta ) )
% 9.00/5.12 = ( abs_abs_real @ Theta ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9319_arccos__lt__bounded,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.00/5.12 & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9320_arccos__bounded,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9321_sin__arccos__nonzero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( sin_real @ ( arccos @ X2 ) )
% 9.00/5.12 != zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9322_arccos__cos2,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 9.00/5.12 => ( ( arccos @ ( cos_real @ X2 ) )
% 9.00/5.12 = ( uminus_uminus_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9323_arccos__minus,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.12 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9324_cos__arcsin__nonzero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( cos_real @ ( arcsin @ X2 ) )
% 9.00/5.12 != zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9325_arccos,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arccos @ Y3 ) @ pi )
% 9.00/5.12 & ( ( cos_real @ ( arccos @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9326_arccos__minus__abs,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( ( arccos @ ( uminus_uminus_real @ X2 ) )
% 9.00/5.12 = ( minus_minus_real @ pi @ ( arccos @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9327_arccos__le__pi2,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( arccos @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9328_cosh__ln__real,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( cosh_real @ ( ln_ln_real @ X2 ) )
% 9.00/5.12 = ( divide_divide_real @ ( plus_plus_real @ X2 @ ( inverse_inverse_real @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9329_arcsin__lt__bounded,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9330_arcsin__lbound,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9331_arcsin__ubound,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9332_arcsin__bounded,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9333_arcsin__sin,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( arcsin @ ( sin_real @ X2 ) )
% 9.00/5.12 = X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9334_arcsin,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9335_arcsin__pi,axiom,
% 9.00/5.12 ! [Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arcsin @ Y3 ) @ pi )
% 9.00/5.12 & ( ( sin_real @ ( arcsin @ Y3 ) )
% 9.00/5.12 = Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9336_arcsin__le__iff,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( arcsin @ X2 ) @ Y3 )
% 9.00/5.12 = ( ord_less_eq_real @ X2 @ ( sin_real @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9337_int__ge__less__than2__def,axiom,
% 9.00/5.12 ( int_ge_less_than2
% 9.00/5.12 = ( ^ [D5: int] :
% 9.00/5.12 ( collec213857154873943460nt_int
% 9.00/5.12 @ ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [Z8: int,Z2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ D5 @ Z2 )
% 9.00/5.12 & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9338_int__ge__less__than__def,axiom,
% 9.00/5.12 ( int_ge_less_than
% 9.00/5.12 = ( ^ [D5: int] :
% 9.00/5.12 ( collec213857154873943460nt_int
% 9.00/5.12 @ ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [Z8: int,Z2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ D5 @ Z8 )
% 9.00/5.12 & ( ord_less_int @ Z8 @ Z2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9339_set__decode__0,axiom,
% 9.00/5.12 ! [X2: nat] :
% 9.00/5.12 ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X2 ) )
% 9.00/5.12 = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9340_arctan__def,axiom,
% 9.00/5.12 ( arctan
% 9.00/5.12 = ( ^ [Y: real] :
% 9.00/5.12 ( the_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.00/5.12 & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( tan_real @ X )
% 9.00/5.12 = Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9341_set__decode__zero,axiom,
% 9.00/5.12 ( ( nat_set_decode @ zero_zero_nat )
% 9.00/5.12 = bot_bot_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9342_set__encode__inverse,axiom,
% 9.00/5.12 ! [A3: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ A3 )
% 9.00/5.12 => ( ( nat_set_decode @ ( nat_set_encode @ A3 ) )
% 9.00/5.12 = A3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9343_finite__set__decode,axiom,
% 9.00/5.12 ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9344_ln__neg__is__const,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( ( ln_ln_real @ X2 )
% 9.00/5.12 = ( the_real
% 9.00/5.12 @ ^ [X: real] : $false ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9345_subset__decode__imp__le,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
% 9.00/5.12 => ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9346_arccos__def,axiom,
% 9.00/5.12 ( arccos
% 9.00/5.12 = ( ^ [Y: real] :
% 9.00/5.12 ( the_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.00/5.12 & ( ord_less_eq_real @ X @ pi )
% 9.00/5.12 & ( ( cos_real @ X )
% 9.00/5.12 = Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9347_pi__half,axiom,
% 9.00/5.12 ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 = ( the_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.00/5.12 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 & ( ( cos_real @ X )
% 9.00/5.12 = zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9348_pi__def,axiom,
% 9.00/5.12 ( pi
% 9.00/5.12 = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.00/5.12 @ ( the_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X )
% 9.00/5.12 & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.00/5.12 & ( ( cos_real @ X )
% 9.00/5.12 = zero_zero_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9349_arcsin__def,axiom,
% 9.00/5.12 ( arcsin
% 9.00/5.12 = ( ^ [Y: real] :
% 9.00/5.12 ( the_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.00/5.12 & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.00/5.12 & ( ( sin_real @ X )
% 9.00/5.12 = Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9350_num_Osize__gen_I3_J,axiom,
% 9.00/5.12 ! [X32: num] :
% 9.00/5.12 ( ( size_num @ ( bit1 @ X32 ) )
% 9.00/5.12 = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9351_modulo__int__unfold,axiom,
% 9.00/5.12 ! [L: int,K: int,N: nat,M: nat] :
% 9.00/5.12 ( ( ( ( ( sgn_sgn_int @ L )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( ( sgn_sgn_int @ K )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( N = zero_zero_nat ) )
% 9.00/5.12 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 9.00/5.12 & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( ( sgn_sgn_int @ K )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( N = zero_zero_nat ) )
% 9.00/5.12 => ( ( ( ( sgn_sgn_int @ K )
% 9.00/5.12 = ( sgn_sgn_int @ L ) )
% 9.00/5.12 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) )
% 9.00/5.12 & ( ( ( sgn_sgn_int @ K )
% 9.00/5.12 != ( sgn_sgn_int @ L ) )
% 9.00/5.12 => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = ( times_times_int @ ( sgn_sgn_int @ L )
% 9.00/5.12 @ ( minus_minus_int
% 9.00/5.12 @ ( semiri1314217659103216013at_int
% 9.00/5.12 @ ( times_times_nat @ N
% 9.00/5.12 @ ( zero_n2687167440665602831ol_nat
% 9.00/5.12 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) )
% 9.00/5.12 @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9352_num_Osize__gen_I2_J,axiom,
% 9.00/5.12 ! [X22: num] :
% 9.00/5.12 ( ( size_num @ ( bit0 @ X22 ) )
% 9.00/5.12 = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9353_take__bit__of__Suc__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se2925701944663578781it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9354_Suc__0__mod__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( zero_n2687167440665602831ol_nat
% 9.00/5.12 @ ( N
% 9.00/5.12 != ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9355_take__bit__tightened__less__eq__nat,axiom,
% 9.00/5.12 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9356_take__bit__nat__less__eq__self,axiom,
% 9.00/5.12 ! [N: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9357_take__bit__nat__eq,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) )
% 9.00/5.12 = ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9358_nat__take__bit__eq,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.00/5.12 = ( bit_se2925701944663578781it_nat @ N @ ( nat2 @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9359_take__bit__tightened__less__eq__int,axiom,
% 9.00/5.12 ! [M: nat,N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9360_take__bit__nonnegative,axiom,
% 9.00/5.12 ! [N: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9361_take__bit__int__less__eq__self__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9362_not__take__bit__negative,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ zero_zero_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9363_take__bit__int__greater__self__iff,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.00/5.12 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9364_take__bit__nat__eq__self__iff,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.00/5.12 = M )
% 9.00/5.12 = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9365_take__bit__nat__less__exp,axiom,
% 9.00/5.12 ! [N: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9366_take__bit__nat__eq__self,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ( bit_se2925701944663578781it_nat @ N @ M )
% 9.00/5.12 = M ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9367_take__bit__int__less__exp,axiom,
% 9.00/5.12 ! [N: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9368_num_Osize__gen_I1_J,axiom,
% 9.00/5.12 ( ( size_num @ one )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9369_take__bit__nat__less__self__iff,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M ) @ M )
% 9.00/5.12 = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9370_take__bit__int__greater__eq__self__iff,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N @ K ) )
% 9.00/5.12 = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9371_take__bit__int__less__self__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ K )
% 9.00/5.12 = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9372_take__bit__int__eq__self__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.00/5.12 = K )
% 9.00/5.12 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9373_take__bit__int__eq__self,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ( bit_se2923211474154528505it_int @ N @ K )
% 9.00/5.12 = K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9374_floor__real__def,axiom,
% 9.00/5.12 ( archim6058952711729229775r_real
% 9.00/5.12 = ( ^ [X: real] :
% 9.00/5.12 ( the_int
% 9.00/5.12 @ ^ [Z2: int] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X )
% 9.00/5.12 & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9375_take__bit__int__less__eq,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9376_take__bit__int__greater__eq,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.12 => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9377_take__bit__minus__small__eq,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ K )
% 9.00/5.12 => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K ) )
% 9.00/5.12 = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9378_divide__int__unfold,axiom,
% 9.00/5.12 ! [L: int,K: int,N: nat,M: nat] :
% 9.00/5.12 ( ( ( ( ( sgn_sgn_int @ L )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( ( sgn_sgn_int @ K )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( N = zero_zero_nat ) )
% 9.00/5.12 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = zero_zero_int ) )
% 9.00/5.12 & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( ( sgn_sgn_int @ K )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 | ( N = zero_zero_nat ) )
% 9.00/5.12 => ( ( ( ( sgn_sgn_int @ K )
% 9.00/5.12 = ( sgn_sgn_int @ L ) )
% 9.00/5.12 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N ) ) ) )
% 9.00/5.12 & ( ( ( sgn_sgn_int @ K )
% 9.00/5.12 != ( sgn_sgn_int @ L ) )
% 9.00/5.12 => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
% 9.00/5.12 = ( uminus_uminus_int
% 9.00/5.12 @ ( semiri1314217659103216013at_int
% 9.00/5.12 @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N )
% 9.00/5.12 @ ( zero_n2687167440665602831ol_nat
% 9.00/5.12 @ ~ ( dvd_dvd_nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9379_flip__bit__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N @ K ) )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9380_flip__bit__negative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.12 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9381_and__int_Opsimps,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 9.00/5.12 => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.00/5.12 = ( uminus_uminus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.00/5.12 = ( plus_plus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 9.00/5.12 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9382_and__int_Opelims,axiom,
% 9.00/5.12 ! [X2: int,Xa2: int,Y3: int] :
% 9.00/5.12 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 9.00/5.12 => ~ ( ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( uminus_uminus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( plus_plus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.00/5.12 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.00/5.12 => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9383_floor__rat__def,axiom,
% 9.00/5.12 ( archim3151403230148437115or_rat
% 9.00/5.12 = ( ^ [X: rat] :
% 9.00/5.12 ( the_int
% 9.00/5.12 @ ^ [Z2: int] :
% 9.00/5.12 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X )
% 9.00/5.12 & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9384_and__int_Osimps,axiom,
% 9.00/5.12 ( bit_se725231765392027082nd_int
% 9.00/5.12 = ( ^ [K3: int,L2: int] :
% 9.00/5.12 ( if_int
% 9.00/5.12 @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 @ ( uminus_uminus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 9.00/5.12 @ ( plus_plus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.00/5.12 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9385_and__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
% 9.00/5.12 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9386_and__negative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 9.00/5.12 = ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.12 & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9387_abs__rat__def,axiom,
% 9.00/5.12 ( abs_abs_rat
% 9.00/5.12 = ( ^ [A4: rat] : ( if_rat @ ( ord_less_rat @ A4 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A4 ) @ A4 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9388_sgn__rat__def,axiom,
% 9.00/5.12 ( sgn_sgn_rat
% 9.00/5.12 = ( ^ [A4: rat] : ( if_rat @ ( A4 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A4 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9389_less__eq__rat__def,axiom,
% 9.00/5.12 ( ord_less_eq_rat
% 9.00/5.12 = ( ^ [X: rat,Y: rat] :
% 9.00/5.12 ( ( ord_less_rat @ X @ Y )
% 9.00/5.12 | ( X = Y ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9390_obtain__pos__sum,axiom,
% 9.00/5.12 ! [R2: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.00/5.12 => ~ ! [S3: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ S3 )
% 9.00/5.12 => ! [T6: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 9.00/5.12 => ( R2
% 9.00/5.12 != ( plus_plus_rat @ S3 @ T6 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9391_AND__lower,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9392_AND__upper1,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9393_AND__upper2,axiom,
% 9.00/5.12 ! [Y3: int,X2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9394_AND__upper1_H,axiom,
% 9.00/5.12 ! [Y3: int,Z: int,Ya: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9395_AND__upper2_H,axiom,
% 9.00/5.12 ! [Y3: int,Z: int,X2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( ord_less_eq_int @ Y3 @ Z )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9396_AND__upper2_H_H,axiom,
% 9.00/5.12 ! [Y3: int,Z: int,X2: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( ord_less_int @ Y3 @ Z )
% 9.00/5.12 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X2 @ Y3 ) @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9397_AND__upper1_H_H,axiom,
% 9.00/5.12 ! [Y3: int,Z: int,Ya: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( ord_less_int @ Y3 @ Z )
% 9.00/5.12 => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9398_and__less__eq,axiom,
% 9.00/5.12 ! [L: int,K: int] :
% 9.00/5.12 ( ( ord_less_int @ L @ zero_zero_int )
% 9.00/5.12 => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9399_and__int_Oelims,axiom,
% 9.00/5.12 ! [X2: int,Xa2: int,Y3: int] :
% 9.00/5.12 ( ( ( bit_se725231765392027082nd_int @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( uminus_uminus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( ( member_int @ X2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.00/5.12 & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( plus_plus_int
% 9.00/5.12 @ ( zero_n2684676970156552555ol_int
% 9.00/5.12 @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X2 )
% 9.00/5.12 & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.00/5.12 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9400_normalize__negative,axiom,
% 9.00/5.12 ! [Q3: int,P6: int] :
% 9.00/5.12 ( ( ord_less_int @ Q3 @ zero_zero_int )
% 9.00/5.12 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 9.00/5.12 = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P6 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9401_xor__Suc__0__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.00/5.12 @ ( zero_n2687167440665602831ol_nat
% 9.00/5.12 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9402_Suc__0__xor__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( minus_minus_nat @ ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.00/5.12 @ ( zero_n2687167440665602831ol_nat
% 9.00/5.12 @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9403_and__nat__numerals_I3_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9404_and__nat__numerals_I1_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9405_and__nat__numerals_I4_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = one_one_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9406_and__nat__numerals_I2_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.00/5.12 = one_one_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9407_and__Suc__0__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9408_Suc__0__and__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( modulo_modulo_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9409_xor__nat__numerals_I1_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9410_xor__nat__numerals_I2_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9411_xor__nat__numerals_I3_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9412_xor__nat__numerals_I4_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit0 @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9413_normalize__denom__pos,axiom,
% 9.00/5.12 ! [R2: product_prod_int_int,P6: int,Q3: int] :
% 9.00/5.12 ( ( ( normalize @ R2 )
% 9.00/5.12 = ( product_Pair_int_int @ P6 @ Q3 ) )
% 9.00/5.12 => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9414_and__nat__unfold,axiom,
% 9.00/5.12 ( bit_se727722235901077358nd_nat
% 9.00/5.12 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.12 ( if_nat
% 9.00/5.12 @ ( ( M2 = zero_zero_nat )
% 9.00/5.12 | ( N2 = zero_zero_nat ) )
% 9.00/5.12 @ zero_zero_nat
% 9.00/5.12 @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9415_xor__nat__unfold,axiom,
% 9.00/5.12 ( bit_se6528837805403552850or_nat
% 9.00/5.12 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9416_horner__sum__of__bool__2__less,axiom,
% 9.00/5.12 ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9417_Least__eq__0,axiom,
% 9.00/5.12 ! [P: nat > $o] :
% 9.00/5.12 ( ( P @ zero_zero_nat )
% 9.00/5.12 => ( ( ord_Least_nat @ P )
% 9.00/5.12 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9418_xor__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
% 9.00/5.12 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9419_xor__negative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 9.00/5.12 = ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.12 != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9420_XOR__lower,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9421_Least__Suc2,axiom,
% 9.00/5.12 ! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
% 9.00/5.12 ( ( P @ N )
% 9.00/5.12 => ( ( Q @ M )
% 9.00/5.12 => ( ~ ( P @ zero_zero_nat )
% 9.00/5.12 => ( ! [K2: nat] :
% 9.00/5.12 ( ( P @ ( suc @ K2 ) )
% 9.00/5.12 = ( Q @ K2 ) )
% 9.00/5.12 => ( ( ord_Least_nat @ P )
% 9.00/5.12 = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9422_Least__Suc,axiom,
% 9.00/5.12 ! [P: nat > $o,N: nat] :
% 9.00/5.12 ( ( P @ N )
% 9.00/5.12 => ( ~ ( P @ zero_zero_nat )
% 9.00/5.12 => ( ( ord_Least_nat @ P )
% 9.00/5.12 = ( suc
% 9.00/5.12 @ ( ord_Least_nat
% 9.00/5.12 @ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9423_fact__eq__fact__times,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( ( semiri1408675320244567234ct_nat @ M )
% 9.00/5.12 = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
% 9.00/5.12 @ ( groups708209901874060359at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9424_fact__div__fact,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N ) )
% 9.00/5.12 = ( groups708209901874060359at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9425_XOR__upper,axiom,
% 9.00/5.12 ! [X2: int,N: nat,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9426_or__nat__unfold,axiom,
% 9.00/5.12 ( bit_se1412395901928357646or_nat
% 9.00/5.12 = ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( if_nat @ ( N2 = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9427_Sum__Ico__nat,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ M @ N ) )
% 9.00/5.12 = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9428_finite__atLeastLessThan,axiom,
% 9.00/5.12 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9429_card__atLeastLessThan,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
% 9.00/5.12 = ( minus_minus_nat @ U @ L ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9430_atLeastLessThan__singleton,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 9.00/5.12 = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9431_or__nat__numerals_I4_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9432_or__nat__numerals_I2_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9433_or__nat__numerals_I1_J,axiom,
% 9.00/5.12 ! [Y3: num] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9434_or__nat__numerals_I3_J,axiom,
% 9.00/5.12 ! [X2: num] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X2 ) ) @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( numeral_numeral_nat @ ( bit1 @ X2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9435_all__nat__less__eq,axiom,
% 9.00/5.12 ! [N: nat,P: nat > $o] :
% 9.00/5.12 ( ( ! [M2: nat] :
% 9.00/5.12 ( ( ord_less_nat @ M2 @ N )
% 9.00/5.12 => ( P @ M2 ) ) )
% 9.00/5.12 = ( ! [X: nat] :
% 9.00/5.12 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.00/5.12 => ( P @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9436_ex__nat__less__eq,axiom,
% 9.00/5.12 ! [N: nat,P: nat > $o] :
% 9.00/5.12 ( ( ? [M2: nat] :
% 9.00/5.12 ( ( ord_less_nat @ M2 @ N )
% 9.00/5.12 & ( P @ M2 ) ) )
% 9.00/5.12 = ( ? [X: nat] :
% 9.00/5.12 ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.00/5.12 & ( P @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9437_atLeastLessThanSuc__atLeastAtMost,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 9.00/5.12 = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9438_lessThan__atLeast0,axiom,
% 9.00/5.12 ( set_ord_lessThan_nat
% 9.00/5.12 = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9439_atLeastLessThan0,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 9.00/5.12 = bot_bot_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9440_atLeast0__lessThan__Suc,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9441_prod__Suc__fact,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.00/5.12 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9442_prod__Suc__Suc__fact,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
% 9.00/5.12 = ( semiri1408675320244567234ct_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9443_prod__int__eq,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 9.00/5.12 = ( groups1705073143266064639nt_int
% 9.00/5.12 @ ^ [X: int] : X
% 9.00/5.12 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9444_subset__eq__atLeast0__lessThan__finite,axiom,
% 9.00/5.12 ! [N6: set_nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.00/5.12 => ( finite_finite_nat @ N6 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9445_subset__card__intvl__is__intvl,axiom,
% 9.00/5.12 ! [A3: set_nat,K: nat] :
% 9.00/5.12 ( ( ord_less_eq_set_nat @ A3 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) )
% 9.00/5.12 => ( A3
% 9.00/5.12 = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9446_atLeastLessThan__add__Un,axiom,
% 9.00/5.12 ! [I: nat,J: nat,K: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.12 => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 9.00/5.12 = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9447_prod__int__plus__eq,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 9.00/5.12 = ( groups1705073143266064639nt_int
% 9.00/5.12 @ ^ [X: int] : X
% 9.00/5.12 @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9448_atLeastLessThanSuc,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N ) )
% 9.00/5.12 = bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9449_subset__eq__atLeast0__lessThan__card,axiom,
% 9.00/5.12 ! [N6: set_nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
% 9.00/5.12 => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9450_card__sum__le__nat__sum,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ord_less_eq_nat
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S ) ) )
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ S ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9451_atLeast1__lessThan__eq__remove0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9452_Suc__0__or__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9453_or__Suc__0__eq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se1412395901928357646or_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( plus_plus_nat @ N @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9454_sum__power2,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 9.00/5.12 = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9455_Chebyshev__sum__upper__nat,axiom,
% 9.00/5.12 ! [N: nat,A2: nat > nat,B3: nat > nat] :
% 9.00/5.12 ( ! [I2: nat,J2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ I2 @ J2 )
% 9.00/5.12 => ( ( ord_less_nat @ J2 @ N )
% 9.00/5.12 => ( ord_less_eq_nat @ ( A2 @ I2 ) @ ( A2 @ J2 ) ) ) )
% 9.00/5.12 => ( ! [I2: nat,J2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ I2 @ J2 )
% 9.00/5.12 => ( ( ord_less_nat @ J2 @ N )
% 9.00/5.12 => ( ord_less_eq_nat @ ( B3 @ J2 ) @ ( B3 @ I2 ) ) ) )
% 9.00/5.12 => ( ord_less_eq_nat
% 9.00/5.12 @ ( times_times_nat @ N
% 9.00/5.12 @ ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_nat @ ( A2 @ I4 ) @ ( B3 @ I4 ) )
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 9.00/5.12 @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9456_Cauchy__iff2,axiom,
% 9.00/5.12 ( topolo4055970368930404560y_real
% 9.00/5.12 = ( ^ [X8: nat > real] :
% 9.00/5.12 ! [J3: nat] :
% 9.00/5.12 ? [M8: nat] :
% 9.00/5.12 ! [M2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M8 @ M2 )
% 9.00/5.12 => ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M8 @ N2 )
% 9.00/5.12 => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9457_finite__atLeastLessThan__int,axiom,
% 9.00/5.12 ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9458_or__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
% 9.00/5.12 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9459_or__negative__int__iff,axiom,
% 9.00/5.12 ! [K: int,L: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 9.00/5.12 = ( ( ord_less_int @ K @ zero_zero_int )
% 9.00/5.12 | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9460_card__atLeastLessThan__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
% 9.00/5.12 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9461_OR__lower,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9462_or__greater__eq,axiom,
% 9.00/5.12 ! [L: int,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ L )
% 9.00/5.12 => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9463_finite__atLeastZeroLessThan__int,axiom,
% 9.00/5.12 ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9464_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
% 9.00/5.12 = ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9465_card__atLeastZeroLessThan__int,axiom,
% 9.00/5.12 ! [U: int] :
% 9.00/5.12 ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
% 9.00/5.12 = ( nat2 @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9466_OR__upper,axiom,
% 9.00/5.12 ! [X2: int,N: nat,Y3: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_int @ X2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
% 9.00/5.12 => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X2 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9467_VEBT_Osize__gen_I1_J,axiom,
% 9.00/5.12 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.00/5.12 ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.00/5.12 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9468_VEBT_Osize_I3_J,axiom,
% 9.00/5.12 ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.00/5.12 ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.00/5.12 = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9469_upto_Opsimps,axiom,
% 9.00/5.12 ! [I: int,J: int] :
% 9.00/5.12 ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( upto @ I @ J )
% 9.00/5.12 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( upto @ I @ J )
% 9.00/5.12 = nil_int ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9470_upto_Opelims,axiom,
% 9.00/5.12 ! [X2: int,Xa2: int,Y3: list_int] :
% 9.00/5.12 ( ( ( upto @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) )
% 9.00/5.12 => ~ ( ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 9.00/5.12 => ( Y3 = nil_int ) ) )
% 9.00/5.12 => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9471_upto__rec__numeral_I4_J,axiom,
% 9.00/5.12 ! [M: num,N: num] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 = nil_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9472_upto__empty,axiom,
% 9.00/5.12 ! [J: int,I: int] :
% 9.00/5.12 ( ( ord_less_int @ J @ I )
% 9.00/5.12 => ( ( upto @ I @ J )
% 9.00/5.12 = nil_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9473_upto__Nil2,axiom,
% 9.00/5.12 ! [I: int,J: int] :
% 9.00/5.12 ( ( nil_int
% 9.00/5.12 = ( upto @ I @ J ) )
% 9.00/5.12 = ( ord_less_int @ J @ I ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9474_upto__Nil,axiom,
% 9.00/5.12 ! [I: int,J: int] :
% 9.00/5.12 ( ( ( upto @ I @ J )
% 9.00/5.12 = nil_int )
% 9.00/5.12 = ( ord_less_int @ J @ I ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9475_nth__upto,axiom,
% 9.00/5.12 ! [I: int,K: nat,J: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 9.00/5.12 => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 9.00/5.12 = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9476_upto__rec__numeral_I1_J,axiom,
% 9.00/5.12 ! [M: num,N: num] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 = nil_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9477_upto__rec__numeral_I2_J,axiom,
% 9.00/5.12 ! [M: num,N: num] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
% 9.00/5.12 = nil_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9478_upto__rec__numeral_I3_J,axiom,
% 9.00/5.12 ! [M: num,N: num] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N ) )
% 9.00/5.12 = nil_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9479_upto__split2,axiom,
% 9.00/5.12 ! [I: int,J: int,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( ord_less_eq_int @ J @ K )
% 9.00/5.12 => ( ( upto @ I @ K )
% 9.00/5.12 = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9480_upto__split1,axiom,
% 9.00/5.12 ! [I: int,J: int,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( ord_less_eq_int @ J @ K )
% 9.00/5.12 => ( ( upto @ I @ K )
% 9.00/5.12 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9481_upto__rec2,axiom,
% 9.00/5.12 ! [I: int,J: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( upto @ I @ J )
% 9.00/5.12 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9482_upto__split3,axiom,
% 9.00/5.12 ! [I: int,J: int,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( ord_less_eq_int @ J @ K )
% 9.00/5.12 => ( ( upto @ I @ K )
% 9.00/5.12 = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9483_upto__rec1,axiom,
% 9.00/5.12 ! [I: int,J: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ I @ J )
% 9.00/5.12 => ( ( upto @ I @ J )
% 9.00/5.12 = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9484_upto_Oelims,axiom,
% 9.00/5.12 ! [X2: int,Xa2: int,Y3: list_int] :
% 9.00/5.12 ( ( ( upto @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ X2 @ Xa2 )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( cons_int @ X2 @ ( upto @ ( plus_plus_int @ X2 @ one_one_int ) @ Xa2 ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_int @ X2 @ Xa2 )
% 9.00/5.12 => ( Y3 = nil_int ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9485_upto_Osimps,axiom,
% 9.00/5.12 ( upto
% 9.00/5.12 = ( ^ [I4: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I4 @ J3 ) @ ( cons_int @ I4 @ ( upto @ ( plus_plus_int @ I4 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9486_divmod__step__integer__def,axiom,
% 9.00/5.12 ( unique4921790084139445826nteger
% 9.00/5.12 = ( ^ [L2: num] :
% 9.00/5.12 ( produc6916734918728496179nteger
% 9.00/5.12 @ ^ [Q6: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q6 ) @ R5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9487_less__eq__integer__code_I1_J,axiom,
% 9.00/5.12 ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9488_sgn__integer__code,axiom,
% 9.00/5.12 ( sgn_sgn_Code_integer
% 9.00/5.12 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9489_zero__natural_Orsp,axiom,
% 9.00/5.12 zero_zero_nat = zero_zero_nat ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9490_integer__of__int__code,axiom,
% 9.00/5.12 ( code_integer_of_int
% 9.00/5.12 = ( ^ [K3: int] :
% 9.00/5.12 ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.00/5.12 @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.00/5.12 @ ( if_Code_integer
% 9.00/5.12 @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9491_signed__take__bit__nonnegative__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N @ K ) )
% 9.00/5.12 = ( ~ ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9492_signed__take__bit__negative__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.12 = ( bit_se1146084159140164899it_int @ K @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9493_bit__nat__iff,axiom,
% 9.00/5.12 ! [K: int,N: nat] :
% 9.00/5.12 ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N )
% 9.00/5.12 = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.00/5.12 & ( bit_se1146084159140164899it_int @ K @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9494_less__integer_Oabs__eq,axiom,
% 9.00/5.12 ! [Xa2: int,X2: int] :
% 9.00/5.12 ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 9.00/5.12 = ( ord_less_int @ Xa2 @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9495_less__integer__code_I1_J,axiom,
% 9.00/5.12 ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9496_abs__integer__code,axiom,
% 9.00/5.12 ( abs_abs_Code_integer
% 9.00/5.12 = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9497_not__bit__Suc__0__Suc,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9498_bit__Suc__0__iff,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 = ( N = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9499_less__eq__integer_Oabs__eq,axiom,
% 9.00/5.12 ! [Xa2: int,X2: int] :
% 9.00/5.12 ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X2 ) )
% 9.00/5.12 = ( ord_less_eq_int @ Xa2 @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9500_not__bit__Suc__0__numeral,axiom,
% 9.00/5.12 ! [N: num] :
% 9.00/5.12 ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9501_bit__imp__take__bit__positive,axiom,
% 9.00/5.12 ! [N: nat,M: nat,K: int] :
% 9.00/5.12 ( ( ord_less_nat @ N @ M )
% 9.00/5.12 => ( ( bit_se1146084159140164899it_int @ K @ N )
% 9.00/5.12 => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9502_int__bit__bound,axiom,
% 9.00/5.12 ! [K: int] :
% 9.00/5.12 ~ ! [N3: nat] :
% 9.00/5.12 ( ! [M3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N3 @ M3 )
% 9.00/5.12 => ( ( bit_se1146084159140164899it_int @ K @ M3 )
% 9.00/5.12 = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 9.00/5.12 => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.00/5.12 => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 9.00/5.12 = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9503_binomial__def,axiom,
% 9.00/5.12 ( binomial
% 9.00/5.12 = ( ^ [N2: nat,K3: nat] :
% 9.00/5.12 ( finite_card_set_nat
% 9.00/5.12 @ ( collect_set_nat
% 9.00/5.12 @ ^ [K6: set_nat] :
% 9.00/5.12 ( ( member_set_nat @ K6 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 9.00/5.12 & ( ( finite_card_nat @ K6 )
% 9.00/5.12 = K3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9504_int__of__integer__code,axiom,
% 9.00/5.12 ( code_int_of_integer
% 9.00/5.12 = ( ^ [K3: code_integer] :
% 9.00/5.12 ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 9.00/5.12 @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 9.00/5.12 @ ( produc1553301316500091796er_int
% 9.00/5.12 @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
% 9.00/5.12 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9505_finite__enumerate,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ S )
% 9.00/5.12 => ? [R3: nat > nat] :
% 9.00/5.12 ( ( strict1292158309912662752at_nat @ R3 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S ) ) )
% 9.00/5.12 & ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_nat @ N5 @ ( finite_card_nat @ S ) )
% 9.00/5.12 => ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9506_image__Suc__atLeastAtMost,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 9.00/5.12 = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9507_image__Suc__atLeastLessThan,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 9.00/5.12 = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9508_zero__notin__Suc__image,axiom,
% 9.00/5.12 ! [A3: set_nat] :
% 9.00/5.12 ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9509_less__integer_Orep__eq,axiom,
% 9.00/5.12 ( ord_le6747313008572928689nteger
% 9.00/5.12 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9510_integer__less__iff,axiom,
% 9.00/5.12 ( ord_le6747313008572928689nteger
% 9.00/5.12 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9511_less__eq__integer_Orep__eq,axiom,
% 9.00/5.12 ( ord_le3102999989581377725nteger
% 9.00/5.12 = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9512_integer__less__eq__iff,axiom,
% 9.00/5.12 ( ord_le3102999989581377725nteger
% 9.00/5.12 = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9513_finite__int__iff__bounded,axiom,
% 9.00/5.12 ( finite_finite_int
% 9.00/5.12 = ( ^ [S6: set_int] :
% 9.00/5.12 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9514_finite__int__iff__bounded__le,axiom,
% 9.00/5.12 ( finite_finite_int
% 9.00/5.12 = ( ^ [S6: set_int] :
% 9.00/5.12 ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S6 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9515_image__int__atLeastAtMost,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A2 @ B3 ) )
% 9.00/5.12 = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9516_image__int__atLeastLessThan,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A2 @ B3 ) )
% 9.00/5.12 = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9517_image__Suc__lessThan,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9518_image__Suc__atMost,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
% 9.00/5.12 = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9519_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9520_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9521_lessThan__Suc__eq__insert__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_ord_lessThan_nat @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9522_atMost__Suc__eq__insert__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( set_ord_atMost_nat @ ( suc @ N ) )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9523_image__add__int__atLeastLessThan,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( image_int_int
% 9.00/5.12 @ ^ [X: int] : ( plus_plus_int @ X @ L )
% 9.00/5.12 @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
% 9.00/5.12 = ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9524_image__atLeastZeroLessThan__int,axiom,
% 9.00/5.12 ! [U: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.00/5.12 => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
% 9.00/5.12 = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9525_image__minus__const__atLeastLessThan__nat,axiom,
% 9.00/5.12 ! [C: nat,Y3: nat,X2: nat] :
% 9.00/5.12 ( ( ( ord_less_nat @ C @ Y3 )
% 9.00/5.12 => ( ( image_nat_nat
% 9.00/5.12 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 9.00/5.12 = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X2 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_nat @ C @ Y3 )
% 9.00/5.12 => ( ( ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.12 => ( ( image_nat_nat
% 9.00/5.12 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 9.00/5.12 & ( ~ ( ord_less_nat @ X2 @ Y3 )
% 9.00/5.12 => ( ( image_nat_nat
% 9.00/5.12 @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ X2 @ Y3 ) )
% 9.00/5.12 = bot_bot_set_nat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9526_Sup__nat__empty,axiom,
% 9.00/5.12 ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9527_num__of__integer__code,axiom,
% 9.00/5.12 ( code_num_of_integer
% 9.00/5.12 = ( ^ [K3: code_integer] :
% 9.00/5.12 ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
% 9.00/5.12 @ ( produc7336495610019696514er_num
% 9.00/5.12 @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
% 9.00/5.12 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9528_nat__of__integer__code,axiom,
% 9.00/5.12 ( code_nat_of_integer
% 9.00/5.12 = ( ^ [K3: code_integer] :
% 9.00/5.12 ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 9.00/5.12 @ ( produc1555791787009142072er_nat
% 9.00/5.12 @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
% 9.00/5.12 @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9529_nat__of__integer__non__positive,axiom,
% 9.00/5.12 ! [K: code_integer] :
% 9.00/5.12 ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 9.00/5.12 => ( ( code_nat_of_integer @ K )
% 9.00/5.12 = zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9530_nat__of__integer__code__post_I1_J,axiom,
% 9.00/5.12 ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9531_Inf__nat__def1,axiom,
% 9.00/5.12 ! [K4: set_nat] :
% 9.00/5.12 ( ( K4 != bot_bot_set_nat )
% 9.00/5.12 => ( member_nat @ ( complete_Inf_Inf_nat @ K4 ) @ K4 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9532_nat__not__finite,axiom,
% 9.00/5.12 ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9533_infinite__UNIV__nat,axiom,
% 9.00/5.12 ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9534_UN__lessThan__UNIV,axiom,
% 9.00/5.12 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
% 9.00/5.12 = top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9535_UN__atMost__UNIV,axiom,
% 9.00/5.12 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
% 9.00/5.12 = top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9536_range__enumerate,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ~ ( finite_finite_nat @ S )
% 9.00/5.12 => ( ( image_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat )
% 9.00/5.12 = S ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9537_UNIV__nat__eq,axiom,
% 9.00/5.12 ( top_top_set_nat
% 9.00/5.12 = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9538_range__mod,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( image_nat_nat
% 9.00/5.12 @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N )
% 9.00/5.12 @ top_top_set_nat )
% 9.00/5.12 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9539_suminf__eq__SUP__real,axiom,
% 9.00/5.12 ! [X6: nat > real] :
% 9.00/5.12 ( ( summable_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X6 @ I2 ) )
% 9.00/5.12 => ( ( suminf_real @ X6 )
% 9.00/5.12 = ( comple1385675409528146559p_real
% 9.00/5.12 @ ( image_nat_real
% 9.00/5.12 @ ^ [I4: nat] : ( groups6591440286371151544t_real @ X6 @ ( set_ord_lessThan_nat @ I4 ) )
% 9.00/5.12 @ top_top_set_nat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9540_card__UNIV__unit,axiom,
% 9.00/5.12 ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
% 9.00/5.12 = one_one_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9541_range__mult,axiom,
% 9.00/5.12 ! [A2: real] :
% 9.00/5.12 ( ( ( A2 = zero_zero_real )
% 9.00/5.12 => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
% 9.00/5.12 = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 9.00/5.12 & ( ( A2 != zero_zero_real )
% 9.00/5.12 => ( ( image_real_real @ ( times_times_real @ A2 ) @ top_top_set_real )
% 9.00/5.12 = top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9542_root__def,axiom,
% 9.00/5.12 ( root
% 9.00/5.12 = ( ^ [N2: nat,X: real] :
% 9.00/5.12 ( if_real @ ( N2 = zero_zero_nat ) @ zero_zero_real
% 9.00/5.12 @ ( the_in5290026491893676941l_real @ top_top_set_real
% 9.00/5.12 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 9.00/5.12 @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9543_DERIV__even__real__root,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9544_DERIV__real__root__generic,axiom,
% 9.00/5.12 ! [N: nat,X2: real,D4: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( X2 != zero_zero_real )
% 9.00/5.12 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( D4
% 9.00/5.12 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 9.00/5.12 => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( D4
% 9.00/5.12 = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 9.00/5.12 => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( D4
% 9.00/5.12 = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9545_DERIV__arctan__series,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X9: real] :
% 9.00/5.12 ( suminf_real
% 9.00/5.12 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X2 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9546_has__real__derivative__pos__inc__left,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,S: set_real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ L )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9547_has__real__derivative__neg__dec__left,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,S: set_real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 9.00/5.12 => ( ( ord_less_real @ L @ zero_zero_real )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( member_real @ ( minus_minus_real @ X2 @ H4 ) @ S )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9548_has__real__derivative__neg__dec__right,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,S: set_real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 9.00/5.12 => ( ( ord_less_real @ L @ zero_zero_real )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9549_has__real__derivative__pos__inc__right,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,S: set_real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ S ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ L )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( member_real @ ( plus_plus_real @ X2 @ H4 ) @ S )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9550_deriv__nonneg__imp__mono,axiom,
% 9.00/5.12 ! [A2: real,B3: real,G3: real > real,G4: real > real] :
% 9.00/5.12 ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( G4 @ X5 ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( G3 @ A2 ) @ ( G3 @ B3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9551_DERIV__nonneg__imp__nondecreasing,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_eq_real @ zero_zero_real @ Y5 ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9552_DERIV__nonpos__imp__nonincreasing,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_eq_real @ Y5 @ zero_zero_real ) ) ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9553_DERIV__neg__imp__decreasing,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9554_DERIV__pos__imp__increasing,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.00/5.12 => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9555_DERIV__neg__dec__right,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ L @ zero_zero_real )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9556_DERIV__pos__inc__right,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ L )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( plus_plus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9557_DERIV__pos__inc__left,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ L )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) @ ( F @ X2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9558_DERIV__neg__dec__left,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ L @ zero_zero_real )
% 9.00/5.12 => ? [D6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D6 )
% 9.00/5.12 & ! [H4: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.00/5.12 => ( ( ord_less_real @ H4 @ D6 )
% 9.00/5.12 => ( ord_less_real @ ( F @ X2 ) @ ( F @ ( minus_minus_real @ X2 @ H4 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9559_MVT2,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,F6: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ? [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 & ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 9.00/5.12 = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ ( F6 @ Z4 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9560_DERIV__local__const,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,D: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ D )
% 9.00/5.12 => ( ! [Y4: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 9.00/5.12 => ( ( F @ X2 )
% 9.00/5.12 = ( F @ Y4 ) ) )
% 9.00/5.12 => ( L = zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9561_DERIV__ln,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9562_DERIV__local__min,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,D: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ D )
% 9.00/5.12 => ( ! [Y4: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y4 ) ) )
% 9.00/5.12 => ( L = zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9563_DERIV__local__max,axiom,
% 9.00/5.12 ! [F: real > real,L: real,X2: real,D: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ D )
% 9.00/5.12 => ( ! [Y4: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X2 @ Y4 ) ) @ D )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ Y4 ) @ ( F @ X2 ) ) )
% 9.00/5.12 => ( L = zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9564_DERIV__ln__divide,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X2 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9565_DERIV__pow,axiom,
% 9.00/5.12 ! [N: nat,X2: real,S2: set_real] :
% 9.00/5.12 ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] : ( power_power_real @ X @ N )
% 9.00/5.12 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X2 @ S2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9566_has__real__derivative__powr,axiom,
% 9.00/5.12 ! [Z: real,R2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ Z )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [Z2: real] : ( powr_real @ Z2 @ R2 )
% 9.00/5.12 @ ( times_times_real @ R2 @ ( powr_real @ Z @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9567_DERIV__log,axiom,
% 9.00/5.12 ! [X2: real,B3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( log @ B3 ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B3 ) @ X2 ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9568_DERIV__fun__powr,axiom,
% 9.00/5.12 ! [G3: real > real,M: real,X2: real,R2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ R2 )
% 9.00/5.12 @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G3 @ X2 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9569_DERIV__powr,axiom,
% 9.00/5.12 ! [G3: real > real,M: real,X2: real,F: real > real,R2: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ G3 @ M @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ ( G3 @ X2 ) )
% 9.00/5.12 => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] : ( powr_real @ ( G3 @ X ) @ ( F @ X ) )
% 9.00/5.12 @ ( times_times_real @ ( powr_real @ ( G3 @ X2 ) @ ( F @ X2 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G3 @ X2 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X2 ) ) @ ( G3 @ X2 ) ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9570_DERIV__real__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9571_DERIV__real__sqrt__generic,axiom,
% 9.00/5.12 ! [X2: real,D4: real] :
% 9.00/5.12 ( ( X2 != zero_zero_real )
% 9.00/5.12 => ( ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( D4
% 9.00/5.12 = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 => ( ( ( ord_less_real @ X2 @ zero_zero_real )
% 9.00/5.12 => ( D4
% 9.00/5.12 = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9572_arcosh__real__has__field__derivative,axiom,
% 9.00/5.12 ! [X2: real,A3: set_real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9573_artanh__real__has__field__derivative,axiom,
% 9.00/5.12 ! [X2: real,A3: set_real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ A3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9574_DERIV__real__root,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9575_DERIV__arccos,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9576_DERIV__arcsin,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9577_Maclaurin__all__le,axiom,
% 9.00/5.12 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 9.00/5.12 ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( F @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9578_Maclaurin__all__le__objl,axiom,
% 9.00/5.12 ! [Diff: nat > real > real,F: real > real,X2: real,N: nat] :
% 9.00/5.12 ( ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 & ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( F @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9579_DERIV__odd__real__root,axiom,
% 9.00/5.12 ! [N: nat,X2: real] :
% 9.00/5.12 ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( ( X2 != zero_zero_real )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X2 ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9580_Maclaurin,axiom,
% 9.00/5.12 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ H2 )
% 9.00/5.12 & ( ( F @ H2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9581_Maclaurin2,axiom,
% 9.00/5.12 ! [H2: real,Diff: nat > real > real,F: real > real,N: nat] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ H2 )
% 9.00/5.12 & ( ( F @ H2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9582_Maclaurin__minus,axiom,
% 9.00/5.12 ! [H2: real,N: nat,Diff: nat > real > real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ H2 @ zero_zero_real )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ H2 @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ H2 @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ zero_zero_real )
% 9.00/5.12 & ( ( F @ H2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9583_Maclaurin__all__lt,axiom,
% 9.00/5.12 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 9.00/5.12 ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( X2 != zero_zero_real )
% 9.00/5.12 => ( ! [M4: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.00/5.12 & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( F @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9584_Maclaurin__bi__le,axiom,
% 9.00/5.12 ! [Diff: nat > real > real,F: real > real,N: nat,X2: real] :
% 9.00/5.12 ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X2 ) )
% 9.00/5.12 & ( ( F @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X2 @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X2 @ N ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9585_Taylor,axiom,
% 9.00/5.12 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real,X2: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ A2 @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ B3 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ A2 @ C )
% 9.00/5.12 => ( ( ord_less_eq_real @ C @ B3 )
% 9.00/5.12 => ( ( ord_less_eq_real @ A2 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ B3 )
% 9.00/5.12 => ( ( X2 != C )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ( ord_less_real @ X2 @ C )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ C ) ) )
% 9.00/5.12 & ( ~ ( ord_less_real @ X2 @ C )
% 9.00/5.12 => ( ( ord_less_real @ C @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ X2 ) ) )
% 9.00/5.12 & ( ( F @ X2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9586_Taylor__up,axiom,
% 9.00/5.12 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ A2 @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ B3 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ A2 @ C )
% 9.00/5.12 => ( ( ord_less_real @ C @ B3 )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ C @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ B3 )
% 9.00/5.12 & ( ( F @ B3 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9587_Taylor__down,axiom,
% 9.00/5.12 ! [N: nat,Diff: nat > real > real,F: real > real,A2: real,B3: real,C: real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( ( Diff @ zero_zero_nat )
% 9.00/5.12 = F )
% 9.00/5.12 => ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ A2 @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ B3 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ( ord_less_real @ A2 @ C )
% 9.00/5.12 => ( ( ord_less_eq_real @ C @ B3 )
% 9.00/5.12 => ? [T6: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ T6 )
% 9.00/5.12 & ( ord_less_real @ T6 @ C )
% 9.00/5.12 & ( ( F @ A2 )
% 9.00/5.12 = ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ M2 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N ) )
% 9.00/5.12 @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A2 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9588_Maclaurin__lemma2,axiom,
% 9.00/5.12 ! [N: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 9.00/5.12 ( ! [M4: nat,T6: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M4 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.00/5.12 & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ( N
% 9.00/5.12 = ( suc @ K ) )
% 9.00/5.12 => ! [M3: nat,T7: real] :
% 9.00/5.12 ( ( ( ord_less_nat @ M3 @ N )
% 9.00/5.12 & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 9.00/5.12 & ( ord_less_eq_real @ T7 @ H2 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [U2: real] :
% 9.00/5.12 ( minus_minus_real @ ( Diff @ M3 @ U2 )
% 9.00/5.12 @ ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
% 9.00/5.12 @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
% 9.00/5.12 @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T7 )
% 9.00/5.12 @ ( plus_plus_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T7 @ P5 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
% 9.00/5.12 @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9589_DERIV__power__series_H,axiom,
% 9.00/5.12 ! [R: real,F: nat > real,X0: real] :
% 9.00/5.12 ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.00/5.12 => ( summable_real
% 9.00/5.12 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X5 @ N2 ) ) ) )
% 9.00/5.12 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ R )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] :
% 9.00/5.12 ( suminf_real
% 9.00/5.12 @ ^ [N2: nat] : ( times_times_real @ ( F @ N2 ) @ ( power_power_real @ X @ ( suc @ N2 ) ) ) )
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [N2: nat] : ( times_times_real @ ( times_times_real @ ( F @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ ( power_power_real @ X0 @ N2 ) ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9590_DERIV__isconst3,axiom,
% 9.00/5.12 ! [A2: real,B3: real,X2: real,Y3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( member_real @ X2 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ( F @ X2 )
% 9.00/5.12 = ( F @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9591_DERIV__series_H,axiom,
% 9.00/5.12 ! [F: real > nat > real,F6: real > nat > real,X0: real,A2: real,B3: real,L5: nat > real] :
% 9.00/5.12 ( ! [N3: nat] :
% 9.00/5.12 ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] : ( F @ X @ N3 )
% 9.00/5.12 @ ( F6 @ X0 @ N3 )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( summable_real @ ( F @ X5 ) ) )
% 9.00/5.12 => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ( summable_real @ ( F6 @ X0 ) )
% 9.00/5.12 => ( ( summable_real @ L5 )
% 9.00/5.12 => ( ! [N3: nat,X5: real,Y4: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ( member_real @ Y4 @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y4 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y4 ) ) ) ) ) )
% 9.00/5.12 => ( has_fi5821293074295781190e_real
% 9.00/5.12 @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
% 9.00/5.12 @ ( suminf_real @ ( F6 @ X0 ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9592_Gcd__eq__Max,axiom,
% 9.00/5.12 ! [M5: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ M5 )
% 9.00/5.12 => ( ( M5 != bot_bot_set_nat )
% 9.00/5.12 => ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 9.00/5.12 => ( ( gcd_Gcd_nat @ M5 )
% 9.00/5.12 = ( lattic8265883725875713057ax_nat
% 9.00/5.12 @ ( comple7806235888213564991et_nat
% 9.00/5.12 @ ( image_nat_set_nat
% 9.00/5.12 @ ^ [M2: nat] :
% 9.00/5.12 ( collect_nat
% 9.00/5.12 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
% 9.00/5.12 @ M5 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9593_finite__greaterThanLessThan,axiom,
% 9.00/5.12 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9594_finite__greaterThanLessThan__int,axiom,
% 9.00/5.12 ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9595_card__greaterThanLessThan,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 9.00/5.12 = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9596_Max__divisors__self__nat,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( N != zero_zero_nat )
% 9.00/5.12 => ( ( lattic8265883725875713057ax_nat
% 9.00/5.12 @ ( collect_nat
% 9.00/5.12 @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N ) ) )
% 9.00/5.12 = N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9597_card__greaterThanLessThan__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
% 9.00/5.12 = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9598_atLeastSucLessThan__greaterThanLessThan,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 9.00/5.12 = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9599_isCont__Lb__Ub,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ B3 ) )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 9.00/5.12 => ? [L6: real,M9: real] :
% 9.00/5.12 ( ! [X4: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ A2 @ X4 )
% 9.00/5.12 & ( ord_less_eq_real @ X4 @ B3 ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ L6 @ ( F @ X4 ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( F @ X4 ) @ M9 ) ) )
% 9.00/5.12 & ! [Y5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ L6 @ Y5 )
% 9.00/5.12 & ( ord_less_eq_real @ Y5 @ M9 ) )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 & ( ( F @ X5 )
% 9.00/5.12 = Y5 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9600_LIM__fun__gt__zero,axiom,
% 9.00/5.12 ! [F: real > real,L: real,C: real] :
% 9.00/5.12 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ L )
% 9.00/5.12 => ? [R3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.00/5.12 & ! [X4: real] :
% 9.00/5.12 ( ( ( X4 != C )
% 9.00/5.12 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 9.00/5.12 => ( ord_less_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9601_LIM__fun__not__zero,axiom,
% 9.00/5.12 ! [F: real > real,L: real,C: real] :
% 9.00/5.12 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.00/5.12 => ( ( L != zero_zero_real )
% 9.00/5.12 => ? [R3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.00/5.12 & ! [X4: real] :
% 9.00/5.12 ( ( ( X4 != C )
% 9.00/5.12 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 9.00/5.12 => ( ( F @ X4 )
% 9.00/5.12 != zero_zero_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9602_LIM__fun__less__zero,axiom,
% 9.00/5.12 ! [F: real > real,L: real,C: real] :
% 9.00/5.12 ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.00/5.12 => ( ( ord_less_real @ L @ zero_zero_real )
% 9.00/5.12 => ? [R3: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ R3 )
% 9.00/5.12 & ! [X4: real] :
% 9.00/5.12 ( ( ( X4 != C )
% 9.00/5.12 & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X4 ) ) @ R3 ) )
% 9.00/5.12 => ( ord_less_real @ ( F @ X4 ) @ zero_zero_real ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9603_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 9.00/5.12 = ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9604_isCont__inverse__function2,axiom,
% 9.00/5.12 ! [A2: real,X2: real,B3: real,G3: real > real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ B3 )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 9.00/5.12 => ( ( G3 @ ( F @ Z4 ) )
% 9.00/5.12 = Z4 ) ) )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9605_Sup__nat__def,axiom,
% 9.00/5.12 ( complete_Sup_Sup_nat
% 9.00/5.12 = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9606_card__le__Suc__Max,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ S )
% 9.00/5.12 => ( ord_less_eq_nat @ ( finite_card_nat @ S ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9607_divide__nat__def,axiom,
% 9.00/5.12 ( divide_divide_nat
% 9.00/5.12 = ( ^ [M2: nat,N2: nat] :
% 9.00/5.12 ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat
% 9.00/5.12 @ ( lattic8265883725875713057ax_nat
% 9.00/5.12 @ ( collect_nat
% 9.00/5.12 @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N2 ) @ M2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9608_isCont__arcosh,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcosh_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9609_DERIV__inverse__function,axiom,
% 9.00/5.12 ! [F: real > real,D4: real,G3: real > real,X2: real,A2: real,B3: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G3 @ X2 ) @ top_top_set_real ) )
% 9.00/5.12 => ( ( D4 != zero_zero_real )
% 9.00/5.12 => ( ( ord_less_real @ A2 @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ B3 )
% 9.00/5.12 => ( ! [Y4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Y4 )
% 9.00/5.12 => ( ( ord_less_real @ Y4 @ B3 )
% 9.00/5.12 => ( ( F @ ( G3 @ Y4 ) )
% 9.00/5.12 = Y4 ) ) )
% 9.00/5.12 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ G3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ G3 @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9610_isCont__arccos,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arccos ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9611_isCont__arcsin,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ arcsin ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9612_LIM__less__bound,axiom,
% 9.00/5.12 ! [B3: real,X2: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ B3 @ X2 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B3 @ X2 ) )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
% 9.00/5.12 => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ F )
% 9.00/5.12 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9613_isCont__artanh,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X2 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9614_isCont__inverse__function,axiom,
% 9.00/5.12 ! [D: real,X2: real,G3: real > real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ D )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
% 9.00/5.12 => ( ( G3 @ ( F @ Z4 ) )
% 9.00/5.12 = Z4 ) )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z4 @ X2 ) ) @ D )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X2 ) @ top_top_set_real ) @ G3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9615_GMVT_H,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,G3: real > real,G4: real > real,F6: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ F ) ) )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_eq_real @ Z4 @ B3 )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) @ G3 ) ) )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ G3 @ ( G4 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ( ! [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 => ( ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ F @ ( F6 @ Z4 ) @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ? [C3: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ C3 )
% 9.00/5.12 & ( ord_less_real @ C3 @ B3 )
% 9.00/5.12 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ ( G4 @ C3 ) )
% 9.00/5.12 = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9616_summable__Leibniz_I3_J,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ( topolo6980174941875973593q_real @ A2 )
% 9.00/5.12 => ( ( ord_less_real @ ( A2 @ zero_zero_nat ) @ zero_zero_real )
% 9.00/5.12 => ! [N5: nat] :
% 9.00/5.12 ( member_real
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 9.00/5.12 @ ( set_or1222579329274155063t_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) )
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9617_summable__Leibniz_I2_J,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ( topolo6980174941875973593q_real @ A2 )
% 9.00/5.12 => ( ( ord_less_real @ zero_zero_real @ ( A2 @ zero_zero_nat ) )
% 9.00/5.12 => ! [N5: nat] :
% 9.00/5.12 ( member_real
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 9.00/5.12 @ ( set_or1222579329274155063t_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9618_summable__Leibniz_H_I4_J,axiom,
% 9.00/5.12 ! [A2: nat > real,N: nat] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ord_less_eq_real
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) )
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9619_trivial__limit__sequentially,axiom,
% 9.00/5.12 at_top_nat != bot_bot_filter_nat ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9620_mult__nat__right__at__top,axiom,
% 9.00/5.12 ! [C: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.12 => ( filterlim_nat_nat
% 9.00/5.12 @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 9.00/5.12 @ at_top_nat
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9621_mult__nat__left__at__top,axiom,
% 9.00/5.12 ! [C: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.00/5.12 => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9622_monoseq__convergent,axiom,
% 9.00/5.12 ! [X6: nat > real,B2: real] :
% 9.00/5.12 ( ( topolo6980174941875973593q_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X6 @ I2 ) ) @ B2 )
% 9.00/5.12 => ~ ! [L6: real] :
% 9.00/5.12 ~ ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9623_nested__sequence__unique,axiom,
% 9.00/5.12 ! [F: nat > real,G3: nat > real] :
% 9.00/5.12 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( G3 @ ( suc @ N3 ) ) @ ( G3 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G3 @ N3 ) )
% 9.00/5.12 => ( ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( minus_minus_real @ ( F @ N2 ) @ ( G3 @ N2 ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.00/5.12 @ at_top_nat )
% 9.00/5.12 => ? [L4: real] :
% 9.00/5.12 ( ! [N5: nat] : ( ord_less_eq_real @ ( F @ N5 ) @ L4 )
% 9.00/5.12 & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 9.00/5.12 & ! [N5: nat] : ( ord_less_eq_real @ L4 @ ( G3 @ N5 ) )
% 9.00/5.12 & ( filterlim_nat_real @ G3 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9624_LIMSEQ__inverse__zero,axiom,
% 9.00/5.12 ! [X6: nat > real] :
% 9.00/5.12 ( ! [R3: real] :
% 9.00/5.12 ? [N8: nat] :
% 9.00/5.12 ! [N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.00/5.12 => ( ord_less_real @ R3 @ ( X6 @ N3 ) ) )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( inverse_inverse_real @ ( X6 @ N2 ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9625_LIMSEQ__root__const,axiom,
% 9.00/5.12 ! [C: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ C )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( root @ N2 @ C )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9626_increasing__LIMSEQ,axiom,
% 9.00/5.12 ! [F: nat > real,L: real] :
% 9.00/5.12 ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 9.00/5.12 => ( ! [E: real] :
% 9.00/5.12 ( ( ord_less_real @ zero_zero_real @ E )
% 9.00/5.12 => ? [N5: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N5 ) @ E ) ) )
% 9.00/5.12 => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9627_LIMSEQ__realpow__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_real @ X2 @ one_one_real )
% 9.00/5.12 => ( filterlim_nat_real @ ( power_power_real @ X2 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9628_LIMSEQ__divide__realpow__zero,axiom,
% 9.00/5.12 ! [X2: real,A2: real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( divide_divide_real @ A2 @ ( power_power_real @ X2 @ N2 ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9629_LIMSEQ__abs__realpow__zero,axiom,
% 9.00/5.12 ! [C: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.00/5.12 => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9630_LIMSEQ__abs__realpow__zero2,axiom,
% 9.00/5.12 ! [C: real] :
% 9.00/5.12 ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.00/5.12 => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9631_LIMSEQ__inverse__realpow__zero,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ X2 )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( inverse_inverse_real @ ( power_power_real @ X2 @ N2 ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9632_summable,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( summable_real
% 9.00/5.12 @ ^ [N2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9633_zeroseq__arctan__series,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ ( abs_abs_real @ X2 ) @ one_one_real )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [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 @ X2 @ ( plus_plus_nat @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.00/5.12 @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9634_summable__Leibniz_H_I3_J,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] :
% 9.00/5.12 ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
% 9.00/5.12 @ at_top_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9635_summable__Leibniz_H_I2_J,axiom,
% 9.00/5.12 ! [A2: nat > real,N: nat] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ord_less_eq_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9636_sums__alternating__upper__lower,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ? [L4: real] :
% 9.00/5.12 ( ! [N5: nat] :
% 9.00/5.12 ( ord_less_eq_real
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) ) )
% 9.00/5.12 @ L4 )
% 9.00/5.12 & ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] :
% 9.00/5.12 ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ L4 )
% 9.00/5.12 @ at_top_nat )
% 9.00/5.12 & ! [N5: nat] :
% 9.00/5.12 ( ord_less_eq_real @ L4
% 9.00/5.12 @ ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N5 ) @ one_one_nat ) ) ) )
% 9.00/5.12 & ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] :
% 9.00/5.12 ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real @ L4 )
% 9.00/5.12 @ at_top_nat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9637_summable__Leibniz_H_I5_J,axiom,
% 9.00/5.12 ! [A2: nat > real] :
% 9.00/5.12 ( ( filterlim_nat_real @ A2 @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A2 @ N3 ) )
% 9.00/5.12 => ( ! [N3: nat] : ( ord_less_eq_real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
% 9.00/5.12 => ( filterlim_nat_real
% 9.00/5.12 @ ^ [N2: nat] :
% 9.00/5.12 ( groups6591440286371151544t_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) )
% 9.00/5.12 @ ( topolo2815343760600316023s_real
% 9.00/5.12 @ ( suminf_real
% 9.00/5.12 @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I4 ) @ ( A2 @ I4 ) ) ) )
% 9.00/5.12 @ at_top_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9638_DERIV__neg__imp__decreasing__at__top,axiom,
% 9.00/5.12 ! [B3: real,F: real > real,Flim: real] :
% 9.00/5.12 ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ B3 @ X5 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ Y5 @ zero_zero_real ) ) )
% 9.00/5.12 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 9.00/5.12 => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9639_filterlim__pow__at__bot__even,axiom,
% 9.00/5.12 ! [N: nat,F: real > real,F2: filter_real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
% 9.00/5.12 => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( filterlim_real_real
% 9.00/5.12 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 9.00/5.12 @ at_top_real
% 9.00/5.12 @ F2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9640_DERIV__pos__imp__increasing__at__bot,axiom,
% 9.00/5.12 ! [B3: real,F: real > real,Flim: real] :
% 9.00/5.12 ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ zero_zero_real @ Y5 ) ) )
% 9.00/5.12 => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 9.00/5.12 => ( ord_less_real @ Flim @ ( F @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9641_filterlim__pow__at__bot__odd,axiom,
% 9.00/5.12 ! [N: nat,F: real > real,F2: filter_real] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( filterlim_real_real @ F @ at_bot_real @ F2 )
% 9.00/5.12 => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
% 9.00/5.12 => ( filterlim_real_real
% 9.00/5.12 @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N )
% 9.00/5.12 @ at_bot_real
% 9.00/5.12 @ F2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9642_GMVT,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,G3: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ B3 ) )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ B3 ) )
% 9.00/5.12 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ( ord_less_eq_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_eq_real @ X5 @ B3 ) )
% 9.00/5.12 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G3 ) )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ B3 ) )
% 9.00/5.12 => ( differ6690327859849518006l_real @ G3 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
% 9.00/5.12 => ? [G_c: real,F_c: real,C3: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ G3 @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 9.00/5.12 & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ A2 @ C3 )
% 9.00/5.12 & ( ord_less_real @ C3 @ B3 )
% 9.00/5.12 & ( ( times_times_real @ ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) ) @ G_c )
% 9.00/5.12 = ( times_times_real @ ( minus_minus_real @ ( G3 @ B3 ) @ ( G3 @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9643_at__bot__le__at__infinity,axiom,
% 9.00/5.12 ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9644_at__top__le__at__infinity,axiom,
% 9.00/5.12 ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9645_MVT,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ? [L4: real,Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 & ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) )
% 9.00/5.12 & ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 9.00/5.12 = ( times_times_real @ ( minus_minus_real @ B3 @ A2 ) @ L4 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9646_eventually__sequentiallyI,axiom,
% 9.00/5.12 ! [C: nat,P: nat > $o] :
% 9.00/5.12 ( ! [X5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ C @ X5 )
% 9.00/5.12 => ( P @ X5 ) )
% 9.00/5.12 => ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9647_eventually__sequentially,axiom,
% 9.00/5.12 ! [P: nat > $o] :
% 9.00/5.12 ( ( eventually_nat @ P @ at_top_nat )
% 9.00/5.12 = ( ? [N4: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N4 @ N2 )
% 9.00/5.12 => ( P @ N2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9648_le__sequentially,axiom,
% 9.00/5.12 ! [F2: filter_nat] :
% 9.00/5.12 ( ( ord_le2510731241096832064er_nat @ F2 @ at_top_nat )
% 9.00/5.12 = ( ! [N4: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N4 ) @ F2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9649_continuous__on__arcosh_H,axiom,
% 9.00/5.12 ! [A3: set_real,F: real > real] :
% 9.00/5.12 ( ( topolo5044208981011980120l_real @ A3 @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ A3 )
% 9.00/5.12 => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
% 9.00/5.12 => ( topolo5044208981011980120l_real @ A3
% 9.00/5.12 @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9650_eventually__at__left__real,axiom,
% 9.00/5.12 ! [B3: real,A2: real] :
% 9.00/5.12 ( ( ord_less_real @ B3 @ A2 )
% 9.00/5.12 => ( eventually_real
% 9.00/5.12 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B3 @ A2 ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5984915006950818249n_real @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9651_continuous__image__closed__interval,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_eq_real @ A2 @ B3 )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ? [C3: real,D6: real] :
% 9.00/5.12 ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 9.00/5.12 = ( set_or1222579329274155063t_real @ C3 @ D6 ) )
% 9.00/5.12 & ( ord_less_eq_real @ C3 @ D6 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9652_continuous__on__artanh,axiom,
% 9.00/5.12 ! [A3: set_real] :
% 9.00/5.12 ( ( ord_less_eq_set_real @ A3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
% 9.00/5.12 => ( topolo5044208981011980120l_real @ A3 @ artanh_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9653_Rolle__deriv,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( ( F @ A2 )
% 9.00/5.12 = ( F @ B3 ) )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ? [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 & ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 & ( ( F6 @ Z4 )
% 9.00/5.12 = ( ^ [V3: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9654_Bseq__eq__bounded,axiom,
% 9.00/5.12 ! [F: nat > real,A2: real,B3: real] :
% 9.00/5.12 ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A2 @ B3 ) )
% 9.00/5.12 => ( bfun_nat_real @ F @ at_top_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9655_mvt,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,F6: real > real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( has_de1759254742604945161l_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ~ ! [Xi: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Xi )
% 9.00/5.12 => ( ( ord_less_real @ Xi @ B3 )
% 9.00/5.12 => ( ( minus_minus_real @ ( F @ B3 ) @ ( F @ A2 ) )
% 9.00/5.12 != ( F6 @ Xi @ ( minus_minus_real @ B3 @ A2 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9656_Bseq__realpow,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( bfun_nat_real @ ( power_power_real @ X2 ) @ at_top_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9657_DERIV__pos__imp__increasing__open,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ zero_zero_real @ Y5 ) ) ) )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ord_less_real @ ( F @ A2 ) @ ( F @ B3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9658_DERIV__neg__imp__decreasing__open,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ? [Y5: real] :
% 9.00/5.12 ( ( has_fi5821293074295781190e_real @ F @ Y5 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 & ( ord_less_real @ Y5 @ zero_zero_real ) ) ) )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ord_less_real @ ( F @ B3 ) @ ( F @ A2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9659_DERIV__isconst__end,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ( ( F @ B3 )
% 9.00/5.12 = ( F @ A2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9660_DERIV__isconst2,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real,X2: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ( ( ord_less_eq_real @ A2 @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ B3 )
% 9.00/5.12 => ( ( F @ X2 )
% 9.00/5.12 = ( F @ A2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9661_Rolle,axiom,
% 9.00/5.12 ! [A2: real,B3: real,F: real > real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( ( ( F @ A2 )
% 9.00/5.12 = ( F @ B3 ) )
% 9.00/5.12 => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A2 @ B3 ) @ F )
% 9.00/5.12 => ( ! [X5: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ X5 )
% 9.00/5.12 => ( ( ord_less_real @ X5 @ B3 )
% 9.00/5.12 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
% 9.00/5.12 => ? [Z4: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ Z4 )
% 9.00/5.12 & ( ord_less_real @ Z4 @ B3 )
% 9.00/5.12 & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z4 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9662_eventually__at__right__real,axiom,
% 9.00/5.12 ! [A2: real,B3: real] :
% 9.00/5.12 ( ( ord_less_real @ A2 @ B3 )
% 9.00/5.12 => ( eventually_real
% 9.00/5.12 @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A2 @ B3 ) )
% 9.00/5.12 @ ( topolo2177554685111907308n_real @ A2 @ ( set_or5849166863359141190n_real @ A2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9663_greaterThan__0,axiom,
% 9.00/5.12 ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 9.00/5.12 = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9664_greaterThan__Suc,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 9.00/5.12 = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9665_INT__greaterThan__UNIV,axiom,
% 9.00/5.12 ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 9.00/5.12 = bot_bot_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9666_finite__greaterThanAtMost,axiom,
% 9.00/5.12 ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9667_card__greaterThanAtMost,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
% 9.00/5.12 = ( minus_minus_nat @ U @ L ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9668_atLeastSucAtMost__greaterThanAtMost,axiom,
% 9.00/5.12 ! [L: nat,U: nat] :
% 9.00/5.12 ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 9.00/5.12 = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9669_GreatestI__ex__nat,axiom,
% 9.00/5.12 ! [P: nat > $o,B3: nat] :
% 9.00/5.12 ( ? [X_12: nat] : ( P @ X_12 )
% 9.00/5.12 => ( ! [Y4: nat] :
% 9.00/5.12 ( ( P @ Y4 )
% 9.00/5.12 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 9.00/5.12 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9670_Greatest__le__nat,axiom,
% 9.00/5.12 ! [P: nat > $o,K: nat,B3: nat] :
% 9.00/5.12 ( ( P @ K )
% 9.00/5.12 => ( ! [Y4: nat] :
% 9.00/5.12 ( ( P @ Y4 )
% 9.00/5.12 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 9.00/5.12 => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9671_GreatestI__nat,axiom,
% 9.00/5.12 ! [P: nat > $o,K: nat,B3: nat] :
% 9.00/5.12 ( ( P @ K )
% 9.00/5.12 => ( ! [Y4: nat] :
% 9.00/5.12 ( ( P @ Y4 )
% 9.00/5.12 => ( ord_less_eq_nat @ Y4 @ B3 ) )
% 9.00/5.12 => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9672_finite__greaterThanAtMost__int,axiom,
% 9.00/5.12 ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9673_atLeast__0,axiom,
% 9.00/5.12 ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 9.00/5.12 = top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9674_card__greaterThanAtMost__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
% 9.00/5.12 = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9675_atLeast__Suc__greaterThan,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.00/5.12 = ( set_or1210151606488870762an_nat @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9676_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
% 9.00/5.12 ! [L: int,U: int] :
% 9.00/5.12 ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
% 9.00/5.12 = ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9677_decseq__bounded,axiom,
% 9.00/5.12 ! [X6: nat > real,B2: real] :
% 9.00/5.12 ( ( order_9091379641038594480t_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
% 9.00/5.12 => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9678_decseq__convergent,axiom,
% 9.00/5.12 ! [X6: nat > real,B2: real] :
% 9.00/5.12 ( ( order_9091379641038594480t_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ B2 @ ( X6 @ I2 ) )
% 9.00/5.12 => ~ ! [L6: real] :
% 9.00/5.12 ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 9.00/5.12 => ~ ! [I3: nat] : ( ord_less_eq_real @ L6 @ ( X6 @ I3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9679_UN__atLeast__UNIV,axiom,
% 9.00/5.12 ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
% 9.00/5.12 = top_top_set_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9680_atLeast__Suc,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.00/5.12 = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9681_continuous__on__arcosh,axiom,
% 9.00/5.12 ! [A3: set_real] :
% 9.00/5.12 ( ( ord_less_eq_set_real @ A3 @ ( set_ord_atLeast_real @ one_one_real ) )
% 9.00/5.12 => ( topolo5044208981011980120l_real @ A3 @ arcosh_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9682_bdd__above__nat,axiom,
% 9.00/5.12 condit2214826472909112428ve_nat = finite_finite_nat ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9683_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.12 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( Xa2 != one_one_nat ) ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( ~ ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9684_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.12 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( Xa2 != one_one_nat ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ~ ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X4: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9685_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.12 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( Xa2 = one_one_nat ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X5: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9686_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 9.00/5.12 ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
% 9.00/5.12 ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
% 9.00/5.12 = ( ( Deg = Deg3 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9687_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.12 ( ~ ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.12 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.00/5.12 => ( Xa2 = one_one_nat ) ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
% 9.00/5.12 => ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X5: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9688_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat] :
% 9.00/5.12 ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.12 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.00/5.12 => ( Xa2 != one_one_nat ) ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
% 9.00/5.12 => ~ ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X4: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9689_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 9.00/5.12 ! [X2: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.00/5.12 ( ( ( vEBT_VEBT_valid @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X2 @ Xa2 ) )
% 9.00/5.12 => ( ! [Uu2: $o,Uv2: $o] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.00/5.12 => ( ( Y3
% 9.00/5.12 = ( Xa2 = one_one_nat ) )
% 9.00/5.12 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.00/5.12 => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.00/5.12 ( ( X2
% 9.00/5.12 = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
% 9.00/5.12 => ( ( Y3
% 9.00/5.12 = ( ( Deg2 = Xa2 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.00/5.12 & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 & ( case_o184042715313410164at_nat
% 9.00/5.12 @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X8 )
% 9.00/5.12 & ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [Mi3: nat,Ma3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.00/5.12 & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 & ! [I4: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.00/5.12 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
% 9.00/5.12 = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.00/5.12 & ( ( Mi3 = Ma3 )
% 9.00/5.12 => ! [X: vEBT_VEBT] :
% 9.00/5.12 ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.00/5.12 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X @ X8 ) ) )
% 9.00/5.12 & ( ( Mi3 != Ma3 )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
% 9.00/5.12 & ! [X: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.00/5.12 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X )
% 9.00/5.12 => ( ( ord_less_nat @ Mi3 @ X )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.00/5.12 @ Mima ) ) )
% 9.00/5.12 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9690_Sup__real__def,axiom,
% 9.00/5.12 ( comple1385675409528146559p_real
% 9.00/5.12 = ( ^ [X8: set_real] :
% 9.00/5.12 ( ord_Least_real
% 9.00/5.12 @ ^ [Z2: real] :
% 9.00/5.12 ! [X: real] :
% 9.00/5.12 ( ( member_real @ X @ X8 )
% 9.00/5.12 => ( ord_less_eq_real @ X @ Z2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9691_Sup__int__def,axiom,
% 9.00/5.12 ( complete_Sup_Sup_int
% 9.00/5.12 = ( ^ [X8: set_int] :
% 9.00/5.12 ( the_int
% 9.00/5.12 @ ^ [X: int] :
% 9.00/5.12 ( ( member_int @ X @ X8 )
% 9.00/5.12 & ! [Y: int] :
% 9.00/5.12 ( ( member_int @ Y @ X8 )
% 9.00/5.12 => ( ord_less_eq_int @ Y @ X ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9692_uniformity__real__def,axiom,
% 9.00/5.12 ( topolo1511823702728130853y_real
% 9.00/5.12 = ( comple2936214249959783750l_real
% 9.00/5.12 @ ( image_2178119161166701260l_real
% 9.00/5.12 @ ^ [E3: real] :
% 9.00/5.12 ( princi6114159922880469582l_real
% 9.00/5.12 @ ( collec3799799289383736868l_real
% 9.00/5.12 @ ( produc5414030515140494994real_o
% 9.00/5.12 @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 9.00/5.12 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9693_uniformity__complex__def,axiom,
% 9.00/5.12 ( topolo896644834953643431omplex
% 9.00/5.12 = ( comple8358262395181532106omplex
% 9.00/5.12 @ ( image_5971271580939081552omplex
% 9.00/5.12 @ ^ [E3: real] :
% 9.00/5.12 ( princi3496590319149328850omplex
% 9.00/5.12 @ ( collec8663557070575231912omplex
% 9.00/5.12 @ ( produc6771430404735790350plex_o
% 9.00/5.12 @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 9.00/5.12 @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9694_bit__cut__integer__code,axiom,
% 9.00/5.12 ( code_bit_cut_integer
% 9.00/5.12 = ( ^ [K3: code_integer] :
% 9.00/5.12 ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 9.00/5.12 @ ( produc9125791028180074456eger_o
% 9.00/5.12 @ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
% 9.00/5.12 @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9695_divmod__integer__code,axiom,
% 9.00/5.12 ( code_divmod_integer
% 9.00/5.12 = ( ^ [K3: code_integer,L2: code_integer] :
% 9.00/5.12 ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.00/5.12 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 9.00/5.12 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.00/5.12 @ ( produc6916734918728496179nteger
% 9.00/5.12 @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S7 ) ) )
% 9.00/5.12 @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 9.00/5.12 @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 9.00/5.12 @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 9.00/5.12 @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.00/5.12 @ ( produc6916734918728496179nteger
% 9.00/5.12 @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S7 ) ) )
% 9.00/5.12 @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9696_eventually__prod__sequentially,axiom,
% 9.00/5.12 ! [P: product_prod_nat_nat > $o] :
% 9.00/5.12 ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 9.00/5.12 = ( ? [N4: nat] :
% 9.00/5.12 ! [M2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N4 @ M2 )
% 9.00/5.12 => ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N4 @ N2 )
% 9.00/5.12 => ( P @ ( product_Pair_nat_nat @ N2 @ M2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9697_card_Ocomp__fun__commute__on,axiom,
% 9.00/5.12 ( ( comp_nat_nat_nat @ suc @ suc )
% 9.00/5.12 = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9698_drop__bit__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N @ K ) )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9699_drop__bit__negative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.12 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9700_drop__bit__of__Suc__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se8570568707652914677it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( zero_n2687167440665602831ol_nat @ ( N = zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9701_mono__Suc,axiom,
% 9.00/5.12 order_mono_nat_nat @ suc ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9702_mono__times__nat,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( order_mono_nat_nat @ ( times_times_nat @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9703_incseq__bounded,axiom,
% 9.00/5.12 ! [X6: nat > real,B2: real] :
% 9.00/5.12 ( ( order_mono_nat_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
% 9.00/5.12 => ( bfun_nat_real @ X6 @ at_top_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9704_incseq__convergent,axiom,
% 9.00/5.12 ! [X6: nat > real,B2: real] :
% 9.00/5.12 ( ( order_mono_nat_real @ X6 )
% 9.00/5.12 => ( ! [I2: nat] : ( ord_less_eq_real @ ( X6 @ I2 ) @ B2 )
% 9.00/5.12 => ~ ! [L6: real] :
% 9.00/5.12 ( ( filterlim_nat_real @ X6 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
% 9.00/5.12 => ~ ! [I3: nat] : ( ord_less_eq_real @ ( X6 @ I3 ) @ L6 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9705_infinite__int__iff__infinite__nat__abs,axiom,
% 9.00/5.12 ! [S: set_int] :
% 9.00/5.12 ( ( ~ ( finite_finite_int @ S ) )
% 9.00/5.12 = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9706_mono__ge2__power__minus__self,axiom,
% 9.00/5.12 ! [K: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.00/5.12 => ( order_mono_nat_nat
% 9.00/5.12 @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9707_take__bit__num__simps_I1_J,axiom,
% 9.00/5.12 ! [M: num] :
% 9.00/5.12 ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 9.00/5.12 = none_num ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9708_nonneg__incseq__Bseq__subseq__iff,axiom,
% 9.00/5.12 ! [F: nat > real,G3: nat > nat] :
% 9.00/5.12 ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.00/5.12 => ( ( order_mono_nat_real @ F )
% 9.00/5.12 => ( ( order_5726023648592871131at_nat @ G3 )
% 9.00/5.12 => ( ( bfun_nat_real
% 9.00/5.12 @ ^ [X: nat] : ( F @ ( G3 @ X ) )
% 9.00/5.12 @ at_top_nat )
% 9.00/5.12 = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9709_strict__mono__imp__increasing,axiom,
% 9.00/5.12 ! [F: nat > nat,N: nat] :
% 9.00/5.12 ( ( order_5726023648592871131at_nat @ F )
% 9.00/5.12 => ( ord_less_eq_nat @ N @ ( F @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9710_infinite__enumerate,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ~ ( finite_finite_nat @ S )
% 9.00/5.12 => ? [R3: nat > nat] :
% 9.00/5.12 ( ( order_5726023648592871131at_nat @ R3 )
% 9.00/5.12 & ! [N5: nat] : ( member_nat @ ( R3 @ N5 ) @ S ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9711_strict__mono__enumerate,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ~ ( finite_finite_nat @ S )
% 9.00/5.12 => ( order_5726023648592871131at_nat @ ( infini8530281810654367211te_nat @ S ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9712_take__bit__num__def,axiom,
% 9.00/5.12 ( bit_take_bit_num
% 9.00/5.12 = ( ^ [N2: nat,M2: num] :
% 9.00/5.12 ( if_option_num
% 9.00/5.12 @ ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) )
% 9.00/5.12 = zero_zero_nat )
% 9.00/5.12 @ none_num
% 9.00/5.12 @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9713_num__of__nat_Osimps_I1_J,axiom,
% 9.00/5.12 ( ( num_of_nat @ zero_zero_nat )
% 9.00/5.12 = one ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9714_numeral__num__of__nat,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( numeral_numeral_nat @ ( num_of_nat @ N ) )
% 9.00/5.12 = N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9715_num__of__nat__One,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N @ one_one_nat )
% 9.00/5.12 => ( ( num_of_nat @ N )
% 9.00/5.12 = one ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9716_num__of__nat__double,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( num_of_nat @ ( plus_plus_nat @ N @ N ) )
% 9.00/5.12 = ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9717_num__of__nat__plus__distrib,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( num_of_nat @ ( plus_plus_nat @ M @ N ) )
% 9.00/5.12 = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9718_pos__deriv__imp__strict__mono,axiom,
% 9.00/5.12 ! [F: real > real,F6: real > real] :
% 9.00/5.12 ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F6 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
% 9.00/5.12 => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F6 @ X5 ) )
% 9.00/5.12 => ( order_7092887310737990675l_real @ F ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9719_Suc__funpow,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( compow_nat_nat @ N @ suc )
% 9.00/5.12 = ( plus_plus_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9720_inj__sgn__power,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( inj_on_real_real
% 9.00/5.12 @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 9.00/5.12 @ top_top_set_real ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9721_log__inj,axiom,
% 9.00/5.12 ! [B3: real] :
% 9.00/5.12 ( ( ord_less_real @ one_one_real @ B3 )
% 9.00/5.12 => ( inj_on_real_real @ ( log @ B3 ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9722_inj__Suc,axiom,
% 9.00/5.12 ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9723_inj__on__diff__nat,axiom,
% 9.00/5.12 ! [N6: set_nat,K: nat] :
% 9.00/5.12 ( ! [N3: nat] :
% 9.00/5.12 ( ( member_nat @ N3 @ N6 )
% 9.00/5.12 => ( ord_less_eq_nat @ K @ N3 ) )
% 9.00/5.12 => ( inj_on_nat_nat
% 9.00/5.12 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ K )
% 9.00/5.12 @ N6 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9724_inj__on__set__encode,axiom,
% 9.00/5.12 inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9725_summable__reindex,axiom,
% 9.00/5.12 ! [F: nat > real,G3: nat > nat] :
% 9.00/5.12 ( ( summable_real @ F )
% 9.00/5.12 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 9.00/5.12 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.00/5.12 => ( summable_real @ ( comp_nat_real_nat @ F @ G3 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9726_suminf__reindex__mono,axiom,
% 9.00/5.12 ! [F: nat > real,G3: nat > nat] :
% 9.00/5.12 ( ( summable_real @ F )
% 9.00/5.12 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 9.00/5.12 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.00/5.12 => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9727_suminf__reindex,axiom,
% 9.00/5.12 ! [F: nat > real,G3: nat > nat] :
% 9.00/5.12 ( ( summable_real @ F )
% 9.00/5.12 => ( ( inj_on_nat_nat @ G3 @ top_top_set_nat )
% 9.00/5.12 => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
% 9.00/5.12 => ( ! [X5: nat] :
% 9.00/5.12 ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G3 @ top_top_set_nat ) )
% 9.00/5.12 => ( ( F @ X5 )
% 9.00/5.12 = zero_zero_real ) )
% 9.00/5.12 => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G3 ) )
% 9.00/5.12 = ( suminf_real @ F ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9728_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.00/5.12 ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 9.00/5.12 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 9.00/5.12 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9729_push__bit__of__Suc__0,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( bit_se547839408752420682it_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9730_push__bit__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N @ K ) )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9731_push__bit__negative__int__iff,axiom,
% 9.00/5.12 ! [N: nat,K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N @ K ) @ zero_zero_int )
% 9.00/5.12 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9732_bit__push__bit__iff__int,axiom,
% 9.00/5.12 ! [M: nat,K: int,N: nat] :
% 9.00/5.12 ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N )
% 9.00/5.12 = ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9733_bit__push__bit__iff__nat,axiom,
% 9.00/5.12 ! [M: nat,Q3: nat,N: nat] :
% 9.00/5.12 ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N )
% 9.00/5.12 = ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9734_powr__real__of__int_H,axiom,
% 9.00/5.12 ! [X2: real,N: int] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ( X2 != zero_zero_real )
% 9.00/5.12 | ( ord_less_int @ zero_zero_int @ N ) )
% 9.00/5.12 => ( ( powr_real @ X2 @ ( ring_1_of_int_real @ N ) )
% 9.00/5.12 = ( power_int_real @ X2 @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9735_sorted__list__of__set__greaterThanAtMost,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 9.00/5.12 => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 9.00/5.12 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9736_sorted__list__of__set__greaterThanLessThan,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( suc @ I ) @ J )
% 9.00/5.12 => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 9.00/5.12 = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9737_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 9.00/5.12 ! [N: nat,J: nat,I: nat] :
% 9.00/5.12 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ I ) )
% 9.00/5.12 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N )
% 9.00/5.12 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9738_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 9.00/5.12 ! [N: nat,J: nat,I: nat] :
% 9.00/5.12 ( ( ord_less_nat @ N @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 9.00/5.12 => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N )
% 9.00/5.12 = ( suc @ ( plus_plus_nat @ I @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9739_num__of__nat_Osimps_I2_J,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( num_of_nat @ ( suc @ N ) )
% 9.00/5.12 = ( inc @ ( num_of_nat @ N ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( num_of_nat @ ( suc @ N ) )
% 9.00/5.12 = one ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9740_Rats__eq__int__div__nat,axiom,
% 9.00/5.12 ( field_5140801741446780682s_real
% 9.00/5.12 = ( collect_real
% 9.00/5.12 @ ^ [Uu3: real] :
% 9.00/5.12 ? [I4: int,N2: nat] :
% 9.00/5.12 ( ( Uu3
% 9.00/5.12 = ( divide_divide_real @ ( ring_1_of_int_real @ I4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.00/5.12 & ( N2 != zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9741_rat__floor__lemma,axiom,
% 9.00/5.12 ! [A2: int,B3: int] :
% 9.00/5.12 ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A2 @ B3 ) ) @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 & ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A2 @ B3 ) @ one_one_int ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9742_less__rat,axiom,
% 9.00/5.12 ! [B3: int,D: int,A2: int,C: int] :
% 9.00/5.12 ( ( B3 != zero_zero_int )
% 9.00/5.12 => ( ( D != zero_zero_int )
% 9.00/5.12 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
% 9.00/5.12 = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9743_le__rat,axiom,
% 9.00/5.12 ! [B3: int,D: int,A2: int,C: int] :
% 9.00/5.12 ( ( B3 != zero_zero_int )
% 9.00/5.12 => ( ( D != zero_zero_int )
% 9.00/5.12 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ ( fract @ C @ D ) )
% 9.00/5.12 = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A2 @ D ) @ ( times_times_int @ B3 @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B3 ) @ ( times_times_int @ B3 @ D ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9744_Rats__dense__in__real,axiom,
% 9.00/5.12 ! [X2: real,Y3: real] :
% 9.00/5.12 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.12 => ? [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.00/5.12 & ( ord_less_real @ X2 @ X5 )
% 9.00/5.12 & ( ord_less_real @ X5 @ Y3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9745_Rats__no__bot__less,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ? [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.00/5.12 & ( ord_less_real @ X5 @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9746_Rat__induct__pos,axiom,
% 9.00/5.12 ! [P: rat > $o,Q3: rat] :
% 9.00/5.12 ( ! [A: int,B: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B )
% 9.00/5.12 => ( P @ ( fract @ A @ B ) ) )
% 9.00/5.12 => ( P @ Q3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9747_Rats__no__top__le,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ? [X5: real] :
% 9.00/5.12 ( ( member_real @ X5 @ field_5140801741446780682s_real )
% 9.00/5.12 & ( ord_less_eq_real @ X2 @ X5 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9748_zero__less__Fract__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 = ( ord_less_int @ zero_zero_int @ A2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9749_Fract__less__zero__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.12 = ( ord_less_int @ A2 @ zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9750_one__less__Fract__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 = ( ord_less_int @ B3 @ A2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9751_Fract__less__one__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
% 9.00/5.12 = ( ord_less_int @ A2 @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9752_zero__le__Fract__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9753_Fract__le__zero__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ zero_zero_rat )
% 9.00/5.12 = ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9754_one__le__Fract__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 = ( ord_less_eq_int @ B3 @ A2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9755_Fract__le__one__iff,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ( ord_less_eq_rat @ ( fract @ A2 @ B3 ) @ one_one_rat )
% 9.00/5.12 = ( ord_less_eq_int @ A2 @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9756_positive__rat,axiom,
% 9.00/5.12 ! [A2: int,B3: int] :
% 9.00/5.12 ( ( positive @ ( fract @ A2 @ B3 ) )
% 9.00/5.12 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A2 @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9757_less__rat__def,axiom,
% 9.00/5.12 ( ord_less_rat
% 9.00/5.12 = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9758_min__Suc__Suc,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N ) )
% 9.00/5.12 = ( suc @ ( ord_min_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9759_min__0L,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_min_nat @ zero_zero_nat @ N )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9760_min__0R,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_min_nat @ N @ zero_zero_nat )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9761_nat__mult__min__right,axiom,
% 9.00/5.12 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.12 ( ( times_times_nat @ M @ ( ord_min_nat @ N @ Q3 ) )
% 9.00/5.12 = ( ord_min_nat @ ( times_times_nat @ M @ N ) @ ( times_times_nat @ M @ Q3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9762_nat__mult__min__left,axiom,
% 9.00/5.12 ! [M: nat,N: nat,Q3: nat] :
% 9.00/5.12 ( ( times_times_nat @ ( ord_min_nat @ M @ N ) @ Q3 )
% 9.00/5.12 = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9763_min__diff,axiom,
% 9.00/5.12 ! [M: nat,I: nat,N: nat] :
% 9.00/5.12 ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N @ I ) )
% 9.00/5.12 = ( minus_minus_nat @ ( ord_min_nat @ M @ N ) @ I ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9764_inf__nat__def,axiom,
% 9.00/5.12 inf_inf_nat = ord_min_nat ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9765_Arg__bounded,axiom,
% 9.00/5.12 ! [Z: complex] :
% 9.00/5.12 ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9766_Arg__correct,axiom,
% 9.00/5.12 ! [Z: complex] :
% 9.00/5.12 ( ( Z != zero_zero_complex )
% 9.00/5.12 => ( ( ( sgn_sgn_complex @ Z )
% 9.00/5.12 = ( cis @ ( arg @ Z ) ) )
% 9.00/5.12 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
% 9.00/5.12 & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9767_bij__betw__roots__unity,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( bij_betw_nat_complex
% 9.00/5.12 @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.00/5.12 @ ( set_ord_lessThan_nat @ N )
% 9.00/5.12 @ ( collect_complex
% 9.00/5.12 @ ^ [Z2: complex] :
% 9.00/5.12 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.12 = one_one_complex ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9768_cis__Arg__unique,axiom,
% 9.00/5.12 ! [Z: complex,X2: real] :
% 9.00/5.12 ( ( ( sgn_sgn_complex @ Z )
% 9.00/5.12 = ( cis @ X2 ) )
% 9.00/5.12 => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ pi )
% 9.00/5.12 => ( ( arg @ Z )
% 9.00/5.12 = X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9769_bij__betw__nth__root__unity,axiom,
% 9.00/5.12 ! [C: complex,N: nat] :
% 9.00/5.12 ( ( C != zero_zero_complex )
% 9.00/5.12 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
% 9.00/5.12 @ ( collect_complex
% 9.00/5.12 @ ^ [Z2: complex] :
% 9.00/5.12 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.12 = one_one_complex ) )
% 9.00/5.12 @ ( collect_complex
% 9.00/5.12 @ ^ [Z2: complex] :
% 9.00/5.12 ( ( power_power_complex @ Z2 @ N )
% 9.00/5.12 = C ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9770_bij__betw__Suc,axiom,
% 9.00/5.12 ! [M5: set_nat,N6: set_nat] :
% 9.00/5.12 ( ( bij_betw_nat_nat @ suc @ M5 @ N6 )
% 9.00/5.12 = ( ( image_nat_nat @ suc @ M5 )
% 9.00/5.12 = N6 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9771_bij__enumerate,axiom,
% 9.00/5.12 ! [S: set_nat] :
% 9.00/5.12 ( ~ ( finite_finite_nat @ S )
% 9.00/5.12 => ( bij_betw_nat_nat @ ( infini8530281810654367211te_nat @ S ) @ top_top_set_nat @ S ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9772_Arg__def,axiom,
% 9.00/5.12 ( arg
% 9.00/5.12 = ( ^ [Z2: complex] :
% 9.00/5.12 ( if_real @ ( Z2 = zero_zero_complex ) @ zero_zero_real
% 9.00/5.12 @ ( fChoice_real
% 9.00/5.12 @ ^ [A4: real] :
% 9.00/5.12 ( ( ( sgn_sgn_complex @ Z2 )
% 9.00/5.12 = ( cis @ A4 ) )
% 9.00/5.12 & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A4 )
% 9.00/5.12 & ( ord_less_eq_real @ A4 @ pi ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9773_less__eq__int_Orep__eq,axiom,
% 9.00/5.12 ( ord_less_eq_int
% 9.00/5.12 = ( ^ [X: int,Xa4: int] :
% 9.00/5.12 ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [Y: nat,Z2: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 9.00/5.12 @ ( rep_Integ @ X )
% 9.00/5.12 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9774_less__int_Orep__eq,axiom,
% 9.00/5.12 ( ord_less_int
% 9.00/5.12 = ( ^ [X: int,Xa4: int] :
% 9.00/5.12 ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [Y: nat,Z2: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V3 ) @ ( plus_plus_nat @ U2 @ Z2 ) ) )
% 9.00/5.12 @ ( rep_Integ @ X )
% 9.00/5.12 @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9775_less__eq__int_Oabs__eq,axiom,
% 9.00/5.12 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 9.00/5.12 ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 9.00/5.12 = ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.00/5.12 @ Xa2
% 9.00/5.12 @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9776_zero__int__def,axiom,
% 9.00/5.12 ( zero_zero_int
% 9.00/5.12 = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9777_int__def,axiom,
% 9.00/5.12 ( semiri1314217659103216013at_int
% 9.00/5.12 = ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N2 @ zero_zero_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9778_one__int__def,axiom,
% 9.00/5.12 ( one_one_int
% 9.00/5.12 = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9779_less__int_Oabs__eq,axiom,
% 9.00/5.12 ! [Xa2: product_prod_nat_nat,X2: product_prod_nat_nat] :
% 9.00/5.12 ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X2 ) )
% 9.00/5.12 = ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.00/5.12 @ Xa2
% 9.00/5.12 @ X2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9780_card__length__sum__list__rec,axiom,
% 9.00/5.12 ! [M: nat,N6: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.00/5.12 => ( ( finite_card_list_nat
% 9.00/5.12 @ ( collect_list_nat
% 9.00/5.12 @ ^ [L2: list_nat] :
% 9.00/5.12 ( ( ( size_size_list_nat @ L2 )
% 9.00/5.12 = M )
% 9.00/5.12 & ( ( groups4561878855575611511st_nat @ L2 )
% 9.00/5.12 = N6 ) ) ) )
% 9.00/5.12 = ( plus_plus_nat
% 9.00/5.12 @ ( finite_card_list_nat
% 9.00/5.12 @ ( collect_list_nat
% 9.00/5.12 @ ^ [L2: list_nat] :
% 9.00/5.12 ( ( ( size_size_list_nat @ L2 )
% 9.00/5.12 = ( minus_minus_nat @ M @ one_one_nat ) )
% 9.00/5.12 & ( ( groups4561878855575611511st_nat @ L2 )
% 9.00/5.12 = N6 ) ) ) )
% 9.00/5.12 @ ( finite_card_list_nat
% 9.00/5.12 @ ( collect_list_nat
% 9.00/5.12 @ ^ [L2: list_nat] :
% 9.00/5.12 ( ( ( size_size_list_nat @ L2 )
% 9.00/5.12 = M )
% 9.00/5.12 & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 9.00/5.12 = N6 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9781_hd__upt,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I @ J )
% 9.00/5.12 => ( ( hd_nat @ ( upt @ I @ J ) )
% 9.00/5.12 = I ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9782_upt__conv__Nil,axiom,
% 9.00/5.12 ! [J: nat,I: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ J @ I )
% 9.00/5.12 => ( ( upt @ I @ J )
% 9.00/5.12 = nil_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9783_upt__eq__Nil__conv,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ( upt @ I @ J )
% 9.00/5.12 = nil_nat )
% 9.00/5.12 = ( ( J = zero_zero_nat )
% 9.00/5.12 | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9784_take__upt,axiom,
% 9.00/5.12 ! [I: nat,M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N )
% 9.00/5.12 => ( ( take_nat @ M @ ( upt @ I @ N ) )
% 9.00/5.12 = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9785_nth__upt,axiom,
% 9.00/5.12 ! [I: nat,K: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 9.00/5.12 => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 9.00/5.12 = ( plus_plus_nat @ I @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9786_upt__rec__numeral,axiom,
% 9.00/5.12 ! [M: num,N: num] :
% 9.00/5.12 ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.12 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.12 = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.12 => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
% 9.00/5.12 = nil_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9787_sum__list__upt,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N ) )
% 9.00/5.12 = ( groups3542108847815614940at_nat
% 9.00/5.12 @ ^ [X: nat] : X
% 9.00/5.12 @ ( set_or4665077453230672383an_nat @ M @ N ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9788_map__add__upt,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( map_nat_nat
% 9.00/5.12 @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
% 9.00/5.12 @ ( upt @ zero_zero_nat @ M ) )
% 9.00/5.12 = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9789_atLeast__upt,axiom,
% 9.00/5.12 ( set_ord_lessThan_nat
% 9.00/5.12 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9790_upt__0,axiom,
% 9.00/5.12 ! [I: nat] :
% 9.00/5.12 ( ( upt @ I @ zero_zero_nat )
% 9.00/5.12 = nil_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9791_sorted__wrt__upt,axiom,
% 9.00/5.12 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9792_upt__add__eq__append,axiom,
% 9.00/5.12 ! [I: nat,J: nat,K: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.12 => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 9.00/5.12 = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9793_sorted__upt,axiom,
% 9.00/5.12 ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9794_atMost__upto,axiom,
% 9.00/5.12 ( set_ord_atMost_nat
% 9.00/5.12 = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9795_upt__rec,axiom,
% 9.00/5.12 ( upt
% 9.00/5.12 = ( ^ [I4: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I4 @ J3 ) @ ( cons_nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9796_upt__conv__Cons,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I @ J )
% 9.00/5.12 => ( ( upt @ I @ J )
% 9.00/5.12 = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9797_upt__Suc__append,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.12 => ( ( upt @ I @ ( suc @ J ) )
% 9.00/5.12 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9798_upt__Suc,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ( ord_less_eq_nat @ I @ J )
% 9.00/5.12 => ( ( upt @ I @ ( suc @ J ) )
% 9.00/5.12 = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_nat @ I @ J )
% 9.00/5.12 => ( ( upt @ I @ ( suc @ J ) )
% 9.00/5.12 = nil_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9799_map__decr__upt,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( map_nat_nat
% 9.00/5.12 @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
% 9.00/5.12 = ( upt @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9800_upt__eq__Cons__conv,axiom,
% 9.00/5.12 ! [I: nat,J: nat,X2: nat,Xs: list_nat] :
% 9.00/5.12 ( ( ( upt @ I @ J )
% 9.00/5.12 = ( cons_nat @ X2 @ Xs ) )
% 9.00/5.12 = ( ( ord_less_nat @ I @ J )
% 9.00/5.12 & ( I = X2 )
% 9.00/5.12 & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 9.00/5.12 = Xs ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9801_sorted__wrt__less__idx,axiom,
% 9.00/5.12 ! [Ns: list_nat,I: nat] :
% 9.00/5.12 ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 9.00/5.12 => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 9.00/5.12 => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9802_sorted__wrt__upto,axiom,
% 9.00/5.12 ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9803_sorted__upto,axiom,
% 9.00/5.12 ! [M: int,N: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9804_Suc__0__div__numeral,axiom,
% 9.00/5.12 ! [K: num] :
% 9.00/5.12 ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9805_bezw_Oelims,axiom,
% 9.00/5.12 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.00/5.12 ( ( ( bezw @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.00/5.12 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9806_bezw_Osimps,axiom,
% 9.00/5.12 ( bezw
% 9.00/5.12 = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9807_bezw__non__0,axiom,
% 9.00/5.12 ! [Y3: nat,X2: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 9.00/5.12 => ( ( bezw @ X2 @ Y3 )
% 9.00/5.12 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Y3 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9808_bezw_Opelims,axiom,
% 9.00/5.12 ! [X2: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.00/5.12 ( ( ( bezw @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 9.00/5.12 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.00/5.12 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X2 @ Xa2 ) ) ) ) ) ) ) )
% 9.00/5.12 => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9809_Suc__0__mod__numeral,axiom,
% 9.00/5.12 ! [K: num] :
% 9.00/5.12 ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9810_Rat_Opositive_Orep__eq,axiom,
% 9.00/5.12 ( positive
% 9.00/5.12 = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9811_normalize__def,axiom,
% 9.00/5.12 ( normalize
% 9.00/5.12 = ( ^ [P5: product_prod_int_int] :
% 9.00/5.12 ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
% 9.00/5.12 @ ( if_Pro3027730157355071871nt_int
% 9.00/5.12 @ ( ( product_snd_int_int @ P5 )
% 9.00/5.12 = zero_zero_int )
% 9.00/5.12 @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.00/5.12 @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9812_gcd__pos__int,axiom,
% 9.00/5.12 ! [M: int,N: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N ) )
% 9.00/5.12 = ( ( M != zero_zero_int )
% 9.00/5.12 | ( N != zero_zero_int ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9813_gcd__ge__0__int,axiom,
% 9.00/5.12 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X2 @ Y3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9814_gcd__le2__int,axiom,
% 9.00/5.12 ! [B3: int,A2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B3 )
% 9.00/5.12 => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ B3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9815_gcd__le1__int,axiom,
% 9.00/5.12 ! [A2: int,B3: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ A2 )
% 9.00/5.12 => ( ord_less_eq_int @ ( gcd_gcd_int @ A2 @ B3 ) @ A2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9816_gcd__cases__int,axiom,
% 9.00/5.12 ! [X2: int,Y3: int,P: int > $o] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.00/5.12 => ( P @ ( gcd_gcd_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 9.00/5.12 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 9.00/5.12 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 9.00/5.12 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.00/5.12 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.00/5.12 => ( P @ ( gcd_gcd_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9817_gcd__unique__int,axiom,
% 9.00/5.12 ! [D: int,A2: int,B3: int] :
% 9.00/5.12 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 9.00/5.12 & ( dvd_dvd_int @ D @ A2 )
% 9.00/5.12 & ( dvd_dvd_int @ D @ B3 )
% 9.00/5.12 & ! [E3: int] :
% 9.00/5.12 ( ( ( dvd_dvd_int @ E3 @ A2 )
% 9.00/5.12 & ( dvd_dvd_int @ E3 @ B3 ) )
% 9.00/5.12 => ( dvd_dvd_int @ E3 @ D ) ) )
% 9.00/5.12 = ( D
% 9.00/5.12 = ( gcd_gcd_int @ A2 @ B3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9818_gcd__non__0__int,axiom,
% 9.00/5.12 ! [Y3: int,X2: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ Y3 )
% 9.00/5.12 => ( ( gcd_gcd_int @ X2 @ Y3 )
% 9.00/5.12 = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9819_gcd__nat_Oeq__neutr__iff,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( ( gcd_gcd_nat @ A2 @ B3 )
% 9.00/5.12 = zero_zero_nat )
% 9.00/5.12 = ( ( A2 = zero_zero_nat )
% 9.00/5.12 & ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9820_gcd__nat_Oleft__neutral,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ( ( gcd_gcd_nat @ zero_zero_nat @ A2 )
% 9.00/5.12 = A2 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9821_gcd__nat_Oneutr__eq__iff,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( zero_zero_nat
% 9.00/5.12 = ( gcd_gcd_nat @ A2 @ B3 ) )
% 9.00/5.12 = ( ( A2 = zero_zero_nat )
% 9.00/5.12 & ( B3 = zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9822_gcd__nat_Oright__neutral,axiom,
% 9.00/5.12 ! [A2: nat] :
% 9.00/5.12 ( ( gcd_gcd_nat @ A2 @ zero_zero_nat )
% 9.00/5.12 = A2 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9823_gcd__0__nat,axiom,
% 9.00/5.12 ! [X2: nat] :
% 9.00/5.12 ( ( gcd_gcd_nat @ X2 @ zero_zero_nat )
% 9.00/5.12 = X2 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9824_gcd__0__left__nat,axiom,
% 9.00/5.12 ! [X2: nat] :
% 9.00/5.12 ( ( gcd_gcd_nat @ zero_zero_nat @ X2 )
% 9.00/5.12 = X2 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9825_gcd__Suc__0,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.00/5.12 = ( suc @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9826_gcd__pos__nat,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N ) )
% 9.00/5.12 = ( ( M != zero_zero_nat )
% 9.00/5.12 | ( N != zero_zero_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9827_Gcd__in,axiom,
% 9.00/5.12 ! [A3: set_nat] :
% 9.00/5.12 ( ! [A: nat,B: nat] :
% 9.00/5.12 ( ( member_nat @ A @ A3 )
% 9.00/5.12 => ( ( member_nat @ B @ A3 )
% 9.00/5.12 => ( member_nat @ ( gcd_gcd_nat @ A @ B ) @ A3 ) ) )
% 9.00/5.12 => ( ( A3 != bot_bot_set_nat )
% 9.00/5.12 => ( member_nat @ ( gcd_Gcd_nat @ A3 ) @ A3 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9828_gcd__non__0__nat,axiom,
% 9.00/5.12 ! [Y3: nat,X2: nat] :
% 9.00/5.12 ( ( Y3 != zero_zero_nat )
% 9.00/5.12 => ( ( gcd_gcd_nat @ X2 @ Y3 )
% 9.00/5.12 = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X2 @ Y3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9829_gcd__nat_Osimps,axiom,
% 9.00/5.12 ( gcd_gcd_nat
% 9.00/5.12 = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9830_gcd__nat_Oelims,axiom,
% 9.00/5.12 ! [X2: nat,Xa2: nat,Y3: nat] :
% 9.00/5.12 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.12 => ( Y3 = X2 ) )
% 9.00/5.12 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9831_gcd__le2__nat,axiom,
% 9.00/5.12 ! [B3: nat,A2: nat] :
% 9.00/5.12 ( ( B3 != zero_zero_nat )
% 9.00/5.12 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ B3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9832_gcd__le1__nat,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( A2 != zero_zero_nat )
% 9.00/5.12 => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A2 @ B3 ) @ A2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9833_gcd__diff1__nat,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ N @ M )
% 9.00/5.12 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N ) @ N )
% 9.00/5.12 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9834_gcd__diff2__nat,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.12 => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M ) @ N )
% 9.00/5.12 = ( gcd_gcd_nat @ M @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9835_bezout__nat,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat] :
% 9.00/5.12 ( ( A2 != zero_zero_nat )
% 9.00/5.12 => ? [X5: nat,Y4: nat] :
% 9.00/5.12 ( ( times_times_nat @ A2 @ X5 )
% 9.00/5.12 = ( plus_plus_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9836_bezout__gcd__nat_H,axiom,
% 9.00/5.12 ! [B3: nat,A2: nat] :
% 9.00/5.12 ? [X5: nat,Y4: nat] :
% 9.00/5.12 ( ( ( ord_less_eq_nat @ ( times_times_nat @ B3 @ Y4 ) @ ( times_times_nat @ A2 @ X5 ) )
% 9.00/5.12 & ( ( minus_minus_nat @ ( times_times_nat @ A2 @ X5 ) @ ( times_times_nat @ B3 @ Y4 ) )
% 9.00/5.12 = ( gcd_gcd_nat @ A2 @ B3 ) ) )
% 9.00/5.12 | ( ( ord_less_eq_nat @ ( times_times_nat @ A2 @ Y4 ) @ ( times_times_nat @ B3 @ X5 ) )
% 9.00/5.12 & ( ( minus_minus_nat @ ( times_times_nat @ B3 @ X5 ) @ ( times_times_nat @ A2 @ Y4 ) )
% 9.00/5.12 = ( gcd_gcd_nat @ A2 @ B3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9837_Gcd__nat__set__eq__fold,axiom,
% 9.00/5.12 ! [Xs: list_nat] :
% 9.00/5.12 ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs ) )
% 9.00/5.12 = ( fold_nat_nat @ gcd_gcd_nat @ Xs @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9838_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.00/5.12 ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 9.00/5.12 @ ^ [M2: nat,N2: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ M2 @ N2 )
% 9.00/5.12 & ( M2 != N2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9839_gcd__is__Max__divisors__nat,axiom,
% 9.00/5.12 ! [N: nat,M: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( ( gcd_gcd_nat @ M @ N )
% 9.00/5.12 = ( lattic8265883725875713057ax_nat
% 9.00/5.12 @ ( collect_nat
% 9.00/5.12 @ ^ [D5: nat] :
% 9.00/5.12 ( ( dvd_dvd_nat @ D5 @ M )
% 9.00/5.12 & ( dvd_dvd_nat @ D5 @ N ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9840_gcd__nat_Opelims,axiom,
% 9.00/5.12 ! [X2: nat,Xa2: nat,Y3: nat] :
% 9.00/5.12 ( ( ( gcd_gcd_nat @ X2 @ Xa2 )
% 9.00/5.12 = Y3 )
% 9.00/5.12 => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) )
% 9.00/5.12 => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.00/5.12 => ( Y3 = X2 ) )
% 9.00/5.12 & ( ( Xa2 != zero_zero_nat )
% 9.00/5.12 => ( Y3
% 9.00/5.12 = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X2 @ Xa2 ) ) ) ) )
% 9.00/5.12 => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9841_Field__natLeq__on,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( field_nat
% 9.00/5.12 @ ( collec3392354462482085612at_nat
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ N )
% 9.00/5.12 & ( ord_less_nat @ Y @ N )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 9.00/5.12 = ( collect_nat
% 9.00/5.12 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9842_natLess__def,axiom,
% 9.00/5.12 ( bNF_Ca8459412986667044542atLess
% 9.00/5.12 = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9843_wf__less,axiom,
% 9.00/5.12 wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9844_cauchyD,axiom,
% 9.00/5.12 ! [X6: nat > rat,R2: rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.00/5.12 => ? [K2: nat] :
% 9.00/5.12 ! [M3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ M3 )
% 9.00/5.12 => ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9845_cauchy__imp__bounded,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ? [B: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.00/5.12 & ! [N5: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ B ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9846_cauchy__def,axiom,
% 9.00/5.12 ( cauchy
% 9.00/5.12 = ( ^ [X8: nat > rat] :
% 9.00/5.12 ! [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 => ? [K3: nat] :
% 9.00/5.12 ! [M2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ M2 )
% 9.00/5.12 => ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M2 ) @ ( X8 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9847_cauchyI,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ! [R3: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 9.00/5.12 => ? [K7: nat] :
% 9.00/5.12 ! [M4: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K7 @ M4 )
% 9.00/5.12 => ! [N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K7 @ N3 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) ) @ R3 ) ) ) )
% 9.00/5.12 => ( cauchy @ X6 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9848_le__Real,axiom,
% 9.00/5.12 ! [X6: nat > rat,Y7: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ( cauchy @ Y7 )
% 9.00/5.12 => ( ( ord_less_eq_real @ ( real2 @ X6 ) @ ( real2 @ Y7 ) )
% 9.00/5.12 = ( ! [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 => ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( plus_plus_rat @ ( Y7 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9849_cauchy__not__vanishes,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ~ ( vanishes @ X6 )
% 9.00/5.12 => ? [B: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.00/5.12 & ? [K2: nat] :
% 9.00/5.12 ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.00/5.12 => ( ord_less_rat @ B @ ( abs_abs_rat @ ( X6 @ N5 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9850_vanishes__mult__bounded,axiom,
% 9.00/5.12 ! [X6: nat > rat,Y7: nat > rat] :
% 9.00/5.12 ( ? [A9: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ A9 )
% 9.00/5.12 & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ A9 ) )
% 9.00/5.12 => ( ( vanishes @ Y7 )
% 9.00/5.12 => ( vanishes
% 9.00/5.12 @ ^ [N2: nat] : ( times_times_rat @ ( X6 @ N2 ) @ ( Y7 @ N2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9851_vanishes__def,axiom,
% 9.00/5.12 ( vanishes
% 9.00/5.12 = ( ^ [X8: nat > rat] :
% 9.00/5.12 ! [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 => ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N2 ) ) @ R5 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9852_vanishesI,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ! [R3: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R3 )
% 9.00/5.12 => ? [K7: nat] :
% 9.00/5.12 ! [N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K7 @ N3 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N3 ) ) @ R3 ) ) )
% 9.00/5.12 => ( vanishes @ X6 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9853_vanishesD,axiom,
% 9.00/5.12 ! [X6: nat > rat,R2: rat] :
% 9.00/5.12 ( ( vanishes @ X6 )
% 9.00/5.12 => ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.00/5.12 => ? [K2: nat] :
% 9.00/5.12 ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.00/5.12 => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N5 ) ) @ R2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9854_cauchy__not__vanishes__cases,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ~ ( vanishes @ X6 )
% 9.00/5.12 => ? [B: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.00/5.12 & ? [K2: nat] :
% 9.00/5.12 ( ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.00/5.12 => ( ord_less_rat @ B @ ( uminus_uminus_rat @ ( X6 @ N5 ) ) ) )
% 9.00/5.12 | ! [N5: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K2 @ N5 )
% 9.00/5.12 => ( ord_less_rat @ B @ ( X6 @ N5 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9855_not__positive__Real,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ( ~ ( positive2 @ ( real2 @ X6 ) ) )
% 9.00/5.12 = ( ! [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 => ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_eq_rat @ ( X6 @ N2 ) @ R5 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9856_positive__Real,axiom,
% 9.00/5.12 ! [X6: nat > rat] :
% 9.00/5.12 ( ( cauchy @ X6 )
% 9.00/5.12 => ( ( positive2 @ ( real2 @ X6 ) )
% 9.00/5.12 = ( ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( X6 @ N2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9857_less__real__def,axiom,
% 9.00/5.12 ( ord_less_real
% 9.00/5.12 = ( ^ [X: real,Y: real] : ( positive2 @ ( minus_minus_real @ Y @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9858_Real_Opositive_Orep__eq,axiom,
% 9.00/5.12 ( positive2
% 9.00/5.12 = ( ^ [X: real] :
% 9.00/5.12 ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( rep_real @ X @ N2 ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9859_mask__nat__positive__iff,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
% 9.00/5.12 = ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9860_not__mask__negative__int,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N ) @ zero_zero_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9861_mask__nonnegative__int,axiom,
% 9.00/5.12 ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9862_less__eq__mask,axiom,
% 9.00/5.12 ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9863_less__mask,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
% 9.00/5.12 => ( ord_less_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9864_mask__nat__less__exp,axiom,
% 9.00/5.12 ! [N: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9865_rat__less__eq__code,axiom,
% 9.00/5.12 ( ord_less_eq_rat
% 9.00/5.12 = ( ^ [P5: rat,Q6: rat] :
% 9.00/5.12 ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [A4: int,C5: int] :
% 9.00/5.12 ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [B4: int,D5: int] : ( ord_less_eq_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
% 9.00/5.12 @ ( quotient_of @ Q6 ) )
% 9.00/5.12 @ ( quotient_of @ P5 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9866_quotient__of__denom__pos,axiom,
% 9.00/5.12 ! [R2: rat,P6: int,Q3: int] :
% 9.00/5.12 ( ( ( quotient_of @ R2 )
% 9.00/5.12 = ( product_Pair_int_int @ P6 @ Q3 ) )
% 9.00/5.12 => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9867_quotient__of__denom__pos_H,axiom,
% 9.00/5.12 ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9868_rat__less__code,axiom,
% 9.00/5.12 ( ord_less_rat
% 9.00/5.12 = ( ^ [P5: rat,Q6: rat] :
% 9.00/5.12 ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [A4: int,C5: int] :
% 9.00/5.12 ( produc4947309494688390418_int_o
% 9.00/5.12 @ ^ [B4: int,D5: int] : ( ord_less_int @ ( times_times_int @ A4 @ D5 ) @ ( times_times_int @ C5 @ B4 ) )
% 9.00/5.12 @ ( quotient_of @ Q6 ) )
% 9.00/5.12 @ ( quotient_of @ P5 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9869_quotient__of__def,axiom,
% 9.00/5.12 ( quotient_of
% 9.00/5.12 = ( ^ [X: rat] :
% 9.00/5.12 ( the_Pr4378521158711661632nt_int
% 9.00/5.12 @ ^ [Pair: product_prod_int_int] :
% 9.00/5.12 ( ( X
% 9.00/5.12 = ( fract @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) )
% 9.00/5.12 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Pair ) )
% 9.00/5.12 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Pair ) @ ( product_snd_int_int @ Pair ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9870_of__real__sqrt,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( real_V4546457046886955230omplex @ ( sqrt @ X2 ) )
% 9.00/5.12 = ( csqrt @ ( real_V4546457046886955230omplex @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9871_normalize__stable,axiom,
% 9.00/5.12 ! [Q3: int,P6: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ Q3 )
% 9.00/5.12 => ( ( algebr932160517623751201me_int @ P6 @ Q3 )
% 9.00/5.12 => ( ( normalize @ ( product_Pair_int_int @ P6 @ Q3 ) )
% 9.00/5.12 = ( product_Pair_int_int @ P6 @ Q3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9872_Rat__induct,axiom,
% 9.00/5.12 ! [P: rat > $o,Q3: rat] :
% 9.00/5.12 ( ! [A: int,B: int] :
% 9.00/5.12 ( ( ord_less_int @ zero_zero_int @ B )
% 9.00/5.12 => ( ( algebr932160517623751201me_int @ A @ B )
% 9.00/5.12 => ( P @ ( fract @ A @ B ) ) ) )
% 9.00/5.12 => ( P @ Q3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9873_Rat__cases,axiom,
% 9.00/5.12 ! [Q3: rat] :
% 9.00/5.12 ~ ! [A: int,B: int] :
% 9.00/5.12 ( ( Q3
% 9.00/5.12 = ( fract @ A @ B ) )
% 9.00/5.12 => ( ( ord_less_int @ zero_zero_int @ B )
% 9.00/5.12 => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9874_Rat__cases__nonzero,axiom,
% 9.00/5.12 ! [Q3: rat] :
% 9.00/5.12 ( ! [A: int,B: int] :
% 9.00/5.12 ( ( Q3
% 9.00/5.12 = ( fract @ A @ B ) )
% 9.00/5.12 => ( ( ord_less_int @ zero_zero_int @ B )
% 9.00/5.12 => ( ( A != zero_zero_int )
% 9.00/5.12 => ~ ( algebr932160517623751201me_int @ A @ B ) ) ) )
% 9.00/5.12 => ( Q3 = zero_zero_rat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9875_quotient__of__unique,axiom,
% 9.00/5.12 ! [R2: rat] :
% 9.00/5.12 ? [X5: product_prod_int_int] :
% 9.00/5.12 ( ( R2
% 9.00/5.12 = ( fract @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) ) )
% 9.00/5.12 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ X5 ) )
% 9.00/5.12 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ X5 ) @ ( product_snd_int_int @ X5 ) )
% 9.00/5.12 & ! [Y5: product_prod_int_int] :
% 9.00/5.12 ( ( ( R2
% 9.00/5.12 = ( fract @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 9.00/5.12 & ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ Y5 ) )
% 9.00/5.12 & ( algebr932160517623751201me_int @ ( product_fst_int_int @ Y5 ) @ ( product_snd_int_int @ Y5 ) ) )
% 9.00/5.12 => ( Y5 = X5 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9876_coprime__Suc__0__right,axiom,
% 9.00/5.12 ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9877_coprime__Suc__0__left,axiom,
% 9.00/5.12 ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9878_coprime__diff__one__left__nat,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9879_coprime__diff__one__right__nat,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.12 => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9880_Rats__abs__nat__div__natE,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( member_real @ X2 @ field_5140801741446780682s_real )
% 9.00/5.12 => ~ ! [M4: nat,N3: nat] :
% 9.00/5.12 ( ( N3 != zero_zero_nat )
% 9.00/5.12 => ( ( ( abs_abs_real @ X2 )
% 9.00/5.12 = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M4 ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
% 9.00/5.12 => ~ ( algebr934650988132801477me_nat @ M4 @ N3 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9881_atLeastLessThan__nat__numeral,axiom,
% 9.00/5.12 ! [M: nat,K: num] :
% 9.00/5.12 ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.00/5.12 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 9.00/5.12 & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.00/5.12 => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = bot_bot_set_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9882_pred__numeral__simps_I1_J,axiom,
% 9.00/5.12 ( ( pred_numeral @ one )
% 9.00/5.12 = zero_zero_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9883_less__Suc__numeral,axiom,
% 9.00/5.12 ! [N: nat,K: num] :
% 9.00/5.12 ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( ord_less_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9884_less__numeral__Suc,axiom,
% 9.00/5.12 ! [K: num,N: nat] :
% 9.00/5.12 ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.00/5.12 = ( ord_less_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9885_le__numeral__Suc,axiom,
% 9.00/5.12 ! [K: num,N: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N ) )
% 9.00/5.12 = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9886_le__Suc__numeral,axiom,
% 9.00/5.12 ! [N: nat,K: num] :
% 9.00/5.12 ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( ord_less_eq_nat @ N @ ( pred_numeral @ K ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9887_lessThan__nat__numeral,axiom,
% 9.00/5.12 ! [K: num] :
% 9.00/5.12 ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9888_atMost__nat__numeral,axiom,
% 9.00/5.12 ! [K: num] :
% 9.00/5.12 ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
% 9.00/5.12 = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9889_last__upt,axiom,
% 9.00/5.12 ! [I: nat,J: nat] :
% 9.00/5.12 ( ( ord_less_nat @ I @ J )
% 9.00/5.12 => ( ( last_nat @ ( upt @ I @ J ) )
% 9.00/5.12 = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9890_Real_Opositive_Oabs__eq,axiom,
% 9.00/5.12 ! [X2: nat > rat] :
% 9.00/5.12 ( ( realrel @ X2 @ X2 )
% 9.00/5.12 => ( ( positive2 @ ( real2 @ X2 ) )
% 9.00/5.12 = ( ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( X2 @ N2 ) ) ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9891_not__negative__int__iff,axiom,
% 9.00/5.12 ! [K: int] :
% 9.00/5.12 ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 9.00/5.12 = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9892_not__nonnegative__int__iff,axiom,
% 9.00/5.12 ! [K: int] :
% 9.00/5.12 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.00/5.12 = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9893_Real_Opositive_Orsp,axiom,
% 9.00/5.12 ( bNF_re728719798268516973at_o_o @ realrel
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [X8: nat > rat] :
% 9.00/5.12 ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
% 9.00/5.12 @ ^ [X8: nat > rat] :
% 9.00/5.12 ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9894_less__eq__integer_Orsp,axiom,
% 9.00/5.12 ( bNF_re3403563459893282935_int_o
% 9.00/5.12 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 9.00/5.12 @ ( bNF_re5089333283451836215nt_o_o
% 9.00/5.12 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ord_less_eq_int
% 9.00/5.12 @ ord_less_eq_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9895_less__eq__natural_Orsp,axiom,
% 9.00/5.12 ( bNF_re578469030762574527_nat_o
% 9.00/5.12 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.12 @ ( bNF_re4705727531993890431at_o_o
% 9.00/5.12 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ord_less_eq_nat
% 9.00/5.12 @ ord_less_eq_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9896_less__natural_Orsp,axiom,
% 9.00/5.12 ( bNF_re578469030762574527_nat_o
% 9.00/5.12 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.12 @ ( bNF_re4705727531993890431at_o_o
% 9.00/5.12 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ord_less_nat
% 9.00/5.12 @ ord_less_nat ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9897_less__integer_Orsp,axiom,
% 9.00/5.12 ( bNF_re3403563459893282935_int_o
% 9.00/5.12 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 9.00/5.12 @ ( bNF_re5089333283451836215nt_o_o
% 9.00/5.12 @ ^ [Y6: int,Z3: int] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ord_less_int
% 9.00/5.12 @ ord_less_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9898_Real_Opositive_Otransfer,axiom,
% 9.00/5.12 ( bNF_re4297313714947099218al_o_o @ pcr_real
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [X8: nat > rat] :
% 9.00/5.12 ? [R5: rat] :
% 9.00/5.12 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.12 & ? [K3: nat] :
% 9.00/5.12 ! [N2: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.12 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) )
% 9.00/5.12 @ positive2 ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9899_Rat_Opositive_Otransfer,axiom,
% 9.00/5.12 ( bNF_re1494630372529172596at_o_o @ pcr_rat
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 9.00/5.12 @ positive ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9900_less__eq__int_Otransfer,axiom,
% 9.00/5.12 ( bNF_re717283939379294677_int_o @ pcr_int
% 9.00/5.12 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.00/5.12 @ ord_less_eq_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9901_zero__int_Otransfer,axiom,
% 9.00/5.12 pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9902_int__transfer,axiom,
% 9.00/5.12 ( bNF_re6830278522597306478at_int
% 9.00/5.12 @ ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 )
% 9.00/5.12 @ pcr_int
% 9.00/5.12 @ ^ [N2: nat] : ( product_Pair_nat_nat @ N2 @ zero_zero_nat )
% 9.00/5.12 @ semiri1314217659103216013at_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9903_one__int_Otransfer,axiom,
% 9.00/5.12 pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9904_less__int_Otransfer,axiom,
% 9.00/5.12 ( bNF_re717283939379294677_int_o @ pcr_int
% 9.00/5.12 @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.12 @ ( produc8739625826339149834_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.00/5.12 @ ord_less_int ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9905_Rat_Opositive_Orsp,axiom,
% 9.00/5.12 ( bNF_re8699439704749558557nt_o_o @ ratrel
% 9.00/5.12 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 )
% 9.00/5.12 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) )
% 9.00/5.12 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9906_Rat_Opositive_Oabs__eq,axiom,
% 9.00/5.12 ! [X2: product_prod_int_int] :
% 9.00/5.12 ( ( ratrel @ X2 @ X2 )
% 9.00/5.12 => ( ( positive @ ( abs_Rat @ X2 ) )
% 9.00/5.12 = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X2 ) @ ( product_snd_int_int @ X2 ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9907_vimage__Suc__insert__0,axiom,
% 9.00/5.12 ! [A3: set_nat] :
% 9.00/5.12 ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A3 ) )
% 9.00/5.12 = ( vimage_nat_nat @ suc @ A3 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9908_finite__vimage__Suc__iff,axiom,
% 9.00/5.12 ! [F2: set_nat] :
% 9.00/5.12 ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F2 ) )
% 9.00/5.12 = ( finite_finite_nat @ F2 ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9909_pairs__le__eq__Sigma,axiom,
% 9.00/5.12 ! [M: nat] :
% 9.00/5.12 ( ( collec3392354462482085612at_nat
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M ) ) )
% 9.00/5.12 = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 9.00/5.12 @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9910_Restr__natLeq,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.00/5.12 @ ( produc457027306803732586at_nat
% 9.00/5.12 @ ( collect_nat
% 9.00/5.12 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
% 9.00/5.12 @ ^ [Uu3: nat] :
% 9.00/5.12 ( collect_nat
% 9.00/5.12 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) )
% 9.00/5.12 = ( collec3392354462482085612at_nat
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ N )
% 9.00/5.12 & ( ord_less_nat @ Y @ N )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9911_natLeq__def,axiom,
% 9.00/5.12 ( bNF_Ca8665028551170535155natLeq
% 9.00/5.12 = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9912_Restr__natLeq2,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.00/5.12 @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 9.00/5.12 @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
% 9.00/5.12 = ( collec3392354462482085612at_nat
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ N )
% 9.00/5.12 & ( ord_less_nat @ Y @ N )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9913_natLeq__underS__less,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
% 9.00/5.12 = ( collect_nat
% 9.00/5.12 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9914_natLeq__on__wo__rel,axiom,
% 9.00/5.12 ! [N: nat] :
% 9.00/5.12 ( bNF_We3818239936649020644el_nat
% 9.00/5.12 @ ( collec3392354462482085612at_nat
% 9.00/5.12 @ ( produc6081775807080527818_nat_o
% 9.00/5.12 @ ^ [X: nat,Y: nat] :
% 9.00/5.12 ( ( ord_less_nat @ X @ N )
% 9.00/5.12 & ( ord_less_nat @ Y @ N )
% 9.00/5.12 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9915_pred__nat__trancl__eq__le,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 9.00/5.12 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9916_Bseq__monoseq__convergent_H__inc,axiom,
% 9.00/5.12 ! [F: nat > real,M5: nat] :
% 9.00/5.12 ( ( bfun_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 9.00/5.12 @ at_top_nat )
% 9.00/5.12 => ( ! [M4: nat,N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M5 @ M4 )
% 9.00/5.12 => ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ M4 ) @ ( F @ N3 ) ) ) )
% 9.00/5.12 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9917_Bseq__mono__convergent,axiom,
% 9.00/5.12 ! [X6: nat > real] :
% 9.00/5.12 ( ( bfun_nat_real @ X6 @ at_top_nat )
% 9.00/5.12 => ( ! [M4: nat,N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( X6 @ M4 ) @ ( X6 @ N3 ) ) )
% 9.00/5.12 => ( topolo7531315842566124627t_real @ X6 ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9918_convergent__realpow,axiom,
% 9.00/5.12 ! [X2: real] :
% 9.00/5.12 ( ( ord_less_eq_real @ zero_zero_real @ X2 )
% 9.00/5.12 => ( ( ord_less_eq_real @ X2 @ one_one_real )
% 9.00/5.12 => ( topolo7531315842566124627t_real @ ( power_power_real @ X2 ) ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9919_less__eq,axiom,
% 9.00/5.12 ! [M: nat,N: nat] :
% 9.00/5.12 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 9.00/5.12 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9920_Bseq__monoseq__convergent_H__dec,axiom,
% 9.00/5.12 ! [F: nat > real,M5: nat] :
% 9.00/5.12 ( ( bfun_nat_real
% 9.00/5.12 @ ^ [N2: nat] : ( F @ ( plus_plus_nat @ N2 @ M5 ) )
% 9.00/5.12 @ at_top_nat )
% 9.00/5.12 => ( ! [M4: nat,N3: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ M5 @ M4 )
% 9.00/5.12 => ( ( ord_less_eq_nat @ M4 @ N3 )
% 9.00/5.12 => ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ M4 ) ) ) )
% 9.00/5.12 => ( topolo7531315842566124627t_real @ F ) ) ) ).
% 9.00/5.12
% 9.00/5.12 thf(fact_9921_pair__lessI2,axiom,
% 9.00/5.12 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 9.00/5.12 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.12 => ( ( ord_less_nat @ S2 @ T )
% 9.00/5.12 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9922_trans__pair__less,axiom,
% 9.00/5.13 trans_4347625901269045472at_nat @ fun_pair_less ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9923_total__pair__less,axiom,
% 9.00/5.13 ! [A3: set_Pr1261947904930325089at_nat] : ( total_3592101749530773125at_nat @ A3 @ fun_pair_less ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9924_pair__less__iff1,axiom,
% 9.00/5.13 ! [X2: nat,Y3: nat,Z: nat] :
% 9.00/5.13 ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( product_Pair_nat_nat @ X2 @ Z ) ) @ fun_pair_less )
% 9.00/5.13 = ( ord_less_nat @ Y3 @ Z ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9925_wf__pair__less,axiom,
% 9.00/5.13 wf_Pro7803398752247294826at_nat @ fun_pair_less ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9926_pair__lessI1,axiom,
% 9.00/5.13 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 9.00/5.13 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.13 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9927_gcd__nat_Oordering__top__axioms,axiom,
% 9.00/5.13 ( ordering_top_nat @ dvd_dvd_nat
% 9.00/5.13 @ ^ [M2: nat,N2: nat] :
% 9.00/5.13 ( ( dvd_dvd_nat @ M2 @ N2 )
% 9.00/5.13 & ( M2 != N2 ) )
% 9.00/5.13 @ zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9928_bot__nat__0_Oordering__top__axioms,axiom,
% 9.00/5.13 ( ordering_top_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9929_pair__leqI2,axiom,
% 9.00/5.13 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 9.00/5.13 ( ( ord_less_eq_nat @ A2 @ B3 )
% 9.00/5.13 => ( ( ord_less_eq_nat @ S2 @ T )
% 9.00/5.13 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9930_pair__leqI1,axiom,
% 9.00/5.13 ! [A2: nat,B3: nat,S2: nat,T: nat] :
% 9.00/5.13 ( ( ord_less_nat @ A2 @ B3 )
% 9.00/5.13 => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A2 @ S2 ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9931_pair__leq__def,axiom,
% 9.00/5.13 ( fun_pair_leq
% 9.00/5.13 = ( sup_su718114333110466843at_nat @ fun_pair_less @ id_Pro2258643101195443293at_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9932_wmin__insertI,axiom,
% 9.00/5.13 ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( member8440522571783428010at_nat @ X2 @ XS )
% 9.00/5.13 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
% 9.00/5.13 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9933_wmax__insertI,axiom,
% 9.00/5.13 ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( member8440522571783428010at_nat @ Y3 @ YS )
% 9.00/5.13 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_leq )
% 9.00/5.13 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9934_wmin__emptyI,axiom,
% 9.00/5.13 ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9935_wmax__emptyI,axiom,
% 9.00/5.13 ! [X6: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( finite6177210948735845034at_nat @ X6 )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9936_max__weak__def,axiom,
% 9.00/5.13 ( fun_max_weak
% 9.00/5.13 = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9937_min__weak__def,axiom,
% 9.00/5.13 ( fun_min_weak
% 9.00/5.13 = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9938_smin__insertI,axiom,
% 9.00/5.13 ! [X2: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( member8440522571783428010at_nat @ X2 @ XS )
% 9.00/5.13 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
% 9.00/5.13 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9939_smax__insertI,axiom,
% 9.00/5.13 ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X2: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
% 9.00/5.13 => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y3 ) @ fun_pair_less )
% 9.00/5.13 => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y7 ) @ fun_max_strict )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X2 @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9940_smax__emptyI,axiom,
% 9.00/5.13 ! [Y7: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( finite6177210948735845034at_nat @ Y7 )
% 9.00/5.13 => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9941_smin__emptyI,axiom,
% 9.00/5.13 ! [X6: set_Pr1261947904930325089at_nat] :
% 9.00/5.13 ( ( X6 != bot_bo2099793752762293965at_nat )
% 9.00/5.13 => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9942_min__strict__def,axiom,
% 9.00/5.13 ( fun_min_strict
% 9.00/5.13 = ( min_ex6901939911449802026at_nat @ fun_pair_less ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9943_max__strict__def,axiom,
% 9.00/5.13 ( fun_max_strict
% 9.00/5.13 = ( max_ex8135407076693332796at_nat @ fun_pair_less ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9944_min__rpair__set,axiom,
% 9.00/5.13 fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9945_max__rpair__set,axiom,
% 9.00/5.13 fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9946_bit__concat__bit__iff,axiom,
% 9.00/5.13 ! [M: nat,K: int,L: int,N: nat] :
% 9.00/5.13 ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N )
% 9.00/5.13 = ( ( ( ord_less_nat @ N @ M )
% 9.00/5.13 & ( bit_se1146084159140164899it_int @ K @ N ) )
% 9.00/5.13 | ( ( ord_less_eq_nat @ M @ N )
% 9.00/5.13 & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9947_concat__bit__0,axiom,
% 9.00/5.13 ! [K: int,L: int] :
% 9.00/5.13 ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 9.00/5.13 = L ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9948_concat__bit__nonnegative__iff,axiom,
% 9.00/5.13 ! [N: nat,K: int,L: int] :
% 9.00/5.13 ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N @ K @ L ) )
% 9.00/5.13 = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9949_concat__bit__negative__iff,axiom,
% 9.00/5.13 ! [N: nat,K: int,L: int] :
% 9.00/5.13 ( ( ord_less_int @ ( bit_concat_bit @ N @ K @ L ) @ zero_zero_int )
% 9.00/5.13 = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9950_less__eq__int_Orsp,axiom,
% 9.00/5.13 ( bNF_re4202695980764964119_nat_o @ intrel
% 9.00/5.13 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.00/5.13 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9951_zero__int_Orsp,axiom,
% 9.00/5.13 intrel @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9952_one__int_Orsp,axiom,
% 9.00/5.13 intrel @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9953_less__int_Orsp,axiom,
% 9.00/5.13 ( bNF_re4202695980764964119_nat_o @ intrel
% 9.00/5.13 @ ( bNF_re3666534408544137501at_o_o @ intrel
% 9.00/5.13 @ ^ [Y6: $o,Z3: $o] : ( Y6 = Z3 ) )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9954_division__segment__int__def,axiom,
% 9.00/5.13 ( euclid3395696857347342551nt_int
% 9.00/5.13 = ( ^ [K3: int] : ( if_int @ ( ord_less_eq_int @ zero_zero_int @ K3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9955_less__eq__enat__def,axiom,
% 9.00/5.13 ( ord_le2932123472753598470d_enat
% 9.00/5.13 = ( ^ [M2: extended_enat] :
% 9.00/5.13 ( extended_case_enat_o
% 9.00/5.13 @ ^ [N1: nat] :
% 9.00/5.13 ( extended_case_enat_o
% 9.00/5.13 @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
% 9.00/5.13 @ $false
% 9.00/5.13 @ M2 )
% 9.00/5.13 @ $true ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9956_less__enat__def,axiom,
% 9.00/5.13 ( ord_le72135733267957522d_enat
% 9.00/5.13 = ( ^ [M2: extended_enat,N2: extended_enat] :
% 9.00/5.13 ( extended_case_enat_o
% 9.00/5.13 @ ^ [M1: nat] : ( extended_case_enat_o @ ( ord_less_nat @ M1 ) @ $true @ N2 )
% 9.00/5.13 @ $false
% 9.00/5.13 @ M2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9957_of__rat__dense,axiom,
% 9.00/5.13 ! [X2: real,Y3: real] :
% 9.00/5.13 ( ( ord_less_real @ X2 @ Y3 )
% 9.00/5.13 => ? [Q4: rat] :
% 9.00/5.13 ( ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ Q4 ) )
% 9.00/5.13 & ( ord_less_real @ ( field_7254667332652039916t_real @ Q4 ) @ Y3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9958_less__RealD,axiom,
% 9.00/5.13 ! [Y7: nat > rat,X2: real] :
% 9.00/5.13 ( ( cauchy @ Y7 )
% 9.00/5.13 => ( ( ord_less_real @ X2 @ ( real2 @ Y7 ) )
% 9.00/5.13 => ? [N3: nat] : ( ord_less_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9959_Real__leI,axiom,
% 9.00/5.13 ! [X6: nat > rat,Y3: real] :
% 9.00/5.13 ( ( cauchy @ X6 )
% 9.00/5.13 => ( ! [N3: nat] : ( ord_less_eq_real @ ( field_7254667332652039916t_real @ ( X6 @ N3 ) ) @ Y3 )
% 9.00/5.13 => ( ord_less_eq_real @ ( real2 @ X6 ) @ Y3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9960_le__RealI,axiom,
% 9.00/5.13 ! [Y7: nat > rat,X2: real] :
% 9.00/5.13 ( ( cauchy @ Y7 )
% 9.00/5.13 => ( ! [N3: nat] : ( ord_less_eq_real @ X2 @ ( field_7254667332652039916t_real @ ( Y7 @ N3 ) ) )
% 9.00/5.13 => ( ord_less_eq_real @ X2 @ ( real2 @ Y7 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9961_compute__powr__real,axiom,
% 9.00/5.13 ( powr_real2
% 9.00/5.13 = ( ^ [B4: real,I4: real] :
% 9.00/5.13 ( if_real @ ( ord_less_eq_real @ B4 @ zero_zero_real )
% 9.00/5.13 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.13 @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) )
% 9.00/5.13 @ ( if_real
% 9.00/5.13 @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
% 9.00/5.13 = I4 )
% 9.00/5.13 @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B4 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
% 9.00/5.13 @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.00/5.13 @ ^ [Uu3: product_unit] : ( powr_real2 @ B4 @ I4 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9962_UNIV__char__of__nat,axiom,
% 9.00/5.13 ( top_top_set_char
% 9.00/5.13 = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9963_inj__on__char__of__nat,axiom,
% 9.00/5.13 inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9964_range__nat__of__char,axiom,
% 9.00/5.13 ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
% 9.00/5.13 = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9965_nat__of__char__less__256,axiom,
% 9.00/5.13 ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9966_numeral__le__enat__iff,axiom,
% 9.00/5.13 ! [M: num,N: nat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 9.00/5.13 = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9967_enat__ord__simps_I2_J,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 9.00/5.13 = ( ord_less_nat @ M @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9968_enat__ord__simps_I1_J,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
% 9.00/5.13 = ( ord_less_eq_nat @ M @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9969_idiff__enat__0__right,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
% 9.00/5.13 = N ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9970_idiff__enat__0,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
% 9.00/5.13 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9971_numeral__less__enat__iff,axiom,
% 9.00/5.13 ! [M: num,N: nat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( extended_enat2 @ N ) )
% 9.00/5.13 = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9972_zero__enat__def,axiom,
% 9.00/5.13 ( zero_z5237406670263579293d_enat
% 9.00/5.13 = ( extended_enat2 @ zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9973_enat__0__iff_I1_J,axiom,
% 9.00/5.13 ! [X2: nat] :
% 9.00/5.13 ( ( ( extended_enat2 @ X2 )
% 9.00/5.13 = zero_z5237406670263579293d_enat )
% 9.00/5.13 = ( X2 = zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9974_enat__0__iff_I2_J,axiom,
% 9.00/5.13 ! [X2: nat] :
% 9.00/5.13 ( ( zero_z5237406670263579293d_enat
% 9.00/5.13 = ( extended_enat2 @ X2 ) )
% 9.00/5.13 = ( X2 = zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9975_finite__enat__bounded,axiom,
% 9.00/5.13 ! [A3: set_Extended_enat,N: nat] :
% 9.00/5.13 ( ! [Y4: extended_enat] :
% 9.00/5.13 ( ( member_Extended_enat @ Y4 @ A3 )
% 9.00/5.13 => ( ord_le2932123472753598470d_enat @ Y4 @ ( extended_enat2 @ N ) ) )
% 9.00/5.13 => ( finite4001608067531595151d_enat @ A3 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9976_enat__ile,axiom,
% 9.00/5.13 ! [N: extended_enat,M: nat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ N @ ( extended_enat2 @ M ) )
% 9.00/5.13 => ? [K2: nat] :
% 9.00/5.13 ( N
% 9.00/5.13 = ( extended_enat2 @ K2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9977_enat__iless,axiom,
% 9.00/5.13 ! [N: extended_enat,M: nat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 9.00/5.13 => ? [K2: nat] :
% 9.00/5.13 ( N
% 9.00/5.13 = ( extended_enat2 @ K2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9978_less__enatE,axiom,
% 9.00/5.13 ! [N: extended_enat,M: nat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ N @ ( extended_enat2 @ M ) )
% 9.00/5.13 => ~ ! [K2: nat] :
% 9.00/5.13 ( ( N
% 9.00/5.13 = ( extended_enat2 @ K2 ) )
% 9.00/5.13 => ~ ( ord_less_nat @ K2 @ M ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9979_Suc__ile__eq,axiom,
% 9.00/5.13 ! [M: nat,N: extended_enat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
% 9.00/5.13 = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9980_iadd__le__enat__iff,axiom,
% 9.00/5.13 ! [X2: extended_enat,Y3: extended_enat,N: nat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X2 @ Y3 ) @ ( extended_enat2 @ N ) )
% 9.00/5.13 = ( ? [Y8: nat,X9: nat] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( extended_enat2 @ X9 ) )
% 9.00/5.13 & ( Y3
% 9.00/5.13 = ( extended_enat2 @ Y8 ) )
% 9.00/5.13 & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9981_elimnum,axiom,
% 9.00/5.13 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.00/5.13 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.13 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.00/5.13 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9982_times__enat__simps_I3_J,axiom,
% 9.00/5.13 ! [N: nat] :
% 9.00/5.13 ( ( ( N = zero_zero_nat )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 9.00/5.13 = zero_z5237406670263579293d_enat ) )
% 9.00/5.13 & ( ( N != zero_zero_nat )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
% 9.00/5.13 = extend5688581933313929465d_enat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9983_elimcomplete,axiom,
% 9.00/5.13 ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
% 9.00/5.13 ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
% 9.00/5.13 => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
% 9.00/5.13 = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9984_enat__ord__simps_I4_J,axiom,
% 9.00/5.13 ! [Q3: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ Q3 @ extend5688581933313929465d_enat )
% 9.00/5.13 = ( Q3 != extend5688581933313929465d_enat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9985_enat__ord__simps_I6_J,axiom,
% 9.00/5.13 ! [Q3: extended_enat] :
% 9.00/5.13 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ Q3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9986_enat__ord__code_I3_J,axiom,
% 9.00/5.13 ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9987_enat__ord__simps_I5_J,axiom,
% 9.00/5.13 ! [Q3: extended_enat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ Q3 )
% 9.00/5.13 = ( Q3 = extend5688581933313929465d_enat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9988_times__enat__simps_I4_J,axiom,
% 9.00/5.13 ! [M: nat] :
% 9.00/5.13 ( ( ( M = zero_zero_nat )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 9.00/5.13 = zero_z5237406670263579293d_enat ) )
% 9.00/5.13 & ( ( M != zero_zero_nat )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat )
% 9.00/5.13 = extend5688581933313929465d_enat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9989_enat__ord__code_I5_J,axiom,
% 9.00/5.13 ! [N: nat] :
% 9.00/5.13 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9990_infinity__ileE,axiom,
% 9.00/5.13 ! [M: nat] :
% 9.00/5.13 ~ ( ord_le2932123472753598470d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9991_enat__add__left__cancel__less,axiom,
% 9.00/5.13 ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
% 9.00/5.13 = ( ( A2 != extend5688581933313929465d_enat )
% 9.00/5.13 & ( ord_le72135733267957522d_enat @ B3 @ C ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9992_enat__ord__code_I4_J,axiom,
% 9.00/5.13 ! [M: nat] : ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ extend5688581933313929465d_enat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9993_less__infinityE,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ N @ extend5688581933313929465d_enat )
% 9.00/5.13 => ~ ! [K2: nat] :
% 9.00/5.13 ( N
% 9.00/5.13 != ( extended_enat2 @ K2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9994_infinity__ilessE,axiom,
% 9.00/5.13 ! [M: nat] :
% 9.00/5.13 ~ ( ord_le72135733267957522d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9995_enat__add__left__cancel__le,axiom,
% 9.00/5.13 ! [A2: extended_enat,B3: extended_enat,C: extended_enat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ A2 @ B3 ) @ ( plus_p3455044024723400733d_enat @ A2 @ C ) )
% 9.00/5.13 = ( ( A2 = extend5688581933313929465d_enat )
% 9.00/5.13 | ( ord_le2932123472753598470d_enat @ B3 @ C ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9996_enat__ord__simps_I3_J,axiom,
% 9.00/5.13 ! [Q3: extended_enat] : ( ord_le2932123472753598470d_enat @ Q3 @ extend5688581933313929465d_enat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9997_Inf__enat__def,axiom,
% 9.00/5.13 ( comple2295165028678016749d_enat
% 9.00/5.13 = ( ^ [A6: set_Extended_enat] :
% 9.00/5.13 ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ extend5688581933313929465d_enat
% 9.00/5.13 @ ( ord_Le1955565732374568822d_enat
% 9.00/5.13 @ ^ [X: extended_enat] : ( member_Extended_enat @ X @ A6 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9998_Sup__enat__def,axiom,
% 9.00/5.13 ( comple4398354569131411667d_enat
% 9.00/5.13 = ( ^ [A6: set_Extended_enat] : ( if_Extended_enat @ ( A6 = bot_bo7653980558646680370d_enat ) @ zero_z5237406670263579293d_enat @ ( if_Extended_enat @ ( finite4001608067531595151d_enat @ A6 ) @ ( lattic921264341876707157d_enat @ A6 ) @ extend5688581933313929465d_enat ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_9999_imult__infinity,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
% 9.00/5.13 = extend5688581933313929465d_enat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10000_imult__infinity__right,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
% 9.00/5.13 => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
% 9.00/5.13 = extend5688581933313929465d_enat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10001_times__enat__def,axiom,
% 9.00/5.13 ( times_7803423173614009249d_enat
% 9.00/5.13 = ( ^ [M2: extended_enat,N2: extended_enat] :
% 9.00/5.13 ( extend3600170679010898289d_enat
% 9.00/5.13 @ ^ [O: nat] :
% 9.00/5.13 ( extend3600170679010898289d_enat
% 9.00/5.13 @ ^ [P5: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P5 ) )
% 9.00/5.13 @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 9.00/5.13 @ N2 )
% 9.00/5.13 @ ( if_Extended_enat @ ( N2 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
% 9.00/5.13 @ M2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10002_iless__Suc__eq,axiom,
% 9.00/5.13 ! [M: nat,N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
% 9.00/5.13 = ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10003_eSuc__mono,axiom,
% 9.00/5.13 ! [N: extended_enat,M: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 9.00/5.13 = ( ord_le72135733267957522d_enat @ N @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10004_eSuc__ile__mono,axiom,
% 9.00/5.13 ! [N: extended_enat,M: extended_enat] :
% 9.00/5.13 ( ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
% 9.00/5.13 = ( ord_le2932123472753598470d_enat @ N @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10005_iless__eSuc0,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ N @ ( extended_eSuc @ zero_z5237406670263579293d_enat ) )
% 9.00/5.13 = ( N = zero_z5237406670263579293d_enat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10006_eSuc__Max,axiom,
% 9.00/5.13 ! [A3: set_Extended_enat] :
% 9.00/5.13 ( ( finite4001608067531595151d_enat @ A3 )
% 9.00/5.13 => ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.13 => ( ( extended_eSuc @ ( lattic921264341876707157d_enat @ A3 ) )
% 9.00/5.13 = ( lattic921264341876707157d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10007_eSuc__Sup,axiom,
% 9.00/5.13 ! [A3: set_Extended_enat] :
% 9.00/5.13 ( ( A3 != bot_bo7653980558646680370d_enat )
% 9.00/5.13 => ( ( extended_eSuc @ ( comple4398354569131411667d_enat @ A3 ) )
% 9.00/5.13 = ( comple4398354569131411667d_enat @ ( image_80655429650038917d_enat @ extended_eSuc @ A3 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10008_not__eSuc__ilei0,axiom,
% 9.00/5.13 ! [N: extended_enat] :
% 9.00/5.13 ~ ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ N ) @ zero_z5237406670263579293d_enat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10009_ile__eSuc,axiom,
% 9.00/5.13 ! [N: extended_enat] : ( ord_le2932123472753598470d_enat @ N @ ( extended_eSuc @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10010_ileI1,axiom,
% 9.00/5.13 ! [M: extended_enat,N: extended_enat] :
% 9.00/5.13 ( ( ord_le72135733267957522d_enat @ M @ N )
% 9.00/5.13 => ( ord_le2932123472753598470d_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10011_i0__iless__eSuc,axiom,
% 9.00/5.13 ! [N: extended_enat] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( extended_eSuc @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10012_less__than__iff,axiom,
% 9.00/5.13 ! [X2: nat,Y3: nat] :
% 9.00/5.13 ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ less_than )
% 9.00/5.13 = ( ord_less_nat @ X2 @ Y3 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10013_pair__less__def,axiom,
% 9.00/5.13 ( fun_pair_less
% 9.00/5.13 = ( lex_prod_nat_nat @ less_than @ less_than ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10014_natLeq__on__well__order__on,axiom,
% 9.00/5.13 ! [N: nat] :
% 9.00/5.13 ( order_2888998067076097458on_nat
% 9.00/5.13 @ ( collect_nat
% 9.00/5.13 @ ^ [X: nat] : ( ord_less_nat @ X @ N ) )
% 9.00/5.13 @ ( collec3392354462482085612at_nat
% 9.00/5.13 @ ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( ( ord_less_nat @ X @ N )
% 9.00/5.13 & ( ord_less_nat @ Y @ N )
% 9.00/5.13 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10015_natLeq__on__Well__order,axiom,
% 9.00/5.13 ! [N: nat] :
% 9.00/5.13 ( order_2888998067076097458on_nat
% 9.00/5.13 @ ( field_nat
% 9.00/5.13 @ ( collec3392354462482085612at_nat
% 9.00/5.13 @ ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( ( ord_less_nat @ X @ N )
% 9.00/5.13 & ( ord_less_nat @ Y @ N )
% 9.00/5.13 & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 9.00/5.13 @ ( collec3392354462482085612at_nat
% 9.00/5.13 @ ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( ( ord_less_nat @ X @ N )
% 9.00/5.13 & ( ord_less_nat @ Y @ N )
% 9.00/5.13 & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10016_nat_Odisc__eq__case_I2_J,axiom,
% 9.00/5.13 ! [Nat: nat] :
% 9.00/5.13 ( ( Nat != zero_zero_nat )
% 9.00/5.13 = ( case_nat_o @ $false
% 9.00/5.13 @ ^ [Uu3: nat] : $true
% 9.00/5.13 @ Nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10017_nat_Odisc__eq__case_I1_J,axiom,
% 9.00/5.13 ! [Nat: nat] :
% 9.00/5.13 ( ( Nat = zero_zero_nat )
% 9.00/5.13 = ( case_nat_o @ $true
% 9.00/5.13 @ ^ [Uu3: nat] : $false
% 9.00/5.13 @ Nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10018_less__eq__nat_Osimps_I2_J,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
% 9.00/5.13 = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10019_max__Suc1,axiom,
% 9.00/5.13 ! [N: nat,M: nat] :
% 9.00/5.13 ( ( ord_max_nat @ ( suc @ N ) @ M )
% 9.00/5.13 = ( case_nat_nat @ ( suc @ N )
% 9.00/5.13 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
% 9.00/5.13 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10020_max__Suc2,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_max_nat @ M @ ( suc @ N ) )
% 9.00/5.13 = ( case_nat_nat @ ( suc @ N )
% 9.00/5.13 @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
% 9.00/5.13 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10021_diff__Suc,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( minus_minus_nat @ M @ ( suc @ N ) )
% 9.00/5.13 = ( case_nat_nat @ zero_zero_nat
% 9.00/5.13 @ ^ [K3: nat] : K3
% 9.00/5.13 @ ( minus_minus_nat @ M @ N ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10022_min__Suc2,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_min_nat @ M @ ( suc @ N ) )
% 9.00/5.13 = ( case_nat_nat @ zero_zero_nat
% 9.00/5.13 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
% 9.00/5.13 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10023_min__Suc1,axiom,
% 9.00/5.13 ! [N: nat,M: nat] :
% 9.00/5.13 ( ( ord_min_nat @ ( suc @ N ) @ M )
% 9.00/5.13 = ( case_nat_nat @ zero_zero_nat
% 9.00/5.13 @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
% 9.00/5.13 @ M ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10024_pred__def,axiom,
% 9.00/5.13 ( pred
% 9.00/5.13 = ( case_nat_nat @ zero_zero_nat
% 9.00/5.13 @ ^ [X24: nat] : X24 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10025_UNIV__bool,axiom,
% 9.00/5.13 ( top_top_set_o
% 9.00/5.13 = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10026_Rep__unit__induct,axiom,
% 9.00/5.13 ! [Y3: $o,P: $o > $o] :
% 9.00/5.13 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
% 9.00/5.13 => ( P @ Y3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10027_Abs__unit__inject,axiom,
% 9.00/5.13 ! [X2: $o,Y3: $o] :
% 9.00/5.13 ( ( member_o @ X2 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ( ( ( product_Abs_unit @ X2 )
% 9.00/5.13 = ( product_Abs_unit @ Y3 ) )
% 9.00/5.13 = ( X2 = Y3 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10028_Abs__unit__inverse,axiom,
% 9.00/5.13 ! [Y3: $o] :
% 9.00/5.13 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ( ( product_Rep_unit @ ( product_Abs_unit @ Y3 ) )
% 9.00/5.13 = Y3 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10029_Rep__unit,axiom,
% 9.00/5.13 ! [X2: product_unit] : ( member_o @ ( product_Rep_unit @ X2 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10030_Abs__unit__cases,axiom,
% 9.00/5.13 ! [X2: product_unit] :
% 9.00/5.13 ~ ! [Y4: $o] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( product_Abs_unit @ Y4 ) )
% 9.00/5.13 => ~ ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10031_Rep__unit__cases,axiom,
% 9.00/5.13 ! [Y3: $o] :
% 9.00/5.13 ( ( member_o @ Y3 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ~ ! [X5: product_unit] :
% 9.00/5.13 ( Y3
% 9.00/5.13 = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10032_Abs__unit__induct,axiom,
% 9.00/5.13 ! [P: product_unit > $o,X2: product_unit] :
% 9.00/5.13 ( ! [Y4: $o] :
% 9.00/5.13 ( ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) )
% 9.00/5.13 => ( P @ ( product_Abs_unit @ Y4 ) ) )
% 9.00/5.13 => ( P @ X2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10033_type__definition__unit,axiom,
% 9.00/5.13 type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10034_Real_Opositive__def,axiom,
% 9.00/5.13 ( positive2
% 9.00/5.13 = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
% 9.00/5.13 @ ^ [X8: nat > rat] :
% 9.00/5.13 ? [R5: rat] :
% 9.00/5.13 ( ( ord_less_rat @ zero_zero_rat @ R5 )
% 9.00/5.13 & ? [K3: nat] :
% 9.00/5.13 ! [N2: nat] :
% 9.00/5.13 ( ( ord_less_eq_nat @ K3 @ N2 )
% 9.00/5.13 => ( ord_less_rat @ R5 @ ( X8 @ N2 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10035_Rat_Opositive__def,axiom,
% 9.00/5.13 ( positive
% 9.00/5.13 = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 9.00/5.13 @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10036_cmod__plus__Re__le__0__iff,axiom,
% 9.00/5.13 ! [Z: complex] :
% 9.00/5.13 ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
% 9.00/5.13 = ( ( re @ Z )
% 9.00/5.13 = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10037_complex__Re__le__cmod,axiom,
% 9.00/5.13 ! [X2: complex] : ( ord_less_eq_real @ ( re @ X2 ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10038_abs__Re__le__cmod,axiom,
% 9.00/5.13 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10039_Re__csqrt,axiom,
% 9.00/5.13 ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10040_complex__abs__le__norm,axiom,
% 9.00/5.13 ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10041_csqrt__unique,axiom,
% 9.00/5.13 ! [W2: complex,Z: complex] :
% 9.00/5.13 ( ( ( power_power_complex @ W2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.00/5.13 = Z )
% 9.00/5.13 => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W2 ) )
% 9.00/5.13 | ( ( ( re @ W2 )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 & ( ord_less_eq_real @ zero_zero_real @ ( im @ W2 ) ) ) )
% 9.00/5.13 => ( ( csqrt @ Z )
% 9.00/5.13 = W2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10042_csqrt__of__real__nonneg,axiom,
% 9.00/5.13 ! [X2: complex] :
% 9.00/5.13 ( ( ( im @ X2 )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) )
% 9.00/5.13 => ( ( csqrt @ X2 )
% 9.00/5.13 = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X2 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10043_abs__Im__le__cmod,axiom,
% 9.00/5.13 ! [X2: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( real_V1022390504157884413omplex @ X2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10044_cmod__Im__le__iff,axiom,
% 9.00/5.13 ! [X2: complex,Y3: complex] :
% 9.00/5.13 ( ( ( re @ X2 )
% 9.00/5.13 = ( re @ Y3 ) )
% 9.00/5.13 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 9.00/5.13 = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X2 ) ) @ ( abs_abs_real @ ( im @ Y3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10045_cmod__Re__le__iff,axiom,
% 9.00/5.13 ! [X2: complex,Y3: complex] :
% 9.00/5.13 ( ( ( im @ X2 )
% 9.00/5.13 = ( im @ Y3 ) )
% 9.00/5.13 => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X2 ) @ ( real_V1022390504157884413omplex @ Y3 ) )
% 9.00/5.13 = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X2 ) ) @ ( abs_abs_real @ ( re @ Y3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10046_csqrt__principal,axiom,
% 9.00/5.13 ! [Z: complex] :
% 9.00/5.13 ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
% 9.00/5.13 | ( ( ( re @ ( csqrt @ Z ) )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10047_cmod__le,axiom,
% 9.00/5.13 ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10048_complex__neq__0,axiom,
% 9.00/5.13 ! [Z: complex] :
% 9.00/5.13 ( ( Z != zero_zero_complex )
% 9.00/5.13 = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10049_csqrt__square,axiom,
% 9.00/5.13 ! [B3: complex] :
% 9.00/5.13 ( ( ( ord_less_real @ zero_zero_real @ ( re @ B3 ) )
% 9.00/5.13 | ( ( ( re @ B3 )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 & ( ord_less_eq_real @ zero_zero_real @ ( im @ B3 ) ) ) )
% 9.00/5.13 => ( ( csqrt @ ( power_power_complex @ B3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.00/5.13 = B3 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10050_csqrt__of__real__nonpos,axiom,
% 9.00/5.13 ! [X2: complex] :
% 9.00/5.13 ( ( ( im @ X2 )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 => ( ( ord_less_eq_real @ ( re @ X2 ) @ zero_zero_real )
% 9.00/5.13 => ( ( csqrt @ X2 )
% 9.00/5.13 = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X2 ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10051_csqrt__minus,axiom,
% 9.00/5.13 ! [X2: complex] :
% 9.00/5.13 ( ( ( ord_less_real @ ( im @ X2 ) @ zero_zero_real )
% 9.00/5.13 | ( ( ( im @ X2 )
% 9.00/5.13 = zero_zero_real )
% 9.00/5.13 & ( ord_less_eq_real @ zero_zero_real @ ( re @ X2 ) ) ) )
% 9.00/5.13 => ( ( csqrt @ ( uminus1482373934393186551omplex @ X2 ) )
% 9.00/5.13 = ( times_times_complex @ imaginary_unit @ ( csqrt @ X2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10052_complex__div__gt__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) )
% 9.00/5.13 & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10053_Re__complex__div__gt__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10054_Re__complex__div__lt__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 9.00/5.13 = ( ord_less_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10055_Re__complex__div__le__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 9.00/5.13 = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10056_Re__complex__div__ge__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10057_Im__complex__div__gt__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10058_Im__complex__div__lt__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 9.00/5.13 = ( ord_less_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10059_Im__complex__div__le__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) @ zero_zero_real )
% 9.00/5.13 = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) @ zero_zero_real ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10060_Im__complex__div__ge__0,axiom,
% 9.00/5.13 ! [A2: complex,B3: complex] :
% 9.00/5.13 ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A2 @ B3 ) ) )
% 9.00/5.13 = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A2 @ ( cnj @ B3 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10061_less__eq__int__def,axiom,
% 9.00/5.13 ( ord_less_eq_int
% 9.00/5.13 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10062_less__int__def,axiom,
% 9.00/5.13 ( ord_less_int
% 9.00/5.13 = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.00/5.13 @ ( produc8739625826339149834_nat_o
% 9.00/5.13 @ ^ [X: nat,Y: nat] :
% 9.00/5.13 ( produc6081775807080527818_nat_o
% 9.00/5.13 @ ^ [U2: nat,V3: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V3 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10063_MOST__SucD,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( eventually_nat
% 9.00/5.13 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 9.00/5.13 @ cofinite_nat )
% 9.00/5.13 => ( eventually_nat @ P @ cofinite_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10064_MOST__SucI,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( eventually_nat @ P @ cofinite_nat )
% 9.00/5.13 => ( eventually_nat
% 9.00/5.13 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 9.00/5.13 @ cofinite_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10065_MOST__Suc__iff,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( eventually_nat
% 9.00/5.13 @ ^ [N2: nat] : ( P @ ( suc @ N2 ) )
% 9.00/5.13 @ cofinite_nat )
% 9.00/5.13 = ( eventually_nat @ P @ cofinite_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10066_MOST__nat__le,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( eventually_nat @ P @ cofinite_nat )
% 9.00/5.13 = ( ? [M2: nat] :
% 9.00/5.13 ! [N2: nat] :
% 9.00/5.13 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.13 => ( P @ N2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10067_MOST__ge__nat,axiom,
% 9.00/5.13 ! [M: nat] : ( eventually_nat @ ( ord_less_eq_nat @ M ) @ cofinite_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10068_MOST__nat,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( eventually_nat @ P @ cofinite_nat )
% 9.00/5.13 = ( ? [M2: nat] :
% 9.00/5.13 ! [N2: nat] :
% 9.00/5.13 ( ( ord_less_nat @ M2 @ N2 )
% 9.00/5.13 => ( P @ N2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10069_infinity__enat__def,axiom,
% 9.00/5.13 ( extend5688581933313929465d_enat
% 9.00/5.13 = ( extended_Abs_enat @ none_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10070_enat__def,axiom,
% 9.00/5.13 ( extended_enat2
% 9.00/5.13 = ( ^ [N2: nat] : ( extended_Abs_enat @ ( some_nat @ N2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10071_INFM__nat__le,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( frequently_nat @ P @ cofinite_nat )
% 9.00/5.13 = ( ! [M2: nat] :
% 9.00/5.13 ? [N2: nat] :
% 9.00/5.13 ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.00/5.13 & ( P @ N2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10072_INFM__nat,axiom,
% 9.00/5.13 ! [P: nat > $o] :
% 9.00/5.13 ( ( frequently_nat @ P @ cofinite_nat )
% 9.00/5.13 = ( ! [M2: nat] :
% 9.00/5.13 ? [N2: nat] :
% 9.00/5.13 ( ( ord_less_nat @ M2 @ N2 )
% 9.00/5.13 & ( P @ N2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10073_list__encode_Oelims,axiom,
% 9.00/5.13 ! [X2: list_nat,Y3: nat] :
% 9.00/5.13 ( ( ( nat_list_encode @ X2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( ( X2 = nil_nat )
% 9.00/5.13 => ( Y3 != zero_zero_nat ) )
% 9.00/5.13 => ~ ! [X5: nat,Xs3: list_nat] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( cons_nat @ X5 @ Xs3 ) )
% 9.00/5.13 => ( Y3
% 9.00/5.13 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10074_le__prod__encode__1,axiom,
% 9.00/5.13 ! [A2: nat,B3: nat] : ( ord_less_eq_nat @ A2 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10075_le__prod__encode__2,axiom,
% 9.00/5.13 ! [B3: nat,A2: nat] : ( ord_less_eq_nat @ B3 @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A2 @ B3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10076_list__encode_Osimps_I1_J,axiom,
% 9.00/5.13 ( ( nat_list_encode @ nil_nat )
% 9.00/5.13 = zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10077_list__encode_Opelims,axiom,
% 9.00/5.13 ! [X2: list_nat,Y3: nat] :
% 9.00/5.13 ( ( ( nat_list_encode @ X2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( accp_list_nat @ nat_list_encode_rel @ X2 )
% 9.00/5.13 => ( ( ( X2 = nil_nat )
% 9.00/5.13 => ( ( Y3 = zero_zero_nat )
% 9.00/5.13 => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 9.00/5.13 => ~ ! [X5: nat,Xs3: list_nat] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( cons_nat @ X5 @ Xs3 ) )
% 9.00/5.13 => ( ( Y3
% 9.00/5.13 = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 9.00/5.13 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs3 ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10078_prod__decode__def,axiom,
% 9.00/5.13 ( nat_prod_decode
% 9.00/5.13 = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10079_list__decode_Opinduct,axiom,
% 9.00/5.13 ! [A0: nat,P: nat > $o] :
% 9.00/5.13 ( ( accp_nat @ nat_list_decode_rel @ A0 )
% 9.00/5.13 => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 9.00/5.13 => ( P @ zero_zero_nat ) )
% 9.00/5.13 => ( ! [N3: nat] :
% 9.00/5.13 ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
% 9.00/5.13 => ( ! [X4: nat,Y5: nat] :
% 9.00/5.13 ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 9.00/5.13 = ( nat_prod_decode @ N3 ) )
% 9.00/5.13 => ( P @ Y5 ) )
% 9.00/5.13 => ( P @ ( suc @ N3 ) ) ) )
% 9.00/5.13 => ( P @ A0 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10080_list__decode_Oelims,axiom,
% 9.00/5.13 ! [X2: nat,Y3: list_nat] :
% 9.00/5.13 ( ( ( nat_list_decode @ X2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( ( X2 = zero_zero_nat )
% 9.00/5.13 => ( Y3 != nil_nat ) )
% 9.00/5.13 => ~ ! [N3: nat] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( suc @ N3 ) )
% 9.00/5.13 => ( Y3
% 9.00/5.13 != ( produc2761476792215241774st_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 9.00/5.13 @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10081_list__decode_Opsimps_I1_J,axiom,
% 9.00/5.13 ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
% 9.00/5.13 => ( ( nat_list_decode @ zero_zero_nat )
% 9.00/5.13 = nil_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10082_list__decode_Osimps_I1_J,axiom,
% 9.00/5.13 ( ( nat_list_decode @ zero_zero_nat )
% 9.00/5.13 = nil_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10083_list__decode_Opelims,axiom,
% 9.00/5.13 ! [X2: nat,Y3: list_nat] :
% 9.00/5.13 ( ( ( nat_list_decode @ X2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( accp_nat @ nat_list_decode_rel @ X2 )
% 9.00/5.13 => ( ( ( X2 = zero_zero_nat )
% 9.00/5.13 => ( ( Y3 = nil_nat )
% 9.00/5.13 => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
% 9.00/5.13 => ~ ! [N3: nat] :
% 9.00/5.13 ( ( X2
% 9.00/5.13 = ( suc @ N3 ) )
% 9.00/5.13 => ( ( Y3
% 9.00/5.13 = ( produc2761476792215241774st_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( cons_nat @ X @ ( nat_list_decode @ Y ) )
% 9.00/5.13 @ ( nat_prod_decode @ N3 ) ) )
% 9.00/5.13 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10084_unit__factor__simps_I1_J,axiom,
% 9.00/5.13 ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
% 9.00/5.13 = zero_zero_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10085_unit__factor__nat__def,axiom,
% 9.00/5.13 ( unit_f2748546683901255202or_nat
% 9.00/5.13 = ( ^ [N2: nat] : ( if_nat @ ( N2 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10086_Lcm__eq__0__I__nat,axiom,
% 9.00/5.13 ! [A3: set_nat] :
% 9.00/5.13 ( ( member_nat @ zero_zero_nat @ A3 )
% 9.00/5.13 => ( ( gcd_Lcm_nat @ A3 )
% 9.00/5.13 = zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10087_Lcm__0__iff__nat,axiom,
% 9.00/5.13 ! [A3: set_nat] :
% 9.00/5.13 ( ( finite_finite_nat @ A3 )
% 9.00/5.13 => ( ( ( gcd_Lcm_nat @ A3 )
% 9.00/5.13 = zero_zero_nat )
% 9.00/5.13 = ( member_nat @ zero_zero_nat @ A3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10088_Lcm__nat__infinite,axiom,
% 9.00/5.13 ! [M5: set_nat] :
% 9.00/5.13 ( ~ ( finite_finite_nat @ M5 )
% 9.00/5.13 => ( ( gcd_Lcm_nat @ M5 )
% 9.00/5.13 = zero_zero_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10089_Lcm__int__greater__eq__0,axiom,
% 9.00/5.13 ! [K4: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Lcm_int @ K4 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10090_Lcm__nat__empty,axiom,
% 9.00/5.13 ( ( gcd_Lcm_nat @ bot_bot_set_nat )
% 9.00/5.13 = one_one_nat ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10091_Lcm__eq__Max__nat,axiom,
% 9.00/5.13 ! [M5: set_nat] :
% 9.00/5.13 ( ( finite_finite_nat @ M5 )
% 9.00/5.13 => ( ( M5 != bot_bot_set_nat )
% 9.00/5.13 => ( ~ ( member_nat @ zero_zero_nat @ M5 )
% 9.00/5.13 => ( ! [M4: nat,N3: nat] :
% 9.00/5.13 ( ( member_nat @ M4 @ M5 )
% 9.00/5.13 => ( ( member_nat @ N3 @ M5 )
% 9.00/5.13 => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
% 9.00/5.13 => ( ( gcd_Lcm_nat @ M5 )
% 9.00/5.13 = ( lattic8265883725875713057ax_nat @ M5 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10092_lcm__0__iff__nat,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ( gcd_lcm_nat @ M @ N )
% 9.00/5.13 = zero_zero_nat )
% 9.00/5.13 = ( ( M = zero_zero_nat )
% 9.00/5.13 | ( N = zero_zero_nat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10093_lcm__1__iff__nat,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ( gcd_lcm_nat @ M @ N )
% 9.00/5.13 = ( suc @ zero_zero_nat ) )
% 9.00/5.13 = ( ( M
% 9.00/5.13 = ( suc @ zero_zero_nat ) )
% 9.00/5.13 & ( N
% 9.00/5.13 = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10094_Lcm__in__lcm__closed__set__nat,axiom,
% 9.00/5.13 ! [M5: set_nat] :
% 9.00/5.13 ( ( finite_finite_nat @ M5 )
% 9.00/5.13 => ( ( M5 != bot_bot_set_nat )
% 9.00/5.13 => ( ! [M4: nat,N3: nat] :
% 9.00/5.13 ( ( member_nat @ M4 @ M5 )
% 9.00/5.13 => ( ( member_nat @ N3 @ M5 )
% 9.00/5.13 => ( member_nat @ ( gcd_lcm_nat @ M4 @ N3 ) @ M5 ) ) )
% 9.00/5.13 => ( member_nat @ ( gcd_Lcm_nat @ M5 ) @ M5 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10095_lcm__pos__int,axiom,
% 9.00/5.13 ! [M: int,N: int] :
% 9.00/5.13 ( ( M != zero_zero_int )
% 9.00/5.13 => ( ( N != zero_zero_int )
% 9.00/5.13 => ( ord_less_int @ zero_zero_int @ ( gcd_lcm_int @ M @ N ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10096_lcm__pos__nat,axiom,
% 9.00/5.13 ! [M: nat,N: nat] :
% 9.00/5.13 ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.00/5.13 => ( ( ord_less_nat @ zero_zero_nat @ N )
% 9.00/5.13 => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M @ N ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10097_lcm__ge__0__int,axiom,
% 9.00/5.13 ! [X2: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_lcm_int @ X2 @ Y3 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10098_lcm__unique__int,axiom,
% 9.00/5.13 ! [D: int,A2: int,B3: int] :
% 9.00/5.13 ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 9.00/5.13 & ( dvd_dvd_int @ A2 @ D )
% 9.00/5.13 & ( dvd_dvd_int @ B3 @ D )
% 9.00/5.13 & ! [E3: int] :
% 9.00/5.13 ( ( ( dvd_dvd_int @ A2 @ E3 )
% 9.00/5.13 & ( dvd_dvd_int @ B3 @ E3 ) )
% 9.00/5.13 => ( dvd_dvd_int @ D @ E3 ) ) )
% 9.00/5.13 = ( D
% 9.00/5.13 = ( gcd_lcm_int @ A2 @ B3 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10099_lcm__cases__int,axiom,
% 9.00/5.13 ! [X2: int,Y3: int,P: int > $o] :
% 9.00/5.13 ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.13 => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) )
% 9.00/5.13 => ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
% 9.00/5.13 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.00/5.13 => ( P @ ( gcd_lcm_int @ X2 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.00/5.13 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 9.00/5.13 => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.00/5.13 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ Y3 ) ) ) )
% 9.00/5.13 => ( ( ( ord_less_eq_int @ X2 @ zero_zero_int )
% 9.00/5.13 => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.00/5.13 => ( P @ ( gcd_lcm_int @ ( uminus_uminus_int @ X2 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.00/5.13 => ( P @ ( gcd_lcm_int @ X2 @ Y3 ) ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10100_Lcm__nat__def,axiom,
% 9.00/5.13 ( gcd_Lcm_nat
% 9.00/5.13 = ( ^ [M8: set_nat] : ( if_nat @ ( finite_finite_nat @ M8 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M8 ) @ zero_zero_nat ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10101_VEBT__internal_Olesseq_Osimps,axiom,
% 9.00/5.13 ( vEBT_VEBT_lesseq
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10102_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10103_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10104_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10105_VEBT__internal_Oless_Oelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10106_VEBT__internal_Oless_Oelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10107_VEBT__internal_Oless_Oelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10108_VEBT__internal_Oless_Osimps,axiom,
% 9.00/5.13 ( vEBT_VEBT_less
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10109_VEBT__internal_Ogreater_Osimps,axiom,
% 9.00/5.13 ( vEBT_VEBT_greater
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10110_VEBT__internal_Ogreater_Oelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10111_VEBT__internal_Ogreater_Oelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10112_VEBT__internal_Ogreater_Oelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10113_VEBT__internal_Ogreater_Opelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) )
% 9.00/5.13 => ~ ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10114_VEBT__internal_Ogreater_Ocases,axiom,
% 9.00/5.13 ! [X2: produc4953844613479565601on_nat] :
% 9.00/5.13 ~ ! [X5: option_nat,Y4: option_nat] :
% 9.00/5.13 ( X2
% 9.00/5.13 != ( produc5098337634421038937on_nat @ X5 @ Y4 ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10115_VEBT__internal_Ogreater_Opelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10116_VEBT__internal_Ogreater_Opelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_greater @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_V5711637165310795180er_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat
% 9.00/5.13 @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X )
% 9.00/5.13 @ X2
% 9.00/5.13 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10117_VEBT__internal_Olesseq_Opelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10118_VEBT__internal_Olesseq_Opelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10119_VEBT__internal_Olesseq_Opelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_lesseq @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_lesseq_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X2 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10120_VEBT__internal_Oless_Opelims_I3_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ~ ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10121_VEBT__internal_Oless_Opelims_I2_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat] :
% 9.00/5.13 ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(fact_10122_VEBT__internal_Oless_Opelims_I1_J,axiom,
% 9.00/5.13 ! [X2: option_nat,Xa2: option_nat,Y3: $o] :
% 9.00/5.13 ( ( ( vEBT_VEBT_less @ X2 @ Xa2 )
% 9.00/5.13 = Y3 )
% 9.00/5.13 => ( ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( ( Y3
% 9.00/5.13 = ( vEBT_V2881884560877996034ft_nat @ ord_less_nat @ X2 @ Xa2 ) )
% 9.00/5.13 => ~ ( accp_P8646395344606611882on_nat @ vEBT_VEBT_less_rel @ ( produc5098337634421038937on_nat @ X2 @ Xa2 ) ) ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 9.00/5.13 ! [X2: int,Y3: int] :
% 9.00/5.13 ( ( if_int @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 9.00/5.13 ! [X2: int,Y3: int] :
% 9.00/5.13 ( ( if_int @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 9.00/5.13 ! [X2: nat,Y3: nat] :
% 9.00/5.13 ( ( if_nat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 9.00/5.13 ! [X2: nat,Y3: nat] :
% 9.00/5.13 ( ( if_nat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
% 9.00/5.13 ! [X2: num,Y3: num] :
% 9.00/5.13 ( ( if_num @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
% 9.00/5.13 ! [X2: num,Y3: num] :
% 9.00/5.13 ( ( if_num @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 9.00/5.13 ! [X2: rat,Y3: rat] :
% 9.00/5.13 ( ( if_rat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 9.00/5.13 ! [X2: rat,Y3: rat] :
% 9.00/5.13 ( ( if_rat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 9.00/5.13 ! [X2: real,Y3: real] :
% 9.00/5.13 ( ( if_real @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 9.00/5.13 ! [X2: real,Y3: real] :
% 9.00/5.13 ( ( if_real @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 9.00/5.13 ! [P: real > $o] :
% 9.00/5.13 ( ( P @ ( fChoice_real @ P ) )
% 9.00/5.13 = ( ? [X8: real] : ( P @ X8 ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.00/5.13 ! [X2: complex,Y3: complex] :
% 9.00/5.13 ( ( if_complex @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.00/5.13 ! [X2: complex,Y3: complex] :
% 9.00/5.13 ( ( if_complex @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.00/5.13 ! [X2: extended_enat,Y3: extended_enat] :
% 9.00/5.13 ( ( if_Extended_enat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.00/5.13 ! [X2: extended_enat,Y3: extended_enat] :
% 9.00/5.13 ( ( if_Extended_enat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.00/5.13 ! [X2: code_integer,Y3: code_integer] :
% 9.00/5.13 ( ( if_Code_integer @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.00/5.13 ! [X2: code_integer,Y3: code_integer] :
% 9.00/5.13 ( ( if_Code_integer @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: set_int,Y3: set_int] :
% 9.00/5.13 ( ( if_set_int @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: set_int,Y3: set_int] :
% 9.00/5.13 ( ( if_set_int @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.00/5.13 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.00/5.13 ( ( if_VEBT_VEBT @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
% 9.00/5.13 ! [X2: vEBT_VEBT,Y3: vEBT_VEBT] :
% 9.00/5.13 ( ( if_VEBT_VEBT @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: list_int,Y3: list_int] :
% 9.00/5.13 ( ( if_list_int @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: list_int,Y3: list_int] :
% 9.00/5.13 ( ( if_list_int @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: list_nat,Y3: list_nat] :
% 9.00/5.13 ( ( if_list_nat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: list_nat,Y3: list_nat] :
% 9.00/5.13 ( ( if_list_nat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: option_nat,Y3: option_nat] :
% 9.00/5.13 ( ( if_option_nat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: option_nat,Y3: option_nat] :
% 9.00/5.13 ( ( if_option_nat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.00/5.13 ! [X2: option_num,Y3: option_num] :
% 9.00/5.13 ( ( if_option_num @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.00/5.13 ! [X2: option_num,Y3: option_num] :
% 9.00/5.13 ( ( if_option_num @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 9.00/5.13 ( ( if_Pro3027730157355071871nt_int @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.00/5.13 ! [X2: product_prod_int_int,Y3: product_prod_int_int] :
% 9.00/5.13 ( ( if_Pro3027730157355071871nt_int @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.00/5.13 ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.00/5.13 ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.00/5.13 ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.00/5.13 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.00/5.13 ( ( if_Pro5737122678794959658eger_o @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.00/5.13 ! [X2: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.00/5.13 ( ( if_Pro5737122678794959658eger_o @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.00/5.13 ! [P: $o] :
% 9.00/5.13 ( ( P = $true )
% 9.00/5.13 | ( P = $false ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.00/5.13 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.00/5.13 ( ( if_Pro6119634080678213985nteger @ $false @ X2 @ Y3 )
% 9.00/5.13 = Y3 ) ).
% 9.00/5.13
% 9.00/5.13 thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.00/5.13 ! [X2: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.00/5.13 ( ( if_Pro6119634080678213985nteger @ $true @ X2 @ Y3 )
% 9.00/5.13 = X2 ) ).
% 9.00/5.13
% 9.00/5.13 thf(conj_0,hypothesis,
% 9.00/5.13 ~ ( ( ( vEBT_vebt_mint @ ta )
% 9.00/5.13 = none_nat )
% 9.00/5.13 & ( ( vEBT_vebt_mint @ k )
% 9.00/5.13 = ( some_nat @ b ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(conj_1,hypothesis,
% 9.00/5.13 ~ ( ( ( vEBT_vebt_mint @ ta )
% 9.00/5.13 = ( some_nat @ a ) )
% 9.00/5.13 & ( ( vEBT_vebt_mint @ k )
% 9.00/5.13 = none_nat ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(conj_2,hypothesis,
% 9.00/5.13 ( ( ord_less_nat @ a @ b )
% 9.00/5.13 & ( ( some_nat @ a )
% 9.00/5.13 = ( vEBT_vebt_mint @ ta ) )
% 9.00/5.13 & ( ( some_nat @ b )
% 9.00/5.13 = ( vEBT_vebt_mint @ k ) ) ) ).
% 9.00/5.13
% 9.00/5.13 thf(conj_3,conjecture,
% 9.00/5.13 $false ).
% 9.00/5.13 ------------------------
% 9.01/5.66 % (3190)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_3077 for (2994ds/275Mi)
% 9.01/5.66 % (3191)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/18Mi)
% 9.01/5.66 % (3186)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_3077 for (2994ds/4Mi)
% 9.01/5.66 % (3185)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/183Mi)
% 9.01/5.66 % (3187)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_3077 for (2994ds/27Mi)
% 9.01/5.66 % (3188)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/2Mi)
% 9.01/5.66 % (3189)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_3077 for (2994ds/2Mi)
% 9.01/5.66 % (3188)Instruction limit reached!
% 9.01/5.66 % (3188)------------------------------
% 9.01/5.66 % (3188)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.67 % (3188)Termination reason: Unknown
% 9.01/5.67 % (3188)Termination phase: shuffling
% 9.01/5.67
% 9.01/5.67 % (3188)Memory used [KB]: 13688
% 9.01/5.67 % (3188)Time elapsed: 0.003 s
% 9.01/5.67 % (3189)Instruction limit reached!
% 9.01/5.67 % (3189)------------------------------
% 9.01/5.67 % (3189)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.67 % (3188)Instructions burned: 2 (million)
% 9.01/5.67 % (3188)------------------------------
% 9.01/5.67 % (3188)------------------------------
% 9.01/5.67 % (3189)Termination reason: Unknown
% 9.01/5.67 % (3189)Termination phase: shuffling
% 9.01/5.67
% 9.01/5.67 % (3189)Memory used [KB]: 13688
% 9.01/5.67 % (3189)Time elapsed: 0.003 s
% 9.01/5.67 % (3189)Instructions burned: 2 (million)
% 9.01/5.67 % (3189)------------------------------
% 9.01/5.67 % (3189)------------------------------
% 9.01/5.67 % (3186)Instruction limit reached!
% 9.01/5.67 % (3186)------------------------------
% 9.01/5.67 % (3186)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.67 % (3186)Termination reason: Unknown
% 9.01/5.67 % (3186)Termination phase: shuffling
% 9.01/5.67
% 9.01/5.67 % (3186)Memory used [KB]: 13816
% 9.01/5.67 % (3186)Time elapsed: 0.007 s
% 9.01/5.67 % (3186)Instructions burned: 4 (million)
% 9.01/5.67 % (3186)------------------------------
% 9.01/5.67 % (3186)------------------------------
% 9.01/5.68 % (3191)Instruction limit reached!
% 9.01/5.68 % (3191)------------------------------
% 9.01/5.68 % (3191)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.68 % (3191)Termination reason: Unknown
% 9.01/5.68 % (3191)Termination phase: shuffling
% 9.01/5.68
% 9.01/5.68 % (3191)Memory used [KB]: 14583
% 9.01/5.68 % (3191)Time elapsed: 0.021 s
% 9.01/5.68 % (3191)Instructions burned: 19 (million)
% 9.01/5.68 % (3191)------------------------------
% 9.01/5.68 % (3191)------------------------------
% 9.01/5.69 % (3192)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/3Mi)
% 9.01/5.69 % (3193)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_3077 for (2994ds/37Mi)
% 9.01/5.69 % (3194)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_3077 for (2994ds/15Mi)
% 9.01/5.69 % (3192)Instruction limit reached!
% 9.01/5.69 % (3192)------------------------------
% 9.01/5.69 % (3192)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.69 % (3192)Termination reason: Unknown
% 9.01/5.69 % (3192)Termination phase: shuffling
% 9.01/5.69
% 9.01/5.69 % (3192)Memory used [KB]: 13688
% 9.01/5.69 % (3192)Time elapsed: 0.006 s
% 9.01/5.69 % (3192)Instructions burned: 3 (million)
% 9.01/5.69 % (3192)------------------------------
% 9.01/5.69 % (3192)------------------------------
% 9.01/5.69 % (3187)Instruction limit reached!
% 9.01/5.69 % (3187)------------------------------
% 9.01/5.69 % (3187)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.69 % (3187)Termination reason: Unknown
% 9.01/5.69 % (3187)Termination phase: shuffling
% 9.01/5.69
% 9.01/5.69 % (3187)Memory used [KB]: 14711
% 9.01/5.69 % (3187)Time elapsed: 0.030 s
% 9.01/5.69 % (3187)Instructions burned: 27 (million)
% 9.01/5.69 % (3187)------------------------------
% 9.01/5.69 % (3187)------------------------------
% 9.01/5.70 % (3195)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_3077 for (2994ds/3Mi)
% 9.01/5.70 % (3195)Instruction limit reached!
% 9.01/5.70 % (3195)------------------------------
% 9.01/5.70 % (3195)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.70 % (3195)Termination reason: Unknown
% 9.01/5.70 % (3195)Termination phase: shuffling
% 9.01/5.70
% 9.01/5.70 % (3195)Memory used [KB]: 13688
% 9.01/5.70 % (3195)Time elapsed: 0.005 s
% 9.01/5.70 % (3195)Instructions burned: 3 (million)
% 9.01/5.70 % (3195)------------------------------
% 9.01/5.70 % (3195)------------------------------
% 9.01/5.71 % (3194)Instruction limit reached!
% 9.01/5.71 % (3194)------------------------------
% 9.01/5.71 % (3194)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.71 % (3194)Termination reason: Unknown
% 9.01/5.71 % (3194)Termination phase: shuffling
% 9.01/5.71
% 9.01/5.71 % (3194)Memory used [KB]: 14583
% 9.01/5.71 % (3194)Time elapsed: 0.022 s
% 9.01/5.71 % (3194)Instructions burned: 15 (million)
% 9.01/5.71 % (3194)------------------------------
% 9.01/5.71 % (3194)------------------------------
% 9.01/5.71 % (3196)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_3077 for (2994ds/1041Mi)
% 9.01/5.71 % (3197)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/7Mi)
% 9.01/5.72 % (3198)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/16Mi)
% 9.01/5.72 % (3193)Instruction limit reached!
% 9.01/5.72 % (3193)------------------------------
% 9.01/5.72 % (3193)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.72 % (3193)Termination reason: Unknown
% 9.01/5.72 % (3193)Termination phase: shuffling
% 9.01/5.72
% 9.01/5.72 % (3193)Memory used [KB]: 14839
% 9.01/5.72 % (3193)Time elapsed: 0.034 s
% 9.01/5.72 % (3193)Instructions burned: 37 (million)
% 9.01/5.72 % (3193)------------------------------
% 9.01/5.72 % (3193)------------------------------
% 9.01/5.73 % (3197)Instruction limit reached!
% 9.01/5.73 % (3197)------------------------------
% 9.01/5.73 % (3197)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.73 % (3197)Termination reason: Unknown
% 9.01/5.73 % (3197)Termination phase: shuffling
% 9.01/5.73
% 9.01/5.73 % (3197)Memory used [KB]: 14072
% 9.01/5.73 % (3197)Time elapsed: 0.016 s
% 9.01/5.73 % (3197)Instructions burned: 7 (million)
% 9.01/5.73 % (3197)------------------------------
% 9.01/5.73 % (3197)------------------------------
% 9.01/5.73 % (3198)Instruction limit reached!
% 9.01/5.73 % (3198)------------------------------
% 9.01/5.73 % (3198)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.73 % (3198)Termination reason: Unknown
% 9.01/5.73 % (3198)Termination phase: shuffling
% 9.01/5.73
% 9.01/5.73 % (3198)Memory used [KB]: 14583
% 9.01/5.73 % (3198)Time elapsed: 0.014 s
% 9.01/5.73 % (3198)Instructions burned: 16 (million)
% 9.01/5.73 % (3198)------------------------------
% 9.01/5.73 % (3198)------------------------------
% 9.01/5.73 % (3199)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_3077 for (2994ds/3Mi)
% 9.01/5.73 % (3199)Instruction limit reached!
% 9.01/5.73 % (3199)------------------------------
% 9.01/5.73 % (3199)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.73 % (3199)Termination reason: Unknown
% 9.01/5.73 % (3199)Termination phase: shuffling
% 9.01/5.73
% 9.01/5.73 % (3199)Memory used [KB]: 13688
% 9.01/5.73 % (3199)Time elapsed: 0.006 s
% 9.01/5.73 % (3199)Instructions burned: 3 (million)
% 9.01/5.73 % (3199)------------------------------
% 9.01/5.73 % (3199)------------------------------
% 9.01/5.74 % (3200)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_3077 for (2994ds/3Mi)
% 9.01/5.74 % (3201)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/7Mi)
% 9.01/5.74 % (3200)Instruction limit reached!
% 9.01/5.74 % (3200)------------------------------
% 9.01/5.74 % (3200)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.74 % (3200)Termination reason: Unknown
% 9.01/5.74 % (3200)Termination phase: shuffling
% 9.01/5.74
% 9.01/5.74 % (3200)Memory used [KB]: 13688
% 9.01/5.74 % (3200)Time elapsed: 0.006 s
% 9.01/5.74 % (3200)Instructions burned: 3 (million)
% 9.01/5.74 % (3200)------------------------------
% 9.01/5.74 % (3200)------------------------------
% 9.01/5.75 % (3201)Instruction limit reached!
% 9.01/5.75 % (3201)------------------------------
% 9.01/5.75 % (3201)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.75 % (3201)Termination reason: Unknown
% 9.01/5.75 % (3201)Termination phase: shuffling
% 9.01/5.75
% 9.01/5.75 % (3201)Memory used [KB]: 14072
% 9.01/5.75 % (3201)Time elapsed: 0.007 s
% 9.01/5.75 % (3201)Instructions burned: 7 (million)
% 9.01/5.75 % (3201)------------------------------
% 9.01/5.75 % (3201)------------------------------
% 9.01/5.75 % (3203)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/4Mi)
% 9.01/5.76 % (3205)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/710Mi)
% 9.01/5.76 % (3202)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_3077 for (2994ds/3Mi)
% 9.01/5.76 % (3203)Instruction limit reached!
% 9.01/5.76 % (3203)------------------------------
% 9.01/5.76 % (3203)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.76 % (3203)Termination reason: Unknown
% 9.01/5.76 % (3203)Termination phase: shuffling
% 9.01/5.76
% 9.01/5.76 % (3203)Memory used [KB]: 13816
% 9.01/5.76 % (3203)Time elapsed: 0.007 s
% 9.01/5.76 % (3203)Instructions burned: 4 (million)
% 9.01/5.76 % (3203)------------------------------
% 9.01/5.76 % (3203)------------------------------
% 9.01/5.76 % (3190)Instruction limit reached!
% 9.01/5.76 % (3190)------------------------------
% 9.01/5.76 % (3190)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.76 % (3190)Termination reason: Unknown
% 9.01/5.76 % (3190)Termination phase: shuffling
% 9.01/5.76
% 9.01/5.76 % (3190)Memory used [KB]: 18421
% 9.01/5.76 % (3190)Time elapsed: 0.103 s
% 9.01/5.76 % (3190)Instructions burned: 277 (million)
% 9.01/5.76 % (3190)------------------------------
% 9.01/5.76 % (3190)------------------------------
% 9.01/5.76 % (3202)Instruction limit reached!
% 9.01/5.76 % (3202)------------------------------
% 9.01/5.76 % (3202)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.76 % (3202)Termination reason: Unknown
% 9.01/5.76 % (3202)Termination phase: shuffling
% 9.01/5.76
% 9.01/5.76 % (3202)Memory used [KB]: 13688
% 9.01/5.76 % (3202)Time elapsed: 0.005 s
% 9.01/5.76 % (3202)Instructions burned: 3 (million)
% 9.01/5.76 % (3202)------------------------------
% 9.01/5.76 % (3202)------------------------------
% 9.01/5.76 % (3204)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/18Mi)
% 9.01/5.77 % (3207)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_3077 for (2993ds/902Mi)
% 9.01/5.78 % (3208)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_3077 for (2993ds/21Mi)
% 9.01/5.78 % (3206)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/6Mi)
% 9.01/5.78 % (3185)Instruction limit reached!
% 9.01/5.78 % (3185)------------------------------
% 9.01/5.78 % (3185)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.78 % (3185)Termination reason: Unknown
% 9.01/5.78 % (3185)Termination phase: shuffling
% 9.01/5.78
% 9.01/5.78 % (3185)Memory used [KB]: 16886
% 9.01/5.78 % (3185)Time elapsed: 0.123 s
% 9.01/5.78 % (3185)Instructions burned: 184 (million)
% 9.01/5.78 % (3185)------------------------------
% 9.01/5.78 % (3185)------------------------------
% 9.01/5.79 % (3206)Instruction limit reached!
% 9.01/5.79 % (3206)------------------------------
% 9.01/5.79 % (3206)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.79 % (3206)Termination reason: Unknown
% 9.01/5.79 % (3206)Termination phase: shuffling
% 9.01/5.79
% 9.01/5.79 % (3206)Memory used [KB]: 13944
% 9.01/5.79 % (3206)Time elapsed: 0.010 s
% 9.01/5.79 % (3206)Instructions burned: 6 (million)
% 9.01/5.79 % (3206)------------------------------
% 9.01/5.79 % (3206)------------------------------
% 9.01/5.79 % (3208)Instruction limit reached!
% 9.01/5.79 % (3208)------------------------------
% 9.01/5.79 % (3208)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.79 % (3204)Instruction limit reached!
% 9.01/5.79 % (3204)------------------------------
% 9.01/5.79 % (3204)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.79 % (3204)Termination reason: Unknown
% 9.01/5.79 % (3204)Termination phase: shuffling
% 9.01/5.79
% 9.01/5.79 % (3204)Memory used [KB]: 14583
% 9.01/5.79 % (3204)Time elapsed: 0.031 s
% 9.01/5.79 % (3204)Instructions burned: 18 (million)
% 9.01/5.79 % (3204)------------------------------
% 9.01/5.79 % (3204)------------------------------
% 9.01/5.79 % (3208)Termination reason: Unknown
% 9.01/5.79 % (3208)Termination phase: shuffling
% 9.01/5.79
% 9.01/5.79 % (3208)Memory used [KB]: 14583
% 9.01/5.79 % (3208)Time elapsed: 0.016 s
% 9.01/5.79 % (3208)Instructions burned: 21 (million)
% 9.01/5.79 % (3208)------------------------------
% 9.01/5.79 % (3208)------------------------------
% 9.01/5.80 % (3209)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/5Mi)
% 9.01/5.80 % (3210)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_3077 for (2993ds/6Mi)
% 9.01/5.80 % (3209)Instruction limit reached!
% 9.01/5.80 % (3209)------------------------------
% 9.01/5.80 % (3209)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.80 % (3209)Termination reason: Unknown
% 9.01/5.80 % (3209)Termination phase: shuffling
% 9.01/5.80
% 9.01/5.80 % (3209)Memory used [KB]: 13816
% 9.01/5.80 % (3209)Time elapsed: 0.005 s
% 9.01/5.80 % (3209)Instructions burned: 5 (million)
% 9.01/5.80 % (3209)------------------------------
% 9.01/5.80 % (3209)------------------------------
% 9.01/5.80 % (3211)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/377Mi)
% 9.01/5.81 % (3210)Instruction limit reached!
% 9.01/5.81 % (3210)------------------------------
% 9.01/5.81 % (3210)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.81 % (3210)Termination reason: Unknown
% 9.01/5.81 % (3210)Termination phase: shuffling
% 9.01/5.81
% 9.01/5.81 % (3210)Memory used [KB]: 13944
% 9.01/5.81 % (3210)Time elapsed: 0.006 s
% 9.01/5.81 % (3210)Instructions burned: 6 (million)
% 9.01/5.81 % (3210)------------------------------
% 9.01/5.81 % (3210)------------------------------
% 9.01/5.81 % (3213)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_3077 for (2993ds/19Mi)
% 9.01/5.82 % (3214)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/879Mi)
% 9.01/5.82 % (3212)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_3077 for (2993ds/779Mi)
% 9.01/5.83 % (3213)Instruction limit reached!
% 9.01/5.83 % (3213)------------------------------
% 9.01/5.83 % (3213)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.83 % (3213)Termination reason: Unknown
% 9.01/5.83 % (3213)Termination phase: shuffling
% 9.01/5.83
% 9.01/5.83 % (3213)Memory used [KB]: 14583
% 9.01/5.83 % (3213)Time elapsed: 0.016 s
% 9.01/5.83 % (3213)Instructions burned: 19 (million)
% 9.01/5.83 % (3213)------------------------------
% 9.01/5.83 % (3213)------------------------------
% 9.01/5.84 % (3215)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_3077 for (2993ds/17Mi)
% 9.01/5.85 % (3215)Instruction limit reached!
% 9.01/5.85 % (3215)------------------------------
% 9.01/5.85 % (3215)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.85 % (3215)Termination reason: Unknown
% 9.01/5.85 % (3215)Termination phase: shuffling
% 9.01/5.85
% 9.01/5.85 % (3215)Memory used [KB]: 14583
% 9.01/5.85 % (3215)Time elapsed: 0.015 s
% 9.01/5.85 % (3215)Instructions burned: 18 (million)
% 9.01/5.85 % (3215)------------------------------
% 9.01/5.85 % (3215)------------------------------
% 9.01/5.86 % (3216)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_3077 for (2992ds/3Mi)
% 9.01/5.87 % (3216)Instruction limit reached!
% 9.01/5.87 % (3216)------------------------------
% 9.01/5.87 % (3216)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.87 % (3216)Termination reason: Unknown
% 9.01/5.87 % (3216)Termination phase: shuffling
% 9.01/5.87
% 9.01/5.87 % (3216)Memory used [KB]: 13688
% 9.01/5.87 % (3216)Time elapsed: 0.004 s
% 9.01/5.87 % (3216)Instructions burned: 3 (million)
% 9.01/5.87 % (3216)------------------------------
% 9.01/5.87 % (3216)------------------------------
% 9.01/5.88 % (3217)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_3077 for (2992ds/30Mi)
% 9.01/5.90 % (3217)Instruction limit reached!
% 9.01/5.90 % (3217)------------------------------
% 9.01/5.90 % (3217)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.90 % (3217)Termination reason: Unknown
% 9.01/5.90 % (3217)Termination phase: shuffling
% 9.01/5.90
% 9.01/5.90 % (3217)Memory used [KB]: 14839
% 9.01/5.90 % (3217)Time elapsed: 0.020 s
% 9.01/5.90 % (3217)Instructions burned: 32 (million)
% 9.01/5.90 % (3217)------------------------------
% 9.01/5.90 % (3217)------------------------------
% 9.01/5.91 % (3218)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_3077 for (2992ds/127Mi)
% 9.01/5.93 % (3211)Instruction limit reached!
% 9.01/5.93 % (3211)------------------------------
% 9.01/5.93 % (3211)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.93 % (3211)Termination reason: Unknown
% 9.01/5.93 % (3211)Termination phase: shuffling
% 9.01/5.93
% 9.01/5.93 % (3211)Memory used [KB]: 19573
% 9.01/5.93 % (3211)Time elapsed: 0.134 s
% 9.01/5.93 % (3211)Instructions burned: 378 (million)
% 9.01/5.93 % (3211)------------------------------
% 9.01/5.93 % (3211)------------------------------
% 9.01/5.95 % (3219)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_3077 for (2992ds/100Mi)
% 9.01/5.96 % (3218)Instruction limit reached!
% 9.01/5.96 % (3218)------------------------------
% 9.01/5.96 % (3218)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.96 % (3218)Termination reason: Unknown
% 9.01/5.96 % (3218)Termination phase: shuffling
% 9.01/5.96
% 9.01/5.96 % (3218)Memory used [KB]: 16247
% 9.01/5.96 % (3218)Time elapsed: 0.052 s
% 9.01/5.96 % (3218)Instructions burned: 128 (million)
% 9.01/5.96 % (3218)------------------------------
% 9.01/5.96 % (3218)------------------------------
% 9.01/5.97 % (3220)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_3077 for (2991ds/3Mi)
% 9.01/5.97 % (3220)Instruction limit reached!
% 9.01/5.97 % (3220)------------------------------
% 9.01/5.97 % (3220)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.97 % (3220)Termination reason: Unknown
% 9.01/5.97 % (3220)Termination phase: shuffling
% 9.01/5.97
% 9.01/5.97 % (3220)Memory used [KB]: 13688
% 9.01/5.97 % (3220)Time elapsed: 0.003 s
% 9.01/5.97 % (3220)Instructions burned: 3 (million)
% 9.01/5.97 % (3220)------------------------------
% 9.01/5.97 % (3220)------------------------------
% 9.01/5.98 % (3221)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_3077 for (2991ds/20Mi)
% 9.01/5.98 % (3219)Instruction limit reached!
% 9.01/5.98 % (3219)------------------------------
% 9.01/5.98 % (3219)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/5.98 % (3219)Termination reason: Unknown
% 9.01/5.98 % (3219)Termination phase: shuffling
% 9.01/5.98
% 9.01/5.98 % (3219)Memory used [KB]: 15735
% 9.01/5.98 % (3219)Time elapsed: 0.038 s
% 9.01/5.98 % (3219)Instructions burned: 102 (million)
% 9.01/5.98 % (3219)------------------------------
% 9.01/5.98 % (3219)------------------------------
% 9.01/6.00 % (3222)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_3077 for (2991ds/86Mi)
% 9.01/6.00 % (3221)Instruction limit reached!
% 9.01/6.00 % (3221)------------------------------
% 9.01/6.00 % (3221)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/6.00 % (3221)Termination reason: Unknown
% 9.01/6.00 % (3221)Termination phase: shuffling
% 9.01/6.00
% 9.01/6.00 % (3221)Memory used [KB]: 14583
% 9.01/6.00 % (3221)Time elapsed: 0.017 s
% 9.01/6.00 % (3221)Instructions burned: 22 (million)
% 9.01/6.00 % (3221)------------------------------
% 9.01/6.00 % (3221)------------------------------
% 9.01/6.01 % (3205)Instruction limit reached!
% 9.01/6.01 % (3205)------------------------------
% 9.01/6.01 % (3205)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.01/6.01 % (3205)Termination reason: Unknown
% 9.01/6.01 % (3205)Termination phase: shuffling
% 9.01/6.01
% 9.01/6.01 % (3205)Memory used [KB]: 26481
% 9.01/6.01 % (3205)Time elapsed: 0.253 s
% 9.01/6.01 % (3205)Instructions burned: 711 (million)
% 9.01/6.01 % (3205)------------------------------
% 9.01/6.01 % (3205)------------------------------
% 9.01/6.01 % (3223)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_3077 for (2991ds/107Mi)
% 9.01/6.02 % (3224)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_3077 for (2991ds/39Mi)
% 12.02/6.03 % (3222)Instruction limit reached!
% 12.02/6.03 % (3222)------------------------------
% 12.02/6.03 % (3222)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.03 % (3222)Termination reason: Unknown
% 12.02/6.03 % (3222)Termination phase: shuffling
% 12.02/6.03
% 12.02/6.03 % (3222)Memory used [KB]: 15607
% 12.02/6.03 % (3222)Time elapsed: 0.038 s
% 12.02/6.03 % (3222)Instructions burned: 88 (million)
% 12.02/6.03 % (3222)------------------------------
% 12.02/6.03 % (3222)------------------------------
% 12.02/6.04 % (3224)Instruction limit reached!
% 12.02/6.04 % (3224)------------------------------
% 12.02/6.04 % (3224)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.04 % (3224)Termination reason: Unknown
% 12.02/6.04 % (3224)Termination phase: shuffling
% 12.02/6.04
% 12.02/6.04 % (3224)Memory used [KB]: 14967
% 12.02/6.04 % (3224)Time elapsed: 0.022 s
% 12.02/6.04 % (3224)Instructions burned: 41 (million)
% 12.02/6.04 % (3224)------------------------------
% 12.02/6.04 % (3224)------------------------------
% 12.02/6.04 % (3225)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_3077 for (2991ds/448Mi)
% 12.02/6.05 % (3226)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_3077 for (2991ds/46Mi)
% 12.02/6.05 % (3223)Instruction limit reached!
% 12.02/6.05 % (3223)------------------------------
% 12.02/6.05 % (3223)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.05 % (3223)Termination reason: Unknown
% 12.02/6.05 % (3223)Termination phase: shuffling
% 12.02/6.05
% 12.02/6.05 % (3223)Memory used [KB]: 15863
% 12.02/6.05 % (3223)Time elapsed: 0.045 s
% 12.02/6.05 % (3223)Instructions burned: 107 (million)
% 12.02/6.05 % (3223)------------------------------
% 12.02/6.05 % (3223)------------------------------
% 12.02/6.06 % (3227)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_3077 for (2990ds/98Mi)
% 12.02/6.07 % (3226)Instruction limit reached!
% 12.02/6.07 % (3226)------------------------------
% 12.02/6.07 % (3226)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.07 % (3226)Termination reason: Unknown
% 12.02/6.07 % (3226)Termination phase: shuffling
% 12.02/6.07
% 12.02/6.07 % (3226)Memory used [KB]: 14967
% 12.02/6.07 % (3226)Time elapsed: 0.025 s
% 12.02/6.07 % (3226)Instructions burned: 46 (million)
% 12.02/6.07 % (3226)------------------------------
% 12.02/6.07 % (3226)------------------------------
% 12.02/6.08 % (3207)Instruction limit reached!
% 12.02/6.08 % (3207)------------------------------
% 12.02/6.08 % (3207)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.08 % (3207)Termination reason: Unknown
% 12.02/6.08 % (3207)Termination phase: shuffling
% 12.02/6.08
% 12.02/6.08 % (3207)Memory used [KB]: 28656
% 12.02/6.08 % (3207)Time elapsed: 0.315 s
% 12.02/6.08 % (3207)Instructions burned: 902 (million)
% 12.02/6.08 % (3207)------------------------------
% 12.02/6.08 % (3207)------------------------------
% 12.02/6.09 % (3228)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_3077 for (2990ds/507Mi)
% 12.02/6.10 % (3229)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_3077 for (2990ds/149Mi)
% 12.02/6.10 % (3227)Instruction limit reached!
% 12.02/6.10 % (3227)------------------------------
% 12.02/6.10 % (3227)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.02/6.10 % (3227)Termination reason: Unknown
% 12.02/6.10 % (3227)Termination phase: shuffling
% 12.02/6.10
% 12.02/6.10 % (3227)Memory used [KB]: 15607
% 12.02/6.10 % (3227)Time elapsed: 0.037 s
% 12.02/6.10 % (3227)Instructions burned: 98 (million)
% 12.02/6.10 % (3227)------------------------------
% 12.02/6.10 % (3227)------------------------------
% 12.02/6.11 % (3230)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_3077 for (2990ds/39Mi)
% 12.62/6.13 % (3230)Instruction limit reached!
% 12.62/6.13 % (3230)------------------------------
% 12.62/6.13 % (3230)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.13 % (3230)Termination reason: Unknown
% 12.62/6.13 % (3230)Termination phase: shuffling
% 12.62/6.13
% 12.62/6.13 % (3230)Memory used [KB]: 14967
% 12.62/6.13 % (3230)Time elapsed: 0.022 s
% 12.62/6.13 % (3230)Instructions burned: 40 (million)
% 12.62/6.13 % (3230)------------------------------
% 12.62/6.13 % (3230)------------------------------
% 12.62/6.14 % (3231)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_3077 for (2989ds/6Mi)
% 12.62/6.15 % (3214)Instruction limit reached!
% 12.62/6.15 % (3214)------------------------------
% 12.62/6.15 % (3214)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.15 % (3214)Termination reason: Unknown
% 12.62/6.15 % (3214)Termination phase: shuffling
% 12.62/6.15
% 12.62/6.15 % (3214)Memory used [KB]: 28016
% 12.62/6.15 % (3214)Time elapsed: 0.330 s
% 12.62/6.15 % (3214)Instructions burned: 881 (million)
% 12.62/6.15 % (3214)------------------------------
% 12.62/6.15 % (3214)------------------------------
% 12.62/6.15 % (3231)Instruction limit reached!
% 12.62/6.15 % (3231)------------------------------
% 12.62/6.15 % (3231)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.15 % (3231)Termination reason: Unknown
% 12.62/6.15 % (3231)Termination phase: shuffling
% 12.62/6.15
% 12.62/6.15 % (3231)Memory used [KB]: 13944
% 12.62/6.15 % (3231)Time elapsed: 0.007 s
% 12.62/6.15 % (3231)Instructions burned: 6 (million)
% 12.62/6.15 % (3231)------------------------------
% 12.62/6.15 % (3231)------------------------------
% 12.62/6.15 % (3229)Instruction limit reached!
% 12.62/6.15 % (3229)------------------------------
% 12.62/6.15 % (3229)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.15 % (3229)Termination reason: Unknown
% 12.62/6.15 % (3229)Termination phase: shuffling
% 12.62/6.15
% 12.62/6.15 % (3229)Memory used [KB]: 16502
% 12.62/6.15 % (3229)Time elapsed: 0.059 s
% 12.62/6.15 % (3229)Instructions burned: 150 (million)
% 12.62/6.15 % (3229)------------------------------
% 12.62/6.15 % (3229)------------------------------
% 12.62/6.16 % (3232)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_3077 for (2989ds/1087Mi)
% 12.62/6.16 % (3233)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_3077 for (2989ds/47Mi)
% 12.62/6.16 % (3212)Instruction limit reached!
% 12.62/6.16 % (3212)------------------------------
% 12.62/6.16 % (3212)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.16 % (3212)Termination reason: Unknown
% 12.62/6.16 % (3212)Termination phase: shuffling
% 12.62/6.16
% 12.62/6.16 % (3212)Memory used [KB]: 27376
% 12.62/6.16 % (3212)Time elapsed: 0.328 s
% 12.62/6.16 % (3212)Instructions burned: 781 (million)
% 12.62/6.16 % (3212)------------------------------
% 12.62/6.16 % (3212)------------------------------
% 12.62/6.16 % (3234)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_3077 for (2989ds/44Mi)
% 12.62/6.18 % (3225)Instruction limit reached!
% 12.62/6.18 % (3225)------------------------------
% 12.62/6.18 % (3225)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.18 % (3225)Termination reason: Unknown
% 12.62/6.18 % (3225)Termination phase: shuffling
% 12.62/6.18
% 12.62/6.18 % (3225)Memory used [KB]: 22771
% 12.62/6.18 % (3225)Time elapsed: 0.138 s
% 12.62/6.18 % (3225)Instructions burned: 449 (million)
% 12.62/6.18 % (3225)------------------------------
% 12.62/6.18 % (3225)------------------------------
% 12.62/6.18 % (3235)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_3077 for (2989ds/432Mi)
% 12.62/6.18 % (3233)Instruction limit reached!
% 12.62/6.18 % (3233)------------------------------
% 12.62/6.18 % (3233)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.18 % (3233)Termination reason: Unknown
% 12.62/6.18 % (3233)Termination phase: shuffling
% 12.62/6.18
% 12.62/6.18 % (3233)Memory used [KB]: 14967
% 12.62/6.18 % (3233)Time elapsed: 0.025 s
% 12.62/6.18 % (3233)Instructions burned: 47 (million)
% 12.62/6.18 % (3233)------------------------------
% 12.62/6.18 % (3233)------------------------------
% 12.62/6.18 % (3234)Instruction limit reached!
% 12.62/6.18 % (3234)------------------------------
% 12.62/6.18 % (3234)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.62/6.18 % (3234)Termination reason: Unknown
% 12.62/6.18 % (3234)Termination phase: shuffling
% 12.62/6.18
% 12.62/6.19 % (3234)Memory used [KB]: 15095
% 12.62/6.19 % (3234)Time elapsed: 0.023 s
% 12.62/6.19 % (3234)Instructions burned: 46 (million)
% 12.62/6.19 % (3234)------------------------------
% 12.62/6.19 % (3234)------------------------------
% 12.62/6.19 % (3236)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_3077 for (2989ds/62Mi)
% 12.62/6.19 % (3237)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_3077 for (2989ds/31Mi)
% 12.62/6.20 % (3238)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_3077 for (2989ds/96Mi)
% 13.09/6.21 % (3237)Instruction limit reached!
% 13.09/6.21 % (3237)------------------------------
% 13.09/6.21 % (3237)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.21 % (3237)Termination reason: Unknown
% 13.09/6.21 % (3237)Termination phase: shuffling
% 13.09/6.21
% 13.09/6.21 % (3237)Memory used [KB]: 14839
% 13.09/6.21 % (3237)Time elapsed: 0.019 s
% 13.09/6.21 % (3237)Instructions burned: 31 (million)
% 13.09/6.21 % (3237)------------------------------
% 13.09/6.21 % (3237)------------------------------
% 13.09/6.22 % (3236)Instruction limit reached!
% 13.09/6.22 % (3236)------------------------------
% 13.09/6.22 % (3236)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.22 % (3236)Termination reason: Unknown
% 13.09/6.22 % (3236)Termination phase: shuffling
% 13.09/6.22
% 13.09/6.22 % (3236)Memory used [KB]: 15223
% 13.09/6.22 % (3236)Time elapsed: 0.028 s
% 13.09/6.22 % (3236)Instructions burned: 64 (million)
% 13.09/6.22 % (3236)------------------------------
% 13.09/6.22 % (3236)------------------------------
% 13.09/6.22 % (3239)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_3077 for (2989ds/453Mi)
% 13.09/6.23 % (3240)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_3077 for (2989ds/71Mi)
% 13.09/6.23 % (3238)Instruction limit reached!
% 13.09/6.23 % (3238)------------------------------
% 13.09/6.23 % (3238)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.23 % (3238)Termination reason: Unknown
% 13.09/6.23 % (3238)Termination phase: shuffling
% 13.09/6.23
% 13.09/6.23 % (3238)Memory used [KB]: 15351
% 13.09/6.23 % (3238)Time elapsed: 0.039 s
% 13.09/6.23 % (3238)Instructions burned: 98 (million)
% 13.09/6.23 % (3238)------------------------------
% 13.09/6.23 % (3238)------------------------------
% 13.09/6.25 % (3241)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_3077 for (2989ds/83Mi)
% 13.09/6.26 % (3228)Instruction limit reached!
% 13.09/6.26 % (3228)------------------------------
% 13.09/6.26 % (3228)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.26 % (3228)Termination reason: Unknown
% 13.09/6.26 % (3228)Termination phase: shuffling
% 13.09/6.26
% 13.09/6.26 % (3228)Memory used [KB]: 20980
% 13.09/6.26 % (3228)Time elapsed: 0.174 s
% 13.09/6.26 % (3228)Instructions burned: 507 (million)
% 13.09/6.26 % (3228)------------------------------
% 13.09/6.26 % (3228)------------------------------
% 13.09/6.26 % (3240)Instruction limit reached!
% 13.09/6.26 % (3240)------------------------------
% 13.09/6.26 % (3240)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.26 % (3240)Termination reason: Unknown
% 13.09/6.26 % (3240)Termination phase: shuffling
% 13.09/6.26
% 13.09/6.26 % (3240)Memory used [KB]: 15223
% 13.09/6.26 % (3240)Time elapsed: 0.033 s
% 13.09/6.26 % (3240)Instructions burned: 71 (million)
% 13.09/6.26 % (3240)------------------------------
% 13.09/6.26 % (3240)------------------------------
% 13.09/6.27 % (3196)Instruction limit reached!
% 13.09/6.27 % (3196)------------------------------
% 13.09/6.27 % (3196)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.27 % (3196)Termination reason: Unknown
% 13.09/6.27 % (3196)Termination phase: shuffling
% 13.09/6.27
% 13.09/6.27 % (3196)Memory used [KB]: 34924
% 13.09/6.27 % (3196)Time elapsed: 0.564 s
% 13.09/6.27 % (3196)Instructions burned: 1042 (million)
% 13.09/6.27 % (3196)------------------------------
% 13.09/6.27 % (3196)------------------------------
% 13.09/6.27 % (3242)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_3077 for (2988ds/11Mi)
% 13.09/6.27 % (3243)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_3077 for (2988ds/41Mi)
% 13.09/6.28 % (3241)Instruction limit reached!
% 13.09/6.28 % (3241)------------------------------
% 13.09/6.28 % (3241)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.28 % (3241)Termination reason: Unknown
% 13.09/6.28 % (3241)Termination phase: shuffling
% 13.09/6.28
% 13.09/6.28 % (3241)Memory used [KB]: 15479
% 13.09/6.28 % (3241)Time elapsed: 0.037 s
% 13.09/6.28 % (3241)Instructions burned: 86 (million)
% 13.09/6.28 % (3241)------------------------------
% 13.09/6.28 % (3241)------------------------------
% 13.09/6.28 % (3242)Instruction limit reached!
% 13.09/6.28 % (3242)------------------------------
% 13.09/6.28 % (3242)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.28 % (3242)Termination reason: Unknown
% 13.09/6.28 % (3242)Termination phase: shuffling
% 13.09/6.28
% 13.09/6.28 % (3242)Memory used [KB]: 14328
% 13.09/6.28 % (3242)Time elapsed: 0.012 s
% 13.09/6.28 % (3242)Instructions burned: 11 (million)
% 13.09/6.28 % (3242)------------------------------
% 13.09/6.28 % (3242)------------------------------
% 13.09/6.29 % (3244)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_3077 for (2988ds/6Mi)
% 13.09/6.29 % (3245)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_3077 for (2988ds/150Mi)
% 13.09/6.29 % (3246)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_3077 for (2988ds/343Mi)
% 13.09/6.29 % (3243)Instruction limit reached!
% 13.09/6.29 % (3243)------------------------------
% 13.09/6.29 % (3243)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.09/6.29 % (3243)Termination reason: Unknown
% 13.09/6.29 % (3243)Termination phase: shuffling
% 13.09/6.29
% 13.09/6.29 % (3243)Memory used [KB]: 14967
% 13.09/6.29 % (3243)Time elapsed: 0.022 s
% 13.09/6.29 % (3243)Instructions burned: 43 (million)
% 13.09/6.29 % (3243)------------------------------
% 13.09/6.29 % (3243)------------------------------
% 13.62/6.30 % (3244)Instruction limit reached!
% 13.62/6.30 % (3244)------------------------------
% 13.62/6.30 % (3244)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.62/6.30 % (3244)Termination reason: Unknown
% 13.62/6.30 % (3244)Termination phase: shuffling
% 13.62/6.30
% 13.62/6.30 % (3244)Memory used [KB]: 13944
% 13.62/6.30 % (3244)Time elapsed: 0.009 s
% 13.62/6.30 % (3244)Instructions burned: 6 (million)
% 13.62/6.30 % (3244)------------------------------
% 13.62/6.30 % (3244)------------------------------
% 13.62/6.31 % (3247)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_3077 for (2988ds/109Mi)
% 13.62/6.31 % (3248)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_3077 for (2988ds/253Mi)
% 13.62/6.34 % (3235)Instruction limit reached!
% 13.62/6.34 % (3235)------------------------------
% 13.62/6.34 % (3235)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.62/6.34 % (3235)Termination reason: Unknown
% 13.62/6.34 % (3235)Termination phase: shuffling
% 13.62/6.34
% 13.62/6.34 % (3235)Memory used [KB]: 22643
% 13.62/6.34 % (3235)Time elapsed: 0.155 s
% 13.62/6.34 % (3235)Instructions burned: 432 (million)
% 13.62/6.34 % (3235)------------------------------
% 13.62/6.34 % (3235)------------------------------
% 13.62/6.35 % (3249)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_3077 for (2988ds/140Mi)
% 13.62/6.35 % (3245)Instruction limit reached!
% 13.62/6.35 % (3245)------------------------------
% 13.62/6.35 % (3245)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.62/6.35 % (3245)Termination reason: Unknown
% 13.62/6.35 % (3245)Termination phase: shuffling
% 13.62/6.35
% 13.62/6.35 % (3245)Memory used [KB]: 16502
% 13.62/6.35 % (3245)Time elapsed: 0.060 s
% 13.62/6.35 % (3245)Instructions burned: 151 (million)
% 13.62/6.35 % (3245)------------------------------
% 13.62/6.35 % (3245)------------------------------
% 13.62/6.36 % (3239)Instruction limit reached!
% 13.62/6.36 % (3239)------------------------------
% 13.62/6.36 % (3239)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.62/6.36 % (3239)Termination reason: Unknown
% 13.62/6.36 % (3239)Termination phase: shuffling
% 13.62/6.36
% 13.62/6.36 % (3239)Memory used [KB]: 22899
% 13.62/6.36 % (3239)Time elapsed: 0.139 s
% 13.62/6.36 % (3239)Instructions burned: 453 (million)
% 13.62/6.36 % (3239)------------------------------
% 13.62/6.36 % (3239)------------------------------
% 13.62/6.36 % (3250)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_3077 for (2987ds/614Mi)
% 13.62/6.37 % (3251)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_3077 for (2987ds/202Mi)
% 13.62/6.38 % (3247)Instruction limit reached!
% 13.62/6.38 % (3247)------------------------------
% 13.62/6.38 % (3247)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.62/6.38 % (3247)Termination reason: Unknown
% 13.62/6.38 % (3247)Termination phase: shuffling
% 13.62/6.38
% 13.62/6.38 % (3247)Memory used [KB]: 15863
% 13.62/6.38 % (3247)Time elapsed: 0.076 s
% 13.62/6.38 % (3247)Instructions burned: 109 (million)
% 13.62/6.38 % (3247)------------------------------
% 13.62/6.38 % (3247)------------------------------
% 14.13/6.40 % (3248)Instruction limit reached!
% 14.13/6.40 % (3248)------------------------------
% 14.13/6.40 % (3248)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.13/6.40 % (3248)Termination reason: Unknown
% 14.13/6.40 % (3248)Termination phase: shuffling
% 14.13/6.40
% 14.13/6.40 % (3248)Memory used [KB]: 17910
% 14.13/6.40 % (3248)Time elapsed: 0.093 s
% 14.13/6.40 % (3248)Instructions burned: 253 (million)
% 14.13/6.40 % (3248)------------------------------
% 14.13/6.40 % (3248)------------------------------
% 14.13/6.40 % (3249)Instruction limit reached!
% 14.13/6.40 % (3249)------------------------------
% 14.13/6.40 % (3249)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.13/6.40 % (3249)Termination reason: Unknown
% 14.13/6.40 % (3249)Termination phase: shuffling
% 14.13/6.40
% 14.13/6.40 % (3249)Memory used [KB]: 16247
% 14.13/6.40 % (3249)Time elapsed: 0.055 s
% 14.13/6.40 % (3249)Instructions burned: 140 (million)
% 14.13/6.40 % (3249)------------------------------
% 14.13/6.40 % (3249)------------------------------
% 14.13/6.41 % (3252)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_3077 for (2987ds/339Mi)
% 14.13/6.41 % (3246)Instruction limit reached!
% 14.13/6.41 % (3246)------------------------------
% 14.13/6.41 % (3246)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.13/6.41 % (3246)Termination reason: Unknown
% 14.13/6.41 % (3246)Termination phase: shuffling
% 14.13/6.41
% 14.13/6.41 % (3246)Memory used [KB]: 18677
% 14.13/6.41 % (3246)Time elapsed: 0.120 s
% 14.13/6.41 % (3246)Instructions burned: 345 (million)
% 14.13/6.41 % (3246)------------------------------
% 14.13/6.41 % (3246)------------------------------
% 14.13/6.41 % (3253)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_3077 for (2987ds/283Mi)
% 14.13/6.41 % (3254)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_3077 for (2987ds/134Mi)
% 14.13/6.42 % (3255)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_3077 for (2987ds/298Mi)
% 14.13/6.45 % (3251)Instruction limit reached!
% 14.13/6.45 % (3251)------------------------------
% 14.13/6.45 % (3251)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.13/6.45 % (3251)Termination reason: Unknown
% 14.13/6.45 % (3251)Termination phase: shuffling
% 14.13/6.45
% 14.13/6.45 % (3251)Memory used [KB]: 17142
% 14.13/6.45 % (3251)Time elapsed: 0.076 s
% 14.13/6.45 % (3251)Instructions burned: 204 (million)
% 14.13/6.45 % (3251)------------------------------
% 14.13/6.45 % (3251)------------------------------
% 14.13/6.46 % (3256)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_3077 for (2986ds/192Mi)
% 14.13/6.47 % (3254)Instruction limit reached!
% 14.13/6.47 % (3254)------------------------------
% 14.13/6.47 % (3254)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.13/6.47 % (3254)Termination reason: Unknown
% 14.13/6.47 % (3254)Termination phase: shuffling
% 14.13/6.47
% 14.13/6.47 % (3254)Memory used [KB]: 16247
% 14.13/6.47 % (3254)Time elapsed: 0.053 s
% 14.13/6.47 % (3254)Instructions burned: 135 (million)
% 14.13/6.47 % (3254)------------------------------
% 14.13/6.47 % (3254)------------------------------
% 14.13/6.48 % (3257)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_3077 for (2986ds/258Mi)
% 14.59/6.50 % (3253)Instruction limit reached!
% 14.59/6.50 % (3253)------------------------------
% 14.59/6.50 % (3253)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.50 % (3253)Termination reason: Unknown
% 14.59/6.50 % (3253)Termination phase: shuffling
% 14.59/6.50
% 14.59/6.50 % (3253)Memory used [KB]: 20468
% 14.59/6.50 % (3253)Time elapsed: 0.090 s
% 14.59/6.50 % (3253)Instructions burned: 283 (million)
% 14.59/6.50 % (3253)------------------------------
% 14.59/6.50 % (3253)------------------------------
% 14.59/6.52 % (3258)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_3077 for (2986ds/90Mi)
% 14.59/6.52 % (3255)Instruction limit reached!
% 14.59/6.52 % (3255)------------------------------
% 14.59/6.52 % (3255)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.52 % (3255)Termination reason: Unknown
% 14.59/6.52 % (3255)Termination phase: shuffling
% 14.59/6.52
% 14.59/6.52 % (3255)Memory used [KB]: 20596
% 14.59/6.52 % (3255)Time elapsed: 0.095 s
% 14.59/6.52 % (3255)Instructions burned: 299 (million)
% 14.59/6.52 % (3255)------------------------------
% 14.59/6.52 % (3255)------------------------------
% 14.59/6.53 % (3232)Instruction limit reached!
% 14.59/6.53 % (3232)------------------------------
% 14.59/6.53 % (3232)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.53 % (3232)Termination reason: Unknown
% 14.59/6.53 % (3232)Termination phase: Property scanning
% 14.59/6.53
% 14.59/6.53 % (3232)Memory used [KB]: 29551
% 14.59/6.53 % (3232)Time elapsed: 0.367 s
% 14.59/6.53 % (3232)Instructions burned: 1087 (million)
% 14.59/6.53 % (3232)------------------------------
% 14.59/6.53 % (3232)------------------------------
% 14.59/6.53 % (3259)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_3077 for (2986ds/231Mi)
% 14.59/6.53 % (3256)Instruction limit reached!
% 14.59/6.53 % (3256)------------------------------
% 14.59/6.53 % (3256)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.53 % (3256)Termination reason: Unknown
% 14.59/6.53 % (3256)Termination phase: shuffling
% 14.59/6.53
% 14.59/6.53 % (3256)Memory used [KB]: 17142
% 14.59/6.53 % (3256)Time elapsed: 0.073 s
% 14.59/6.53 % (3256)Instructions burned: 192 (million)
% 14.59/6.53 % (3256)------------------------------
% 14.59/6.53 % (3256)------------------------------
% 14.59/6.54 % (3260)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_3077 for (2985ds/458Mi)
% 14.59/6.54 % (3261)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_3077 for (2985ds/31Mi)
% 14.59/6.56 % (3258)Instruction limit reached!
% 14.59/6.56 % (3258)------------------------------
% 14.59/6.56 % (3258)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.56 % (3258)Termination reason: Unknown
% 14.59/6.56 % (3258)Termination phase: shuffling
% 14.59/6.56
% 14.59/6.56 % (3258)Memory used [KB]: 15607
% 14.59/6.56 % (3258)Time elapsed: 0.043 s
% 14.59/6.56 % (3258)Instructions burned: 90 (million)
% 14.59/6.56 % (3258)------------------------------
% 14.59/6.56 % (3258)------------------------------
% 14.59/6.56 % (3261)Instruction limit reached!
% 14.59/6.56 % (3261)------------------------------
% 14.59/6.56 % (3261)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.56 % (3261)Termination reason: Unknown
% 14.59/6.56 % (3261)Termination phase: shuffling
% 14.59/6.56
% 14.59/6.56 % (3261)Memory used [KB]: 14839
% 14.59/6.56 % (3261)Time elapsed: 0.020 s
% 14.59/6.56 % (3261)Instructions burned: 31 (million)
% 14.59/6.56 % (3261)------------------------------
% 14.59/6.56 % (3261)------------------------------
% 14.59/6.57 % (3262)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_3077 for (2985ds/297Mi)
% 14.59/6.57 % (3263)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_3077 for (2985ds/987Mi)
% 14.59/6.58 % (3250)Instruction limit reached!
% 14.59/6.58 % (3250)------------------------------
% 14.59/6.58 % (3250)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.58 % (3250)Termination reason: Unknown
% 14.59/6.58 % (3250)Termination phase: shuffling
% 14.59/6.58
% 14.59/6.58 % (3250)Memory used [KB]: 24690
% 14.59/6.58 % (3250)Time elapsed: 0.213 s
% 14.59/6.58 % (3250)Instructions burned: 615 (million)
% 14.59/6.58 % (3250)------------------------------
% 14.59/6.58 % (3250)------------------------------
% 14.59/6.58 % (3257)Instruction limit reached!
% 14.59/6.58 % (3257)------------------------------
% 14.59/6.58 % (3257)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.58 % (3257)Termination reason: Unknown
% 14.59/6.58 % (3257)Termination phase: shuffling
% 14.59/6.58
% 14.59/6.58 % (3257)Memory used [KB]: 17910
% 14.59/6.58 % (3257)Time elapsed: 0.106 s
% 14.59/6.58 % (3257)Instructions burned: 258 (million)
% 14.59/6.58 % (3257)------------------------------
% 14.59/6.58 % (3257)------------------------------
% 14.59/6.59 % (3252)Instruction limit reached!
% 14.59/6.59 % (3252)------------------------------
% 14.59/6.59 % (3252)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.59/6.59 % (3252)Termination reason: Unknown
% 14.59/6.59 % (3252)Termination phase: shuffling
% 14.59/6.59
% 14.59/6.59 % (3252)Memory used [KB]: 19189
% 14.59/6.59 % (3252)Time elapsed: 0.182 s
% 14.59/6.59 % (3252)Instructions burned: 340 (million)
% 14.59/6.59 % (3252)------------------------------
% 14.59/6.59 % (3252)------------------------------
% 14.59/6.59 % (3264)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_3077 for (2985ds/182Mi)
% 14.59/6.59 % (3265)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_3077 for (2985ds/182Mi)
% 15.88/6.60 % (3259)Instruction limit reached!
% 15.88/6.60 % (3259)------------------------------
% 15.88/6.60 % (3259)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.88/6.60 % (3259)Termination reason: Unknown
% 15.88/6.60 % (3259)Termination phase: shuffling
% 15.88/6.60
% 15.88/6.60 % (3259)Memory used [KB]: 17654
% 15.88/6.60 % (3259)Time elapsed: 0.076 s
% 15.88/6.60 % (3259)Instructions burned: 232 (million)
% 15.88/6.60 % (3259)------------------------------
% 15.88/6.60 % (3259)------------------------------
% 15.88/6.61 % (3266)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_3077 for (2985ds/154Mi)
% 15.88/6.61 % (3267)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_3077 for (2985ds/79Mi)
% 15.88/6.65 % (3267)Instruction limit reached!
% 15.88/6.65 % (3267)------------------------------
% 15.88/6.65 % (3267)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.88/6.65 % (3267)Termination reason: Unknown
% 15.88/6.65 % (3267)Termination phase: shuffling
% 15.88/6.65
% 15.88/6.65 % (3267)Memory used [KB]: 15479
% 15.88/6.65 % (3267)Time elapsed: 0.035 s
% 15.88/6.65 % (3267)Instructions burned: 79 (million)
% 15.88/6.65 % (3267)------------------------------
% 15.88/6.65 % (3267)------------------------------
% 15.88/6.66 % (3264)Instruction limit reached!
% 15.88/6.66 % (3264)------------------------------
% 15.88/6.66 % (3264)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.88/6.66 % (3264)Termination reason: Unknown
% 15.88/6.66 % (3264)Termination phase: shuffling
% 15.88/6.66
% 15.88/6.66 % (3264)Memory used [KB]: 16758
% 15.88/6.66 % (3264)Time elapsed: 0.069 s
% 15.88/6.66 % (3264)Instructions burned: 183 (million)
% 15.88/6.66 % (3264)------------------------------
% 15.88/6.66 % (3264)------------------------------
% 15.88/6.66 % (3268)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_3077 for (2984ds/216Mi)
% 15.88/6.66 % (3265)Instruction limit reached!
% 15.88/6.66 % (3265)------------------------------
% 15.88/6.66 % (3265)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.88/6.66 % (3265)Termination reason: Unknown
% 15.88/6.66 % (3265)Termination phase: shuffling
% 15.88/6.66
% 15.88/6.66 % (3265)Memory used [KB]: 17014
% 15.88/6.66 % (3265)Time elapsed: 0.071 s
% 15.88/6.66 % (3265)Instructions burned: 182 (million)
% 15.88/6.66 % (3265)------------------------------
% 15.88/6.66 % (3265)------------------------------
% 15.88/6.67 % (3269)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_3077 for (2984ds/114Mi)
% 15.88/6.68 % (3270)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_3077 for (2984ds/922Mi)
% 15.88/6.69 % (3266)Instruction limit reached!
% 15.88/6.69 % (3266)------------------------------
% 15.88/6.69 % (3266)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.88/6.69 % (3266)Termination reason: Unknown
% 15.88/6.69 % (3266)Termination phase: shuffling
% 15.88/6.69
% 15.88/6.69 % (3266)Memory used [KB]: 16502
% 15.88/6.69 % (3266)Time elapsed: 0.086 s
% 15.88/6.69 % (3266)Instructions burned: 154 (million)
% 15.88/6.69 % (3266)------------------------------
% 15.88/6.69 % (3266)------------------------------
% 16.27/6.70 % (3260)Instruction limit reached!
% 16.27/6.70 % (3260)------------------------------
% 16.27/6.70 % (3260)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.70 % (3260)Termination reason: Unknown
% 16.27/6.70 % (3260)Termination phase: shuffling
% 16.27/6.70
% 16.27/6.70 % (3260)Memory used [KB]: 20340
% 16.27/6.70 % (3260)Time elapsed: 0.160 s
% 16.27/6.70 % (3260)Instructions burned: 459 (million)
% 16.27/6.70 % (3260)------------------------------
% 16.27/6.70 % (3260)------------------------------
% 16.27/6.70 % (3262)Instruction limit reached!
% 16.27/6.70 % (3262)------------------------------
% 16.27/6.70 % (3262)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.70 % (3262)Termination reason: Unknown
% 16.27/6.70 % (3262)Termination phase: shuffling
% 16.27/6.70
% 16.27/6.70 % (3262)Memory used [KB]: 20596
% 16.27/6.70 % (3262)Time elapsed: 0.135 s
% 16.27/6.70 % (3262)Instructions burned: 297 (million)
% 16.27/6.70 % (3262)------------------------------
% 16.27/6.70 % (3262)------------------------------
% 16.27/6.70 % (3271)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_3077 for (2984ds/182Mi)
% 16.27/6.71 % (3272)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_3077 for (2984ds/164Mi)
% 16.27/6.71 % (3269)Instruction limit reached!
% 16.27/6.71 % (3269)------------------------------
% 16.27/6.71 % (3269)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.71 % (3269)Termination reason: Unknown
% 16.27/6.71 % (3269)Termination phase: shuffling
% 16.27/6.71
% 16.27/6.71 % (3269)Memory used [KB]: 15991
% 16.27/6.71 % (3269)Time elapsed: 0.048 s
% 16.27/6.71 % (3269)Instructions burned: 115 (million)
% 16.27/6.71 % (3269)------------------------------
% 16.27/6.71 % (3269)------------------------------
% 16.27/6.72 % (3273)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_3077 for (2984ds/50Mi)
% 16.27/6.73 % (3274)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_3077 for (2984ds/368Mi)
% 16.27/6.73 % (3268)Instruction limit reached!
% 16.27/6.73 % (3268)------------------------------
% 16.27/6.73 % (3268)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.73 % (3268)Termination reason: Unknown
% 16.27/6.73 % (3268)Termination phase: shuffling
% 16.27/6.73
% 16.27/6.73 % (3268)Memory used [KB]: 17398
% 16.27/6.73 % (3268)Time elapsed: 0.071 s
% 16.27/6.73 % (3268)Instructions burned: 218 (million)
% 16.27/6.73 % (3268)------------------------------
% 16.27/6.73 % (3268)------------------------------
% 16.27/6.74 % (3275)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_3077 for (2984ds/1021Mi)
% 16.27/6.76 % (3273)Instruction limit reached!
% 16.27/6.76 % (3273)------------------------------
% 16.27/6.76 % (3273)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.76 % (3273)Termination reason: Unknown
% 16.27/6.76 % (3273)Termination phase: shuffling
% 16.27/6.76
% 16.27/6.76 % (3273)Memory used [KB]: 15095
% 16.27/6.76 % (3273)Time elapsed: 0.036 s
% 16.27/6.76 % (3273)Instructions burned: 51 (million)
% 16.27/6.76 % (3273)------------------------------
% 16.27/6.76 % (3273)------------------------------
% 16.27/6.76 % (3272)Instruction limit reached!
% 16.27/6.76 % (3272)------------------------------
% 16.27/6.76 % (3272)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.76 % (3272)Termination reason: Unknown
% 16.27/6.76 % (3272)Termination phase: shuffling
% 16.27/6.76
% 16.27/6.76 % (3272)Memory used [KB]: 16758
% 16.27/6.76 % (3272)Time elapsed: 0.057 s
% 16.27/6.76 % (3272)Instructions burned: 167 (million)
% 16.27/6.76 % (3272)------------------------------
% 16.27/6.76 % (3272)------------------------------
% 16.27/6.76 % (3271)Instruction limit reached!
% 16.27/6.76 % (3271)------------------------------
% 16.27/6.76 % (3271)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.76 % (3271)Termination reason: Unknown
% 16.27/6.76 % (3271)Termination phase: shuffling
% 16.27/6.76
% 16.27/6.76 % (3271)Memory used [KB]: 16886
% 16.27/6.76 % (3271)Time elapsed: 0.061 s
% 16.27/6.76 % (3271)Instructions burned: 182 (million)
% 16.27/6.76 % (3271)------------------------------
% 16.27/6.76 % (3271)------------------------------
% 16.27/6.78 % (3277)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_3077 for (2983ds/204Mi)
% 16.27/6.78 % (3278)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_3077 for (2983ds/26Mi)
% 16.27/6.78 % (3276)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_3077 for (2983ds/267Mi)
% 16.27/6.79 % (3278)Instruction limit reached!
% 16.27/6.79 % (3278)------------------------------
% 16.27/6.79 % (3278)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.27/6.79 % (3278)Termination reason: Unknown
% 16.27/6.79 % (3278)Termination phase: shuffling
% 16.27/6.79
% 16.27/6.79 % (3278)Memory used [KB]: 14711
% 16.27/6.79 % (3278)Time elapsed: 0.018 s
% 16.27/6.79 % (3278)Instructions burned: 27 (million)
% 16.27/6.79 % (3278)------------------------------
% 16.27/6.79 % (3278)------------------------------
% 16.94/6.80 % (3279)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_3077 for (2983ds/53Mi)
% 16.94/6.83 % (3279)Instruction limit reached!
% 16.94/6.83 % (3279)------------------------------
% 16.94/6.83 % (3279)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.94/6.83 % (3279)Termination reason: Unknown
% 16.94/6.83 % (3279)Termination phase: shuffling
% 16.94/6.83
% 16.94/6.83 % (3279)Memory used [KB]: 15223
% 16.94/6.83 % (3279)Time elapsed: 0.025 s
% 16.94/6.83 % (3279)Instructions burned: 56 (million)
% 16.94/6.83 % (3279)------------------------------
% 16.94/6.83 % (3279)------------------------------
% 16.94/6.84 % (3280)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_3077 for (2983ds/4436Mi)
% 16.94/6.85 % (3277)Instruction limit reached!
% 16.94/6.85 % (3277)------------------------------
% 16.94/6.85 % (3277)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.94/6.85 % (3277)Termination reason: Unknown
% 16.94/6.85 % (3277)Termination phase: shuffling
% 16.94/6.85
% 16.94/6.85 % (3277)Memory used [KB]: 17142
% 16.94/6.85 % (3277)Time elapsed: 0.076 s
% 16.94/6.85 % (3277)Instructions burned: 205 (million)
% 16.94/6.85 % (3277)------------------------------
% 16.94/6.85 % (3277)------------------------------
% 16.94/6.85 % (3274)Instruction limit reached!
% 16.94/6.85 % (3274)------------------------------
% 16.94/6.85 % (3274)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.94/6.85 % (3274)Termination reason: Unknown
% 16.94/6.85 % (3274)Termination phase: shuffling
% 16.94/6.85
% 16.94/6.85 % (3274)Memory used [KB]: 19445
% 16.94/6.85 % (3274)Time elapsed: 0.131 s
% 16.94/6.85 % (3274)Instructions burned: 370 (million)
% 16.94/6.85 % (3274)------------------------------
% 16.94/6.85 % (3274)------------------------------
% 16.94/6.86 % (3281)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_3077 for (2982ds/1539Mi)
% 16.94/6.87 % (3282)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_3077 for (2982ds/408Mi)
% 16.94/6.87 % (3263)Instruction limit reached!
% 16.94/6.87 % (3263)------------------------------
% 16.94/6.87 % (3263)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.94/6.87 % (3263)Termination reason: Unknown
% 16.94/6.87 % (3263)Termination phase: shuffling
% 16.94/6.87
% 16.94/6.87 % (3263)Memory used [KB]: 34157
% 16.94/6.87 % (3263)Time elapsed: 0.298 s
% 16.94/6.87 % (3263)Instructions burned: 987 (million)
% 16.94/6.87 % (3263)------------------------------
% 16.94/6.87 % (3263)------------------------------
% 16.94/6.88 % (3283)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_3077 for (2982ds/5205Mi)
% 17.53/6.91 % (3276)Instruction limit reached!
% 17.53/6.91 % (3276)------------------------------
% 17.53/6.91 % (3276)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.53/6.91 % (3276)Termination reason: Unknown
% 17.53/6.91 % (3276)Termination phase: shuffling
% 17.53/6.91
% 17.53/6.91 % (3276)Memory used [KB]: 20212
% 17.53/6.91 % (3276)Time elapsed: 0.133 s
% 17.53/6.91 % (3276)Instructions burned: 268 (million)
% 17.53/6.91 % (3276)------------------------------
% 17.53/6.91 % (3276)------------------------------
% 17.53/6.93 % (3284)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_3077 for (2982ds/1913Mi)
% 17.53/7.00 % (3270)Instruction limit reached!
% 17.53/7.00 % (3270)------------------------------
% 17.53/7.00 % (3270)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.53/7.00 % (3270)Termination reason: Unknown
% 17.53/7.00 % (3270)Termination phase: shuffling
% 17.53/7.00
% 17.53/7.00 % (3270)Memory used [KB]: 28912
% 17.53/7.00 % (3270)Time elapsed: 0.322 s
% 17.53/7.00 % (3270)Instructions burned: 923 (million)
% 17.53/7.00 % (3270)------------------------------
% 17.53/7.00 % (3270)------------------------------
% 17.53/7.01 % (3285)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_3077 for (2981ds/4350Mi)
% 17.53/7.01 % (3282)Instruction limit reached!
% 17.53/7.01 % (3282)------------------------------
% 17.53/7.01 % (3282)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.53/7.01 % (3282)Termination reason: Unknown
% 17.53/7.01 % (3282)Termination phase: shuffling
% 17.53/7.01
% 17.53/7.01 % (3282)Memory used [KB]: 20212
% 17.53/7.01 % (3282)Time elapsed: 0.147 s
% 17.53/7.01 % (3282)Instructions burned: 409 (million)
% 17.53/7.01 % (3282)------------------------------
% 17.53/7.01 % (3282)------------------------------
% 17.53/7.02 % (3286)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_3077 for (2981ds/68Mi)
% 18.93/7.05 % (3286)Instruction limit reached!
% 18.93/7.05 % (3286)------------------------------
% 18.93/7.05 % (3286)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.93/7.05 % (3286)Termination reason: Unknown
% 18.93/7.05 % (3286)Termination phase: shuffling
% 18.93/7.05
% 18.93/7.05 % (3286)Memory used [KB]: 15351
% 18.93/7.05 % (3286)Time elapsed: 0.032 s
% 18.93/7.05 % (3286)Instructions burned: 69 (million)
% 18.93/7.06 % (3286)------------------------------
% 18.93/7.06 % (3286)------------------------------
% 18.93/7.07 % (3287)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_3077 for (2980ds/358Mi)
% 18.93/7.09 % (3275)Instruction limit reached!
% 18.93/7.09 % (3275)------------------------------
% 18.93/7.09 % (3275)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.93/7.09 % (3275)Termination reason: Unknown
% 18.93/7.09 % (3275)Termination phase: Property scanning
% 18.93/7.09
% 18.93/7.09 % (3275)Memory used [KB]: 29551
% 18.93/7.09 % (3275)Time elapsed: 0.350 s
% 18.93/7.09 % (3275)Instructions burned: 1022 (million)
% 18.93/7.09 % (3275)------------------------------
% 18.93/7.09 % (3275)------------------------------
% 18.93/7.10 % (3288)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_3077 for (2980ds/31Mi)
% 18.93/7.12 % (3288)Instruction limit reached!
% 18.93/7.12 % (3288)------------------------------
% 18.93/7.12 % (3288)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.93/7.12 % (3288)Termination reason: Unknown
% 18.93/7.12 % (3288)Termination phase: shuffling
% 18.93/7.12
% 18.93/7.12 % (3288)Memory used [KB]: 14839
% 18.93/7.12 % (3288)Time elapsed: 0.020 s
% 18.93/7.12 % (3288)Instructions burned: 32 (million)
% 18.93/7.12 % (3288)------------------------------
% 18.93/7.12 % (3288)------------------------------
% 19.55/7.14 % (3289)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_3077 for (2980ds/279Mi)
% 19.55/7.19 % (3287)Instruction limit reached!
% 19.55/7.19 % (3287)------------------------------
% 19.55/7.19 % (3287)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.55/7.19 % (3287)Termination reason: Unknown
% 19.55/7.19 % (3287)Termination phase: shuffling
% 19.55/7.19
% 19.55/7.19 % (3287)Memory used [KB]: 19317
% 19.55/7.19 % (3287)Time elapsed: 0.127 s
% 19.55/7.19 % (3287)Instructions burned: 360 (million)
% 19.55/7.19 % (3287)------------------------------
% 19.55/7.19 % (3287)------------------------------
% 20.00/7.21 % (3290)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_3077 for (2979ds/258Mi)
% 20.00/7.22 % (3289)Instruction limit reached!
% 20.00/7.22 % (3289)------------------------------
% 20.00/7.22 % (3289)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/7.22 % (3289)Termination reason: Unknown
% 20.00/7.22 % (3289)Termination phase: shuffling
% 20.00/7.22
% 20.00/7.22 % (3289)Memory used [KB]: 20340
% 20.00/7.22 % (3289)Time elapsed: 0.089 s
% 20.00/7.22 % (3289)Instructions burned: 279 (million)
% 20.00/7.22 % (3289)------------------------------
% 20.00/7.22 % (3289)------------------------------
% 20.00/7.24 % (3291)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_3077 for (2979ds/26Mi)
% 20.00/7.25 % (3291)Instruction limit reached!
% 20.00/7.25 % (3291)------------------------------
% 20.00/7.25 % (3291)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/7.25 % (3291)Termination reason: Unknown
% 20.00/7.25 % (3291)Termination phase: shuffling
% 20.00/7.25
% 20.00/7.25 % (3291)Memory used [KB]: 14711
% 20.00/7.25 % (3291)Time elapsed: 0.017 s
% 20.00/7.25 % (3291)Instructions burned: 26 (million)
% 20.00/7.25 % (3291)------------------------------
% 20.00/7.25 % (3291)------------------------------
% 20.00/7.26 % (3292)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_3077 for (2978ds/995Mi)
% 20.00/7.30 % (3290)Instruction limit reached!
% 20.00/7.30 % (3290)------------------------------
% 20.00/7.30 % (3290)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.00/7.30 % (3290)Termination reason: Unknown
% 20.00/7.30 % (3290)Termination phase: shuffling
% 20.00/7.30
% 20.00/7.30 % (3290)Memory used [KB]: 17782
% 20.00/7.30 % (3290)Time elapsed: 0.092 s
% 20.00/7.30 % (3290)Instructions burned: 259 (million)
% 20.00/7.30 % (3290)------------------------------
% 20.00/7.30 % (3290)------------------------------
% 20.54/7.31 % (3293)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_3077 for (2978ds/373Mi)
% 20.54/7.38 % (3281)Instruction limit reached!
% 20.54/7.38 % (3281)------------------------------
% 20.54/7.38 % (3281)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.54/7.38 % (3281)Termination reason: Unknown
% 20.54/7.38 % (3281)Termination phase: Property scanning
% 20.54/7.38
% 20.54/7.38 % (3281)Memory used [KB]: 29551
% 20.54/7.38 % (3281)Time elapsed: 0.499 s
% 20.54/7.38 % (3281)Instructions burned: 1542 (million)
% 20.54/7.38 % (3281)------------------------------
% 20.54/7.38 % (3281)------------------------------
% 20.54/7.39 % (3294)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_3077 for (2977ds/253Mi)
% 21.02/7.44 % (3293)Instruction limit reached!
% 21.02/7.44 % (3293)------------------------------
% 21.02/7.44 % (3293)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.02/7.44 % (3293)Termination reason: Unknown
% 21.02/7.44 % (3293)Termination phase: shuffling
% 21.02/7.44
% 21.02/7.44 % (3293)Memory used [KB]: 19701
% 21.02/7.44 % (3293)Time elapsed: 0.133 s
% 21.02/7.44 % (3293)Instructions burned: 375 (million)
% 21.02/7.44 % (3293)------------------------------
% 21.02/7.44 % (3293)------------------------------
% 21.02/7.45 % (3295)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_3077 for (2976ds/832Mi)
% 21.02/7.50 % (3294)Instruction limit reached!
% 21.02/7.50 % (3294)------------------------------
% 21.02/7.50 % (3294)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.02/7.50 % (3294)Termination reason: Unknown
% 21.02/7.50 % (3294)Termination phase: shuffling
% 21.02/7.50
% 21.02/7.50 % (3294)Memory used [KB]: 19957
% 21.02/7.50 % (3294)Time elapsed: 0.108 s
% 21.02/7.50 % (3294)Instructions burned: 253 (million)
% 21.02/7.50 % (3294)------------------------------
% 21.02/7.50 % (3294)------------------------------
% 21.47/7.52 % (3296)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_3077 for (2976ds/668Mi)
% 21.47/7.56 % (3292)Instruction limit reached!
% 21.47/7.56 % (3292)------------------------------
% 21.47/7.56 % (3292)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.47/7.56 % (3292)Termination reason: Unknown
% 21.47/7.56 % (3292)Termination phase: shuffling
% 21.47/7.56
% 21.47/7.56 % (3292)Memory used [KB]: 34285
% 21.47/7.56 % (3292)Time elapsed: 0.298 s
% 21.47/7.56 % (3292)Instructions burned: 995 (million)
% 21.47/7.56 % (3292)------------------------------
% 21.47/7.56 % (3292)------------------------------
% 21.47/7.57 % (3297)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_3077 for (2975ds/40Mi)
% 21.47/7.58 % (3284)Instruction limit reached!
% 21.47/7.58 % (3284)------------------------------
% 21.47/7.58 % (3284)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.47/7.58 % (3284)Termination reason: Unknown
% 21.47/7.58 % (3284)Termination phase: SInE selection
% 21.47/7.58
% 21.47/7.58 % (3284)Memory used [KB]: 29935
% 21.47/7.58 % (3284)Time elapsed: 0.658 s
% 21.47/7.58 % (3284)Instructions burned: 1913 (million)
% 21.47/7.58 % (3284)------------------------------
% 21.47/7.58 % (3284)------------------------------
% 21.47/7.59 % (3297)Instruction limit reached!
% 21.47/7.59 % (3297)------------------------------
% 21.47/7.59 % (3297)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 21.47/7.59 % (3297)Termination reason: Unknown
% 21.47/7.59 % (3297)Termination phase: shuffling
% 21.47/7.59
% 21.47/7.59 % (3297)Memory used [KB]: 14839
% 21.47/7.59 % (3297)Time elapsed: 0.021 s
% 21.47/7.59 % (3297)Instructions burned: 41 (million)
% 21.47/7.59 % (3297)------------------------------
% 21.47/7.59 % (3297)------------------------------
% 21.47/7.59 % (3298)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_3077 for (2975ds/294Mi)
% 22.22/7.60 % (3299)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_3077 for (2975ds/187Mi)
% 22.22/7.67 % (3299)Instruction limit reached!
% 22.22/7.67 % (3299)------------------------------
% 22.22/7.67 % (3299)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.22/7.67 % (3299)Termination reason: Unknown
% 22.22/7.67 % (3299)Termination phase: shuffling
% 22.22/7.67
% 22.22/7.67 % (3299)Memory used [KB]: 16886
% 22.22/7.67 % (3299)Time elapsed: 0.071 s
% 22.22/7.67 % (3299)Instructions burned: 187 (million)
% 22.22/7.67 % (3299)------------------------------
% 22.22/7.67 % (3299)------------------------------
% 22.22/7.69 % (3300)lrs+1010_2:3_au=on:fsr=off:sd=4:ss=axioms:st=1.5:i=371:si=on:rtra=on_0 on DTF2THF_3077 for (2974ds/371Mi)
% 22.22/7.69 % (3298)Instruction limit reached!
% 22.22/7.69 % (3298)------------------------------
% 22.22/7.69 % (3298)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.22/7.69 % (3298)Termination reason: Unknown
% 22.22/7.69 % (3298)Termination phase: shuffling
% 22.22/7.69
% 22.22/7.69 % (3298)Memory used [KB]: 20596
% 22.22/7.69 % (3298)Time elapsed: 0.094 s
% 22.22/7.69 % (3298)Instructions burned: 296 (million)
% 22.22/7.69 % (3298)------------------------------
% 22.22/7.69 % (3298)------------------------------
% 22.61/7.70 % (3301)dis+1003_1:1_au=on:bd=preordered:fde=none:ixr=off:nwc=10.0:i=1827:si=on:rtra=on_0 on DTF2THF_3077 for (2974ds/1827Mi)
% 22.61/7.74 % (3295)Instruction limit reached!
% 22.61/7.74 % (3295)------------------------------
% 22.61/7.74 % (3295)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.61/7.74 % (3295)Termination reason: Unknown
% 22.61/7.74 % (3295)Termination phase: shuffling
% 22.61/7.74
% 22.61/7.74 % (3295)Memory used [KB]: 27632
% 22.61/7.74 % (3295)Time elapsed: 0.289 s
% 22.61/7.74 % (3295)Instructions burned: 833 (million)
% 22.61/7.74 % (3295)------------------------------
% 22.61/7.74 % (3295)------------------------------
% 22.61/7.76 % (3302)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_3077 for (2973ds/70Mi)
% 22.61/7.79 % (3302)Instruction limit reached!
% 22.61/7.79 % (3302)------------------------------
% 22.61/7.79 % (3302)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 22.61/7.79 % (3302)Termination reason: Unknown
% 22.61/7.79 % (3302)Termination phase: shuffling
% 22.61/7.79
% 22.61/7.79 % (3302)Memory used [KB]: 15479
% 22.61/7.79 % (3302)Time elapsed: 0.033 s
% 22.61/7.79 % (3302)Instructions burned: 72 (million)
% 22.61/7.79 % (3302)------------------------------
% 22.61/7.79 % (3302)------------------------------
% 23.17/7.80 % (3296)Instruction limit reached!
% 23.17/7.80 % (3296)------------------------------
% 23.17/7.80 % (3296)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.17/7.80 % (3296)Termination reason: Unknown
% 23.17/7.80 % (3296)Termination phase: shuffling
% 23.17/7.80
% 23.17/7.80 % (3296)Memory used [KB]: 25841
% 23.17/7.80 % (3296)Time elapsed: 0.289 s
% 23.17/7.80 % (3296)Instructions burned: 671 (million)
% 23.17/7.80 % (3296)------------------------------
% 23.17/7.80 % (3296)------------------------------
% 23.17/7.81 % (3303)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_3077 for (2973ds/316Mi)
% 23.17/7.81 % (3300)Instruction limit reached!
% 23.17/7.81 % (3300)------------------------------
% 23.17/7.81 % (3300)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.17/7.81 % (3300)Termination reason: Unknown
% 23.17/7.81 % (3300)Termination phase: shuffling
% 23.17/7.81
% 23.17/7.81 % (3300)Memory used [KB]: 19445
% 23.17/7.81 % (3300)Time elapsed: 0.130 s
% 23.17/7.81 % (3300)Instructions burned: 373 (million)
% 23.17/7.81 % (3300)------------------------------
% 23.17/7.81 % (3300)------------------------------
% 23.17/7.82 % (3304)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_3077 for (2973ds/932Mi)
% 23.17/7.83 % (3305)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_3077 for (2973ds/4714Mi)
% 23.82/7.91 % (3303)Instruction limit reached!
% 23.82/7.91 % (3303)------------------------------
% 23.82/7.91 % (3303)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 23.82/7.91 % (3303)Termination reason: Unknown
% 23.82/7.91 % (3303)Termination phase: shuffling
% 23.82/7.91
% 23.82/7.91 % (3303)Memory used [KB]: 20724
% 23.82/7.91 % (3303)Time elapsed: 0.105 s
% 23.82/7.91 % (3303)Instructions burned: 316 (million)
% 23.82/7.91 % (3303)------------------------------
% 23.82/7.91 % (3303)------------------------------
% 23.82/7.93 % (3306)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_3077 for (2972ds/8250Mi)
% 26.01/8.09 % (3304)Instruction limit reached!
% 26.01/8.09 % (3304)------------------------------
% 26.01/8.09 % (3304)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 26.01/8.09 % (3304)Termination reason: Unknown
% 26.01/8.09 % (3304)Termination phase: shuffling
% 26.01/8.09
% 26.01/8.09 % (3304)Memory used [KB]: 33389
% 26.01/8.09 % (3304)Time elapsed: 0.280 s
% 26.01/8.09 % (3304)Instructions burned: 933 (million)
% 26.01/8.09 % (3304)------------------------------
% 26.01/8.09 % (3304)------------------------------
% 26.01/8.11 % (3307)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_3077 for (2970ds/770Mi)
% 27.15/8.29 % (3301)Instruction limit reached!
% 27.15/8.29 % (3301)------------------------------
% 27.15/8.29 % (3301)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.15/8.29 % (3301)Termination reason: Unknown
% 27.15/8.29 % (3301)Termination phase: Preprocessing 3
% 27.15/8.29
% 27.15/8.29 % (3301)Memory used [KB]: 35180
% 27.15/8.29 % (3301)Time elapsed: 0.588 s
% 27.15/8.29 % (3301)Instructions burned: 1829 (million)
% 27.15/8.29 % (3301)------------------------------
% 27.15/8.29 % (3301)------------------------------
% 27.75/8.30 % (3308)ott+1010_5:1_av=off:fe=off:ins=1:kws=precedence:sp=reverse_arity:i=794:si=on:rtra=on_0 on DTF2THF_3077 for (2968ds/794Mi)
% 27.75/8.32 % (3283)Instruction limit reached!
% 27.75/8.32 % (3283)------------------------------
% 27.75/8.32 % (3283)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.75/8.32 % (3283)Termination reason: Unknown
% 27.75/8.32 % (3283)Termination phase: Property scanning
% 27.75/8.32
% 27.75/8.32 % (3283)Memory used [KB]: 61150
% 27.75/8.32 % (3283)Time elapsed: 1.441 s
% 27.75/8.32 % (3283)Instructions burned: 5207 (million)
% 27.75/8.32 % (3283)------------------------------
% 27.75/8.32 % (3283)------------------------------
% 27.75/8.34 % (3309)dis+1002_1:1_au=on:avsq=on:avsqr=1,16:cond=on:sp=frequency:i=464:si=on:rtra=on_0 on DTF2THF_3077 for (2968ds/464Mi)
% 27.75/8.37 % (3307)Instruction limit reached!
% 27.75/8.37 % (3307)------------------------------
% 27.75/8.37 % (3307)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 27.75/8.37 % (3307)Termination reason: Unknown
% 27.75/8.37 % (3307)Termination phase: shuffling
% 27.75/8.37
% 27.75/8.37 % (3307)Memory used [KB]: 26609
% 27.75/8.37 % (3307)Time elapsed: 0.263 s
% 27.75/8.37 % (3307)Instructions burned: 771 (million)
% 27.75/8.37 % (3307)------------------------------
% 27.75/8.37 % (3307)------------------------------
% 27.75/8.38 % (3310)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_3077 for (2967ds/1376Mi)
% 28.46/8.43 % (3285)Instruction limit reached!
% 28.46/8.43 % (3285)------------------------------
% 28.46/8.43 % (3285)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.46/8.43 % (3285)Termination reason: Unknown
% 28.46/8.43 % (3285)Termination phase: Saturation
% 28.46/8.43
% 28.46/8.43 % (3285)Memory used [KB]: 36715
% 28.46/8.43 % (3285)Time elapsed: 1.446 s
% 28.46/8.43 % (3285)Instructions burned: 4351 (million)
% 28.46/8.43 % (3285)------------------------------
% 28.46/8.43 % (3285)------------------------------
% 28.46/8.45 % (3311)dis+10_1:1_etr=on:plsq=on:plsqc=1:plsqr=32,1:i=906:si=on:rtra=on_0 on DTF2THF_3077 for (2967ds/906Mi)
% 28.46/8.47 % (3280)Instruction limit reached!
% 28.46/8.47 % (3280)------------------------------
% 28.46/8.47 % (3280)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.46/8.47 % (3280)Termination reason: Unknown
% 28.46/8.47 % (3280)Termination phase: Saturation
% 28.46/8.47
% 28.46/8.47 % (3280)Memory used [KB]: 37867
% 28.46/8.47 % (3280)Time elapsed: 1.630 s
% 28.46/8.47 % (3280)Instructions burned: 4437 (million)
% 28.46/8.47 % (3280)------------------------------
% 28.46/8.47 % (3280)------------------------------
% 28.46/8.48 % (3312)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_3077 for (2966ds/434Mi)
% 28.46/8.50 % (3309)Instruction limit reached!
% 28.46/8.50 % (3309)------------------------------
% 28.46/8.50 % (3309)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.46/8.50 % (3309)Termination reason: Unknown
% 28.46/8.50 % (3309)Termination phase: shuffling
% 28.46/8.50
% 28.46/8.50 % (3309)Memory used [KB]: 20724
% 28.46/8.50 % (3309)Time elapsed: 0.161 s
% 28.46/8.50 % (3309)Instructions burned: 464 (million)
% 28.79/8.50 % (3309)------------------------------
% 28.79/8.50 % (3309)------------------------------
% 28.79/8.51 % (3313)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_3077 for (2966ds/865Mi)
% 28.79/8.58 % (3308)Instruction limit reached!
% 28.79/8.58 % (3308)------------------------------
% 28.79/8.58 % (3308)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 28.79/8.58 % (3308)Termination reason: Unknown
% 28.79/8.58 % (3308)Termination phase: shuffling
% 28.79/8.58
% 28.79/8.58 % (3308)Memory used [KB]: 27121
% 28.79/8.58 % (3308)Time elapsed: 0.285 s
% 28.79/8.58 % (3308)Instructions burned: 794 (million)
% 28.79/8.58 % (3308)------------------------------
% 28.79/8.58 % (3308)------------------------------
% 29.45/8.60 % (3314)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_3077 for (2965ds/593Mi)
% 29.45/8.66 % (3312)Instruction limit reached!
% 29.45/8.66 % (3312)------------------------------
% 29.45/8.66 % (3312)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.45/8.66 % (3312)Termination reason: Unknown
% 29.45/8.66 % (3312)Termination phase: shuffling
% 29.45/8.66
% 29.45/8.66 % (3312)Memory used [KB]: 20468
% 29.45/8.66 % (3312)Time elapsed: 0.178 s
% 29.45/8.66 % (3312)Instructions burned: 434 (million)
% 29.45/8.66 % (3312)------------------------------
% 29.45/8.66 % (3312)------------------------------
% 29.45/8.68 % (3315)dis+10_1:1_sd=2:sgt=20:slsq=on:ss=axioms:st=6.0:i=443:si=on:rtra=on_0 on DTF2THF_3077 for (2964ds/443Mi)
% 29.85/8.76 % (3311)Instruction limit reached!
% 29.85/8.76 % (3311)------------------------------
% 29.85/8.76 % (3311)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.85/8.76 % (3311)Termination reason: Unknown
% 29.85/8.76 % (3311)Termination phase: shuffling
% 29.85/8.76
% 29.85/8.76 % (3311)Memory used [KB]: 28400
% 29.85/8.76 % (3311)Time elapsed: 0.310 s
% 29.85/8.76 % (3311)Instructions burned: 908 (million)
% 29.85/8.76 % (3311)------------------------------
% 29.85/8.76 % (3311)------------------------------
% 29.85/8.77 % (3316)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_3077 for (2963ds/594Mi)
% 29.85/8.77 % (3313)Instruction limit reached!
% 29.85/8.77 % (3313)------------------------------
% 29.85/8.77 % (3313)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.85/8.77 % (3313)Termination reason: Unknown
% 29.85/8.77 % (3313)Termination phase: shuffling
% 29.85/8.77
% 29.85/8.77 % (3313)Memory used [KB]: 32494
% 29.85/8.77 % (3313)Time elapsed: 0.261 s
% 29.85/8.77 % (3313)Instructions burned: 867 (million)
% 29.85/8.77 % (3313)------------------------------
% 29.85/8.77 % (3313)------------------------------
% 29.85/8.78 % (3314)Instruction limit reached!
% 29.85/8.78 % (3314)------------------------------
% 29.85/8.78 % (3314)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 29.85/8.78 % (3314)Termination reason: Unknown
% 29.85/8.78 % (3314)Termination phase: shuffling
% 29.85/8.78
% 29.85/8.78 % (3314)Memory used [KB]: 24562
% 29.85/8.78 % (3314)Time elapsed: 0.180 s
% 29.85/8.78 % (3314)Instructions burned: 593 (million)
% 29.85/8.78 % (3314)------------------------------
% 29.85/8.78 % (3314)------------------------------
% 29.85/8.78 % (3317)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_3077 for (2963ds/327Mi)
% 29.85/8.79 % (3318)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_3077 for (2963ds/5411Mi)
% 30.84/8.85 % (3310)Instruction limit reached!
% 30.84/8.85 % (3310)------------------------------
% 30.84/8.85 % (3310)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.84/8.85 % (3310)Termination reason: Unknown
% 30.84/8.85 % (3310)Termination phase: shuffling
% 30.84/8.85
% 30.84/8.85 % (3310)Memory used [KB]: 39530
% 30.84/8.85 % (3310)Time elapsed: 0.466 s
% 30.84/8.85 % (3310)Instructions burned: 1378 (million)
% 30.84/8.85 % (3310)------------------------------
% 30.84/8.85 % (3310)------------------------------
% 30.84/8.86 % (3319)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_3077 for (2962ds/1762Mi)
% 30.84/8.88 % (3315)Instruction limit reached!
% 30.84/8.88 % (3315)------------------------------
% 30.84/8.88 % (3315)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 30.84/8.88 % (3315)Termination reason: Unknown
% 30.84/8.88 % (3315)Termination phase: shuffling
% 30.84/8.88
% 30.84/8.88 % (3315)Memory used [KB]: 20468
% 30.84/8.88 % (3315)Time elapsed: 0.200 s
% 30.84/8.88 % (3315)Instructions burned: 444 (million)
% 30.84/8.88 % (3315)------------------------------
% 30.84/8.88 % (3315)------------------------------
% 30.84/8.89 % (3320)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_3077 for (2962ds/96Mi)
% 31.75/8.90 % (3317)Instruction limit reached!
% 31.75/8.90 % (3317)------------------------------
% 31.75/8.90 % (3317)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.75/8.90 % (3317)Termination reason: Unknown
% 31.75/8.90 % (3317)Termination phase: shuffling
% 31.75/8.90
% 31.75/8.90 % (3317)Memory used [KB]: 18805
% 31.75/8.90 % (3317)Time elapsed: 0.117 s
% 31.75/8.90 % (3317)Instructions burned: 327 (million)
% 31.75/8.90 % (3317)------------------------------
% 31.75/8.90 % (3317)------------------------------
% 31.75/8.91 % (3321)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_3077 for (2962ds/503Mi)
% 31.75/8.93 % (3320)Instruction limit reached!
% 31.75/8.93 % (3320)------------------------------
% 31.75/8.93 % (3320)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.75/8.93 % (3320)Termination reason: Unknown
% 31.75/8.93 % (3320)Termination phase: shuffling
% 31.75/8.93
% 31.75/8.93 % (3320)Memory used [KB]: 15735
% 31.75/8.93 % (3320)Time elapsed: 0.038 s
% 31.75/8.93 % (3320)Instructions burned: 99 (million)
% 31.75/8.93 % (3320)------------------------------
% 31.75/8.93 % (3320)------------------------------
% 31.75/8.94 % (3322)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_3077 for (2962ds/90Mi)
% 31.75/8.97 % (3316)Instruction limit reached!
% 31.75/8.97 % (3316)------------------------------
% 31.75/8.97 % (3316)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.75/8.97 % (3316)Termination reason: Unknown
% 31.75/8.97 % (3316)Termination phase: shuffling
% 31.75/8.97
% 31.75/8.97 % (3316)Memory used [KB]: 24306
% 31.75/8.97 % (3316)Time elapsed: 0.206 s
% 31.75/8.97 % (3316)Instructions burned: 594 (million)
% 31.75/8.97 % (3316)------------------------------
% 31.75/8.97 % (3316)------------------------------
% 31.75/8.98 % (3322)Instruction limit reached!
% 31.75/8.98 % (3322)------------------------------
% 31.75/8.98 % (3322)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 31.75/8.98 % (3322)Termination reason: Unknown
% 31.75/8.98 % (3322)Termination phase: shuffling
% 31.75/8.98
% 31.75/8.98 % (3322)Memory used [KB]: 15607
% 31.75/8.98 % (3322)Time elapsed: 0.040 s
% 31.75/8.98 % (3322)Instructions burned: 92 (million)
% 31.75/8.98 % (3322)------------------------------
% 31.75/8.98 % (3322)------------------------------
% 31.75/8.99 % (3323)lrs+10_1:64_hud=5:plsq=on:plsqr=2011171,524288:i=593:si=on:rtra=on_0 on DTF2THF_3077 for (2961ds/593Mi)
% 31.75/8.99 % (3324)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_3077 for (2961ds/1512Mi)
% 32.92/9.09 % (3321)Instruction limit reached!
% 32.92/9.09 % (3321)------------------------------
% 32.92/9.09 % (3321)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 32.92/9.09 % (3321)Termination reason: Unknown
% 32.92/9.09 % (3321)Termination phase: shuffling
% 32.92/9.09
% 32.92/9.09 % (3321)Memory used [KB]: 21364
% 32.92/9.09 % (3321)Time elapsed: 0.177 s
% 32.92/9.09 % (3321)Instructions burned: 505 (million)
% 32.92/9.09 % (3321)------------------------------
% 32.92/9.09 % (3321)------------------------------
% 32.92/9.10 % (3325)lrs+1002_1:128_cbe=off:cnfonf=lazy_not_gen_be_off:ins=1:i=569:si=on:rtra=on_0 on DTF2THF_3077 for (2960ds/569Mi)
% 33.64/9.19 % (3323)Instruction limit reached!
% 33.64/9.19 % (3323)------------------------------
% 33.64/9.19 % (3323)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 33.64/9.19 % (3323)Termination reason: Unknown
% 33.64/9.19 % (3323)Termination phase: shuffling
% 33.64/9.19
% 33.64/9.19 % (3323)Memory used [KB]: 22259
% 33.64/9.19 % (3323)Time elapsed: 0.207 s
% 33.64/9.19 % (3323)Instructions burned: 593 (million)
% 33.64/9.19 % (3323)------------------------------
% 33.64/9.19 % (3323)------------------------------
% 34.14/9.20 % (3326)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_3077 for (2959ds/501Mi)
% 34.14/9.27 % (3325)Instruction limit reached!
% 34.14/9.27 % (3325)------------------------------
% 34.14/9.27 % (3325)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.14/9.27 % (3325)Termination reason: Unknown
% 34.14/9.27 % (3325)Termination phase: shuffling
% 34.14/9.27
% 34.14/9.27 % (3325)Memory used [KB]: 24562
% 34.14/9.27 % (3325)Time elapsed: 0.174 s
% 34.14/9.27 % (3325)Instructions burned: 571 (million)
% 34.14/9.27 % (3325)------------------------------
% 34.14/9.27 % (3325)------------------------------
% 34.14/9.28 % (3327)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_3077 for (2958ds/562Mi)
% 34.82/9.38 % (3326)Instruction limit reached!
% 34.82/9.38 % (3326)------------------------------
% 34.82/9.38 % (3326)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.82/9.38 % (3326)Termination reason: Unknown
% 34.82/9.38 % (3326)Termination phase: shuffling
% 34.82/9.38
% 34.82/9.38 % (3326)Memory used [KB]: 21108
% 34.82/9.38 % (3326)Time elapsed: 0.176 s
% 34.82/9.38 % (3326)Instructions burned: 503 (million)
% 34.82/9.38 % (3326)------------------------------
% 34.82/9.38 % (3326)------------------------------
% 34.82/9.38 % (3305)Instruction limit reached!
% 34.82/9.38 % (3305)------------------------------
% 34.82/9.38 % (3305)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.82/9.38 % (3305)Termination reason: Unknown
% 34.82/9.38 % (3305)Termination phase: Property scanning
% 34.82/9.38
% 34.82/9.38 % (3305)Memory used [KB]: 61534
% 34.82/9.38 % (3305)Time elapsed: 1.559 s
% 34.82/9.38 % (3305)Instructions burned: 4714 (million)
% 34.82/9.38 % (3305)------------------------------
% 34.82/9.38 % (3305)------------------------------
% 34.82/9.38 % (3319)Instruction limit reached!
% 34.82/9.38 % (3319)------------------------------
% 34.82/9.38 % (3319)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 34.82/9.38 % (3319)Termination reason: Unknown
% 34.82/9.38 % (3319)Termination phase: Property scanning
% 34.82/9.38
% 34.82/9.38 % (3319)Memory used [KB]: 43624
% 34.82/9.38 % (3319)Time elapsed: 0.523 s
% 34.82/9.38 % (3319)Instructions burned: 1765 (million)
% 34.82/9.38 % (3319)------------------------------
% 34.82/9.38 % (3319)------------------------------
% 34.82/9.39 % (3328)lrs+10_1:1_au=on:cnfonf=off:fde=unused:kws=precedence:tgt=full:i=406:si=on:rtra=on_0 on DTF2THF_3077 for (2957ds/406Mi)
% 35.52/9.40 % (3329)dis+1004_1:128_au=on:cbe=off:e2e=on:sac=on:i=582:si=on:rtra=on_0 on DTF2THF_3077 for (2957ds/582Mi)
% 35.52/9.41 % (3330)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_3077 for (2957ds/456Mi)
% 35.52/9.46 % (3327)Instruction limit reached!
% 35.52/9.46 % (3327)------------------------------
% 35.52/9.46 % (3327)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.52/9.46 % (3327)Termination reason: Unknown
% 35.52/9.46 % (3327)Termination phase: shuffling
% 35.52/9.46
% 35.52/9.46 % (3327)Memory used [KB]: 24434
% 35.52/9.46 % (3327)Time elapsed: 0.173 s
% 35.52/9.46 % (3327)Instructions burned: 563 (million)
% 35.52/9.46 % (3327)------------------------------
% 35.52/9.46 % (3327)------------------------------
% 35.52/9.47 % (3331)lrs+1010_1:1_au=on:cbe=off:cnfonf=conj_eager:tgt=ground:i=393:si=on:rtra=on_0 on DTF2THF_3077 for (2956ds/393Mi)
% 35.99/9.53 % (3328)Instruction limit reached!
% 35.99/9.53 % (3328)------------------------------
% 35.99/9.53 % (3328)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.99/9.53 % (3328)Termination reason: Unknown
% 35.99/9.53 % (3328)Termination phase: shuffling
% 35.99/9.53
% 35.99/9.53 % (3328)Memory used [KB]: 19957
% 35.99/9.53 % (3328)Time elapsed: 0.143 s
% 35.99/9.53 % (3328)Instructions burned: 409 (million)
% 35.99/9.53 % (3328)------------------------------
% 35.99/9.53 % (3328)------------------------------
% 35.99/9.55 % (3332)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_3077 for (2955ds/1335Mi)
% 35.99/9.56 % (3324)Instruction limit reached!
% 35.99/9.56 % (3324)------------------------------
% 35.99/9.56 % (3324)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.99/9.56 % (3324)Termination reason: Unknown
% 35.99/9.56 % (3324)Termination phase: Property scanning
% 35.99/9.56
% 35.99/9.56 % (3324)Memory used [KB]: 31086
% 35.99/9.56 % (3324)Time elapsed: 0.558 s
% 35.99/9.56 % (3324)Instructions burned: 1512 (million)
% 35.99/9.56 % (3324)------------------------------
% 35.99/9.56 % (3324)------------------------------
% 35.99/9.57 % (3333)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_3077 for (2955ds/3373Mi)
% 35.99/9.59 % (3331)Instruction limit reached!
% 35.99/9.59 % (3331)------------------------------
% 35.99/9.59 % (3331)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 35.99/9.59 % (3331)Termination reason: Unknown
% 35.99/9.59 % (3331)Termination phase: shuffling
% 35.99/9.59
% 35.99/9.59 % (3331)Memory used [KB]: 21875
% 35.99/9.59 % (3331)Time elapsed: 0.123 s
% 35.99/9.59 % (3331)Instructions burned: 394 (million)
% 35.99/9.59 % (3331)------------------------------
% 35.99/9.59 % (3331)------------------------------
% 36.56/9.60 % (3329)Instruction limit reached!
% 36.56/9.60 % (3329)------------------------------
% 36.56/9.60 % (3329)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.56/9.60 % (3329)Termination reason: Unknown
% 36.56/9.60 % (3329)Termination phase: shuffling
% 36.56/9.60
% 36.56/9.60 % (3329)Memory used [KB]: 22259
% 36.56/9.60 % (3329)Time elapsed: 0.204 s
% 36.56/9.60 % (3329)Instructions burned: 582 (million)
% 36.56/9.60 % (3329)------------------------------
% 36.56/9.60 % (3329)------------------------------
% 36.56/9.60 % (3334)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_3077 for (2955ds/474Mi)
% 36.56/9.61 % (3335)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_3077 for (2955ds/157Mi)
% 36.56/9.66 % (3330)Instruction limit reached!
% 36.56/9.66 % (3330)------------------------------
% 36.56/9.66 % (3330)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.56/9.66 % (3330)Termination reason: Unknown
% 36.56/9.66 % (3330)Termination phase: shuffling
% 36.56/9.66
% 36.56/9.66 % (3330)Memory used [KB]: 20596
% 36.56/9.66 % (3330)Time elapsed: 0.255 s
% 36.56/9.66 % (3330)Instructions burned: 456 (million)
% 36.56/9.66 % (3330)------------------------------
% 36.56/9.66 % (3330)------------------------------
% 36.56/9.67 % (3335)Instruction limit reached!
% 36.56/9.67 % (3335)------------------------------
% 36.56/9.67 % (3335)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 36.56/9.67 % (3335)Termination reason: Unknown
% 36.56/9.67 % (3335)Termination phase: shuffling
% 36.56/9.67
% 36.56/9.67 % (3335)Memory used [KB]: 16630
% 36.56/9.67 % (3335)Time elapsed: 0.055 s
% 36.56/9.67 % (3335)Instructions burned: 159 (million)
% 36.56/9.67 % (3335)------------------------------
% 36.56/9.67 % (3335)------------------------------
% 36.56/9.68 % (3337)dis+10_1:1024_kws=precedence:prag=on:rawr=on:sos=all:sp=unary_first:i=104:si=on:rtra=on_0 on DTF2THF_3077 for (2954ds/104Mi)
% 36.56/9.68 % (3336)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_3077 for (2954ds/590Mi)
% 37.31/9.72 % (3337)Instruction limit reached!
% 37.31/9.72 % (3337)------------------------------
% 37.31/9.72 % (3337)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.31/9.72 % (3337)Termination reason: Unknown
% 37.31/9.72 % (3337)Termination phase: shuffling
% 37.31/9.72
% 37.31/9.72 % (3337)Memory used [KB]: 15863
% 37.31/9.72 % (3337)Time elapsed: 0.045 s
% 37.31/9.72 % (3337)Instructions burned: 105 (million)
% 37.31/9.72 % (3337)------------------------------
% 37.31/9.72 % (3337)------------------------------
% 37.31/9.73 % (3338)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=414:si=on:rtra=on_0 on DTF2THF_3077 for (2954ds/414Mi)
% 37.31/9.75 % (3334)Instruction limit reached!
% 37.31/9.75 % (3334)------------------------------
% 37.31/9.75 % (3334)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.31/9.75 % (3334)Termination reason: Unknown
% 37.31/9.75 % (3334)Termination phase: shuffling
% 37.31/9.75
% 37.31/9.75 % (3334)Memory used [KB]: 23155
% 37.31/9.75 % (3334)Time elapsed: 0.146 s
% 37.31/9.75 % (3334)Instructions burned: 477 (million)
% 37.31/9.75 % (3334)------------------------------
% 37.31/9.75 % (3334)------------------------------
% 37.31/9.76 % (3339)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_3077 for (2953ds/715Mi)
% 37.72/9.86 % (3338)Instruction limit reached!
% 37.72/9.86 % (3338)------------------------------
% 37.72/9.86 % (3338)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 37.72/9.86 % (3338)Termination reason: Unknown
% 37.72/9.86 % (3338)Termination phase: shuffling
% 37.72/9.86
% 37.72/9.86 % (3338)Memory used [KB]: 22387
% 37.72/9.86 % (3338)Time elapsed: 0.128 s
% 37.72/9.86 % (3338)Instructions burned: 416 (million)
% 37.72/9.86 % (3338)------------------------------
% 37.72/9.86 % (3338)------------------------------
% 37.72/9.87 % (3340)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_3077 for (2952ds/961Mi)
% 38.06/9.94 % (3332)Instruction limit reached!
% 38.06/9.94 % (3332)------------------------------
% 38.06/9.94 % (3332)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.06/9.94 % (3332)Termination reason: Unknown
% 38.06/9.94 % (3332)Termination phase: shuffling
% 38.06/9.94
% 38.06/9.94 % (3332)Memory used [KB]: 38890
% 38.06/9.94 % (3332)Time elapsed: 0.400 s
% 38.06/9.94 % (3332)Instructions burned: 1335 (million)
% 38.06/9.94 % (3332)------------------------------
% 38.06/9.94 % (3332)------------------------------
% 38.06/9.96 % (3341)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_3077 for (2951ds/1213Mi)
% 38.06/10.01 % (3339)Instruction limit reached!
% 38.06/10.01 % (3339)------------------------------
% 38.06/10.01 % (3339)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 38.06/10.01 % (3339)Termination reason: Unknown
% 38.06/10.01 % (3339)Termination phase: shuffling
% 38.06/10.01
% 38.06/10.01 % (3339)Memory used [KB]: 26097
% 38.06/10.01 % (3339)Time elapsed: 0.248 s
% 38.06/10.01 % (3339)Instructions burned: 717 (million)
% 38.06/10.01 % (3339)------------------------------
% 38.06/10.01 % (3339)------------------------------
% 39.94/10.02 % (3336)Instruction limit reached!
% 39.94/10.02 % (3336)------------------------------
% 39.94/10.02 % (3336)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 39.94/10.02 % (3336)Termination reason: Unknown
% 39.94/10.02 % (3336)Termination phase: shuffling
% 39.94/10.02
% 39.94/10.02 % (3336)Memory used [KB]: 22131
% 39.94/10.02 % (3336)Time elapsed: 0.336 s
% 39.94/10.02 % (3336)Instructions burned: 590 (million)
% 39.94/10.02 % (3336)------------------------------
% 39.94/10.02 % (3336)------------------------------
% 39.94/10.02 % (3342)lrs+1002_1:12_bd=off:hud=1:ntd=on:piset=equals:prag=on:i=1979:si=on:rtra=on_0 on DTF2THF_3077 for (2951ds/1979Mi)
% 39.94/10.04 % (3343)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_3077 for (2951ds/481Mi)
% 40.86/10.20 % (3340)Instruction limit reached!
% 40.86/10.20 % (3340)------------------------------
% 40.86/10.20 % (3340)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.86/10.20 % (3340)Termination reason: Unknown
% 40.86/10.20 % (3340)Termination phase: shuffling
% 40.86/10.20
% 40.86/10.20 % (3340)Memory used [KB]: 29167
% 40.86/10.20 % (3340)Time elapsed: 0.328 s
% 40.86/10.20 % (3340)Instructions burned: 961 (million)
% 40.86/10.20 % (3340)------------------------------
% 40.86/10.20 % (3340)------------------------------
% 40.86/10.21 % (3344)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_3077 for (2949ds/783Mi)
% 40.86/10.26 % (3343)Instruction limit reached!
% 40.86/10.26 % (3343)------------------------------
% 40.86/10.26 % (3343)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.86/10.26 % (3343)Termination reason: Unknown
% 40.86/10.26 % (3343)Termination phase: shuffling
% 40.86/10.26
% 40.86/10.26 % (3343)Memory used [KB]: 23155
% 40.86/10.26 % (3343)Time elapsed: 0.215 s
% 40.86/10.26 % (3343)Instructions burned: 482 (million)
% 40.86/10.26 % (3343)------------------------------
% 40.86/10.26 % (3343)------------------------------
% 40.86/10.27 % (3345)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_3077 for (2948ds/432Mi)
% 41.52/10.36 % (3341)Instruction limit reached!
% 41.52/10.36 % (3341)------------------------------
% 41.52/10.36 % (3341)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.52/10.36 % (3341)Termination reason: Unknown
% 41.52/10.36 % (3341)Termination phase: Property scanning
% 41.52/10.36
% 41.52/10.36 % (3341)Memory used [KB]: 29551
% 41.52/10.36 % (3341)Time elapsed: 0.400 s
% 41.52/10.36 % (3341)Instructions burned: 1214 (million)
% 41.52/10.36 % (3341)------------------------------
% 41.52/10.36 % (3341)------------------------------
% 41.52/10.37 % (3346)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_3077 for (2947ds/1670Mi)
% 41.89/10.42 % (3345)Instruction limit reached!
% 41.89/10.42 % (3345)------------------------------
% 41.89/10.42 % (3345)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.89/10.42 % (3345)Termination reason: Unknown
% 41.89/10.42 % (3345)Termination phase: shuffling
% 41.89/10.42
% 41.89/10.42 % (3345)Memory used [KB]: 19957
% 41.89/10.42 % (3345)Time elapsed: 0.149 s
% 41.89/10.42 % (3345)Instructions burned: 432 (million)
% 41.89/10.42 % (3345)------------------------------
% 41.89/10.42 % (3345)------------------------------
% 41.89/10.43 % (3347)dis+2_1:1_au=on:hfaw=0:hfsq=on:sp=weighted_frequency:i=90:si=on:rtra=on_0 on DTF2THF_3077 for (2947ds/90Mi)
% 41.89/10.47 % (3318)Instruction limit reached!
% 41.89/10.47 % (3318)------------------------------
% 41.89/10.47 % (3318)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.89/10.47 % (3318)Termination reason: Unknown
% 41.89/10.47 % (3318)Termination phase: Saturation
% 41.89/10.47
% 41.89/10.47 % (3318)Memory used [KB]: 37611
% 41.89/10.47 % (3318)Time elapsed: 1.684 s
% 41.89/10.47 % (3318)Instructions burned: 5412 (million)
% 41.89/10.47 % (3318)------------------------------
% 41.89/10.47 % (3318)------------------------------
% 41.89/10.48 % (3344)Instruction limit reached!
% 41.89/10.48 % (3344)------------------------------
% 41.89/10.48 % (3344)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.89/10.48 % (3344)Termination reason: Unknown
% 41.89/10.48 % (3344)Termination phase: shuffling
% 41.89/10.48
% 41.89/10.48 % (3344)Memory used [KB]: 26865
% 41.89/10.48 % (3344)Time elapsed: 0.270 s
% 41.89/10.48 % (3344)Instructions burned: 785 (million)
% 41.89/10.48 % (3344)------------------------------
% 41.89/10.48 % (3344)------------------------------
% 41.89/10.48 % (3348)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_3077 for (2946ds/346Mi)
% 41.89/10.49 % (3347)Instruction limit reached!
% 41.89/10.49 % (3347)------------------------------
% 41.89/10.49 % (3347)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 41.89/10.49 % (3347)Termination reason: Unknown
% 41.89/10.49 % (3347)Termination phase: shuffling
% 41.89/10.49
% 41.89/10.49 % (3347)Memory used [KB]: 15479
% 41.89/10.49 % (3347)Time elapsed: 0.058 s
% 41.89/10.49 % (3347)Instructions burned: 91 (million)
% 41.89/10.49 % (3347)------------------------------
% 41.89/10.49 % (3347)------------------------------
% 41.89/10.49 % (3349)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_3077 for (2946ds/401Mi)
% 42.69/10.51 % (3350)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=6454:si=on:rtra=on_0 on DTF2THF_3077 for (2946ds/6454Mi)
% 42.69/10.59 % (3348)Instruction limit reached!
% 42.69/10.59 % (3348)------------------------------
% 42.69/10.59 % (3348)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.69/10.59 % (3348)Termination reason: Unknown
% 42.69/10.59 % (3348)Termination phase: shuffling
% 42.69/10.59
% 42.69/10.59 % (3348)Memory used [KB]: 21364
% 42.69/10.59 % (3348)Time elapsed: 0.109 s
% 42.69/10.59 % (3348)Instructions burned: 347 (million)
% 42.69/10.59 % (3348)------------------------------
% 42.69/10.59 % (3348)------------------------------
% 42.69/10.59 % (3306)Instruction limit reached!
% 42.69/10.59 % (3306)------------------------------
% 42.69/10.59 % (3306)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 42.69/10.59 % (3306)Termination reason: Unknown
% 42.69/10.59 % (3306)Termination phase: Saturation
% 42.69/10.59
% 42.69/10.59 % (3306)Memory used [KB]: 63069
% 42.69/10.59 % (3306)Time elapsed: 2.665 s
% 42.69/10.59 % (3306)Instructions burned: 8250 (million)
% 42.69/10.59 % (3306)------------------------------
% 42.69/10.59 % (3306)------------------------------
% 40.33/10.60 % (3351)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_3077 for (2945ds/283Mi)
% 40.33/10.61 % (3352)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_3077 for (2945ds/865Mi)
% 40.33/10.62 % (3349)Instruction limit reached!
% 40.33/10.62 % (3349)------------------------------
% 40.33/10.62 % (3349)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.33/10.62 % (3349)Termination reason: Unknown
% 40.33/10.62 % (3349)Termination phase: shuffling
% 40.33/10.62
% 40.33/10.62 % (3349)Memory used [KB]: 22003
% 40.33/10.62 % (3349)Time elapsed: 0.124 s
% 40.33/10.62 % (3349)Instructions burned: 401 (million)
% 40.33/10.62 % (3349)------------------------------
% 40.33/10.62 % (3349)------------------------------
% 40.33/10.63 % (3353)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_3077 for (2945ds/4528Mi)
% 40.33/10.66 % (3342)Instruction limit reached!
% 40.33/10.66 % (3342)------------------------------
% 40.33/10.66 % (3342)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 40.33/10.66 % (3342)Termination reason: Unknown
% 40.33/10.66 % (3342)Termination phase: Preprocessing 3
% 40.33/10.66
% 40.33/10.66 % (3342)Memory used [KB]: 37099
% 40.33/10.66 % (3342)Time elapsed: 0.637 s
% 40.33/10.66 % (3342)Instructions burned: 1981 (million)
% 40.33/10.66 % (3342)------------------------------
% 40.33/10.66 % (3342)------------------------------
% 40.33/10.67 % (3354)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_3077 for (2944ds/1001Mi)
% 43.91/10.71 % (3351)Instruction limit reached!
% 43.91/10.71 % (3351)------------------------------
% 43.91/10.71 % (3351)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.91/10.71 % (3351)Termination reason: Unknown
% 43.91/10.71 % (3351)Termination phase: shuffling
% 43.91/10.71
% 43.91/10.71 % (3351)Memory used [KB]: 18293
% 43.91/10.71 % (3351)Time elapsed: 0.102 s
% 43.91/10.71 % (3351)Instructions burned: 284 (million)
% 43.91/10.71 % (3351)------------------------------
% 43.91/10.71 % (3351)------------------------------
% 43.91/10.72 % (3355)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=102:si=on:rtra=on_0 on DTF2THF_3077 for (2944ds/102Mi)
% 43.91/10.74 % (3333)Instruction limit reached!
% 43.91/10.74 % (3333)------------------------------
% 43.91/10.74 % (3333)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.91/10.74 % (3333)Termination reason: Unknown
% 43.91/10.74 % (3333)Termination phase: Function definition elimination
% 43.91/10.74
% 43.91/10.74 % (3333)Memory used [KB]: 48613
% 43.91/10.74 % (3333)Time elapsed: 1.160 s
% 43.91/10.74 % (3333)Instructions burned: 3373 (million)
% 43.91/10.74 % (3333)------------------------------
% 43.91/10.74 % (3333)------------------------------
% 43.91/10.76 % (3355)Instruction limit reached!
% 43.91/10.76 % (3355)------------------------------
% 43.91/10.76 % (3355)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 43.91/10.76 % (3355)Termination reason: Unknown
% 43.91/10.76 % (3355)Termination phase: shuffling
% 43.91/10.76
% 43.91/10.76 % (3355)Memory used [KB]: 15735
% 43.91/10.76 % (3355)Time elapsed: 0.039 s
% 43.91/10.76 % (3355)Instructions burned: 103 (million)
% 43.91/10.76 % (3355)------------------------------
% 43.91/10.76 % (3355)------------------------------
% 43.91/10.76 % (3356)ott+1010_3:1_nm=20:s2a=on:s2at=3.0:sp=const_frequency:i=952:si=on:rtra=on_0 on DTF2THF_3077 for (2943ds/952Mi)
% 43.91/10.77 % (3357)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_3077 for (2943ds/578Mi)
% 44.13/10.87 % (3346)Instruction limit reached!
% 44.13/10.87 % (3346)------------------------------
% 44.13/10.87 % (3346)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.13/10.87 % (3346)Termination reason: Unknown
% 44.13/10.87 % (3346)Termination phase: shuffling
% 44.13/10.87
% 44.13/10.87 % (3346)Memory used [KB]: 43368
% 44.13/10.87 % (3346)Time elapsed: 0.496 s
% 44.13/10.87 % (3346)Instructions burned: 1671 (million)
% 44.13/10.87 % (3346)------------------------------
% 44.13/10.87 % (3346)------------------------------
% 44.13/10.88 % (3358)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_3077 for (2942ds/643Mi)
% 44.63/10.90 % (3352)Instruction limit reached!
% 44.63/10.90 % (3352)------------------------------
% 44.63/10.90 % (3352)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.63/10.90 % (3352)Termination reason: Unknown
% 44.63/10.90 % (3352)Termination phase: shuffling
% 44.63/10.90
% 44.63/10.90 % (3352)Memory used [KB]: 28144
% 44.63/10.90 % (3352)Time elapsed: 0.298 s
% 44.63/10.90 % (3352)Instructions burned: 867 (million)
% 44.63/10.90 % (3352)------------------------------
% 44.63/10.90 % (3352)------------------------------
% 44.63/10.92 % (3359)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_3077 for (2942ds/865Mi)
% 44.63/10.94 % (3357)Instruction limit reached!
% 44.63/10.94 % (3357)------------------------------
% 44.63/10.94 % (3357)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 44.63/10.94 % (3357)Termination reason: Unknown
% 44.63/10.94 % (3357)Termination phase: shuffling
% 44.63/10.94
% 44.63/10.94 % (3357)Memory used [KB]: 24562
% 44.63/10.94 % (3357)Time elapsed: 0.177 s
% 44.63/10.94 % (3357)Instructions burned: 578 (million)
% 44.63/10.94 % (3357)------------------------------
% 44.63/10.94 % (3357)------------------------------
% 44.63/10.96 % (3360)lrs+10_1:1_au=on:erd=off:s2a=on:s2at=3.0:sos=all:i=1174:si=on:rtra=on_0 on DTF2THF_3077 for (2941ds/1174Mi)
% 46.96/11.08 % (3356)Instruction limit reached!
% 46.96/11.08 % (3356)------------------------------
% 46.96/11.08 % (3356)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.96/11.08 % (3356)Termination reason: Unknown
% 46.96/11.08 % (3356)Termination phase: shuffling
% 46.96/11.08
% 46.96/11.08 % (3356)Memory used [KB]: 29040
% 46.96/11.08 % (3356)Time elapsed: 0.325 s
% 46.96/11.08 % (3356)Instructions burned: 954 (million)
% 46.96/11.08 % (3356)------------------------------
% 46.96/11.08 % (3356)------------------------------
% 46.96/11.09 % (3354)Instruction limit reached!
% 46.96/11.09 % (3354)------------------------------
% 46.96/11.09 % (3354)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.96/11.09 % (3354)Termination reason: Unknown
% 46.96/11.09 % (3354)Termination phase: Property scanning
% 46.96/11.09
% 46.96/11.09 % (3354)Memory used [KB]: 29551
% 46.96/11.09 % (3354)Time elapsed: 0.426 s
% 46.96/11.09 % (3354)Instructions burned: 1003 (million)
% 46.96/11.09 % (3354)------------------------------
% 46.96/11.09 % (3354)------------------------------
% 46.96/11.09 % (3361)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_3077 for (2940ds/644Mi)
% 46.96/11.11 % (3358)Instruction limit reached!
% 46.96/11.11 % (3358)------------------------------
% 46.96/11.11 % (3358)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 46.96/11.11 % (3358)Termination reason: Unknown
% 46.96/11.11 % (3358)Termination phase: shuffling
% 46.96/11.11
% 46.96/11.11 % (3358)Memory used [KB]: 25458
% 46.96/11.11 % (3358)Time elapsed: 0.229 s
% 46.96/11.11 % (3358)Instructions burned: 644 (million)
% 46.96/11.11 % (3358)------------------------------
% 46.96/11.11 % (3358)------------------------------
% 46.96/11.12 % (3362)lrs+10_1:1_alpa=true:au=on:cs=on:tgt=ground:i=1201:si=on:rtra=on_0 on DTF2THF_3077 for (2940ds/1201Mi)
% 46.96/11.12 % (3363)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_3077 for (2940ds/1046Mi)
% 47.93/11.21 % (3359)Instruction limit reached!
% 47.93/11.21 % (3359)------------------------------
% 47.93/11.21 % (3359)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 47.93/11.21 % (3359)Termination reason: Unknown
% 47.93/11.21 % (3359)Termination phase: shuffling
% 47.93/11.21
% 47.93/11.21 % (3359)Memory used [KB]: 27888
% 47.93/11.21 % (3359)Time elapsed: 0.296 s
% 47.93/11.21 % (3359)Instructions burned: 866 (million)
% 47.93/11.21 % (3359)------------------------------
% 47.93/11.21 % (3359)------------------------------
% 47.93/11.23 % (3364)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_3077 for (2939ds/222Mi)
% 48.27/11.30 % (3364)Instruction limit reached!
% 48.27/11.30 % (3364)------------------------------
% 48.27/11.30 % (3364)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.27/11.30 % (3364)Termination reason: Unknown
% 48.27/11.30 % (3364)Termination phase: shuffling
% 48.27/11.30
% 48.27/11.30 % (3364)Memory used [KB]: 17526
% 48.27/11.30 % (3364)Time elapsed: 0.073 s
% 48.27/11.30 % (3364)Instructions burned: 224 (million)
% 48.27/11.30 % (3364)------------------------------
% 48.27/11.30 % (3364)------------------------------
% 48.27/11.31 % (3365)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_3077 for (2938ds/68Mi)
% 48.27/11.31 % (3361)Instruction limit reached!
% 48.27/11.31 % (3361)------------------------------
% 48.27/11.31 % (3361)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.27/11.31 % (3361)Termination reason: Unknown
% 48.27/11.31 % (3361)Termination phase: shuffling
% 48.27/11.31
% 48.27/11.31 % (3361)Memory used [KB]: 24946
% 48.27/11.31 % (3361)Time elapsed: 0.221 s
% 48.27/11.31 % (3361)Instructions burned: 645 (million)
% 48.27/11.31 % (3361)------------------------------
% 48.27/11.31 % (3361)------------------------------
% 48.27/11.33 % (3366)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_3077 for (2938ds/454Mi)
% 48.27/11.34 % (3365)Instruction limit reached!
% 48.27/11.34 % (3365)------------------------------
% 48.27/11.34 % (3365)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.27/11.34 % (3365)Termination reason: Unknown
% 48.27/11.34 % (3365)Termination phase: shuffling
% 48.27/11.34
% 48.27/11.34 % (3365)Memory used [KB]: 15223
% 48.27/11.34 % (3365)Time elapsed: 0.032 s
% 48.27/11.34 % (3365)Instructions burned: 68 (million)
% 48.27/11.34 % (3365)------------------------------
% 48.27/11.34 % (3365)------------------------------
% 48.27/11.34 % (3360)Instruction limit reached!
% 48.27/11.34 % (3360)------------------------------
% 48.27/11.34 % (3360)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.27/11.34 % (3360)Termination reason: Unknown
% 48.27/11.34 % (3360)Termination phase: Property scanning
% 48.27/11.34
% 48.27/11.34 % (3360)Memory used [KB]: 29551
% 48.27/11.34 % (3360)Time elapsed: 0.386 s
% 48.27/11.34 % (3360)Instructions burned: 1175 (million)
% 48.27/11.34 % (3360)------------------------------
% 48.27/11.34 % (3360)------------------------------
% 48.27/11.35 % (3367)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_3077 for (2937ds/360Mi)
% 48.27/11.35 % (3368)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_3077 for (2937ds/157Mi)
% 48.89/11.41 % (3368)Instruction limit reached!
% 48.89/11.41 % (3368)------------------------------
% 48.89/11.41 % (3368)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.89/11.41 % (3368)Termination reason: Unknown
% 48.89/11.41 % (3368)Termination phase: shuffling
% 48.89/11.41
% 48.89/11.41 % (3368)Memory used [KB]: 16630
% 48.89/11.41 % (3368)Time elapsed: 0.055 s
% 48.89/11.41 % (3368)Instructions burned: 160 (million)
% 48.89/11.41 % (3368)------------------------------
% 48.89/11.41 % (3368)------------------------------
% 48.89/11.42 % (3369)ott+1004_1:24_anc=none:cbe=off:cnfonf=lazy_not_gen:fe=off:piset=not:prag=on:rp=on:i=5061:si=on:rtra=on_0 on DTF2THF_3077 for (2937ds/5061Mi)
% 48.89/11.46 % (3367)Instruction limit reached!
% 48.89/11.46 % (3367)------------------------------
% 48.89/11.46 % (3367)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.89/11.46 % (3367)Termination reason: Unknown
% 48.89/11.46 % (3367)Termination phase: shuffling
% 48.89/11.46
% 48.89/11.46 % (3367)Memory used [KB]: 21492
% 48.89/11.46 % (3367)Time elapsed: 0.112 s
% 48.89/11.46 % (3367)Instructions burned: 360 (million)
% 48.89/11.46 % (3367)------------------------------
% 48.89/11.46 % (3367)------------------------------
% 48.89/11.46 % (3366)Instruction limit reached!
% 48.89/11.46 % (3366)------------------------------
% 48.89/11.46 % (3366)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 48.89/11.46 % (3366)Termination reason: Unknown
% 48.89/11.46 % (3366)Termination phase: shuffling
% 48.89/11.46
% 48.89/11.46 % (3366)Memory used [KB]: 22771
% 48.89/11.46 % (3366)Time elapsed: 0.138 s
% 48.89/11.46 % (3366)Instructions burned: 454 (million)
% 48.89/11.46 % (3366)------------------------------
% 48.89/11.46 % (3366)------------------------------
% 48.89/11.47 % (3370)lrs+1002_1:8_av=off:fd=off:hud=5:tnu=1:i=2401:si=on:rtra=on_0 on DTF2THF_3077 for (2936ds/2401Mi)
% 48.89/11.48 % (3371)dis+10_1:1_av=off:e2e=on:nm=4:s2a=on:i=5798:si=on:rtra=on_0 on DTF2THF_3077 for (2936ds/5798Mi)
% 49.42/11.53 % (3363)Instruction limit reached!
% 49.42/11.53 % (3363)------------------------------
% 49.42/11.53 % (3363)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.42/11.53 % (3363)Termination reason: Unknown
% 49.42/11.53 % (3363)Termination phase: shuffling
% 49.42/11.53
% 49.42/11.53 % (3363)Memory used [KB]: 34924
% 49.42/11.53 % (3363)Time elapsed: 0.414 s
% 49.42/11.53 % (3363)Instructions burned: 1047 (million)
% 49.42/11.53 % (3363)------------------------------
% 49.42/11.53 % (3363)------------------------------
% 49.42/11.55 % (3372)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_3077 for (2935ds/346Mi)
% 49.42/11.59 % (3362)Instruction limit reached!
% 49.42/11.59 % (3362)------------------------------
% 49.42/11.59 % (3362)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 49.42/11.59 % (3362)Termination reason: Unknown
% 49.42/11.59 % (3362)Termination phase: Property scanning
% 49.42/11.59
% 49.42/11.59 % (3362)Memory used [KB]: 29551
% 49.42/11.59 % (3362)Time elapsed: 0.481 s
% 49.42/11.59 % (3362)Instructions burned: 1201 (million)
% 49.42/11.59 % (3362)------------------------------
% 49.42/11.59 % (3362)------------------------------
% 50.04/11.61 % (3373)lrs+1002_1:1_av=off:bd=off:bet=on:e2e=on:fde=unused:fe=off:plsq=on:prag=on:sos=on:i=1321:si=on:rtra=on_0 on DTF2THF_3077 for (2935ds/1321Mi)
% 50.04/11.66 % (3372)Instruction limit reached!
% 50.04/11.66 % (3372)------------------------------
% 50.04/11.66 % (3372)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 50.04/11.66 % (3372)Termination reason: Unknown
% 50.04/11.66 % (3372)Termination phase: shuffling
% 50.04/11.66
% 50.04/11.66 % (3372)Memory used [KB]: 21364
% 50.04/11.66 % (3372)Time elapsed: 0.108 s
% 50.04/11.66 % (3372)Instructions burned: 348 (million)
% 50.04/11.66 % (3372)------------------------------
% 50.04/11.66 % (3372)------------------------------
% 50.04/11.67 % (3374)lrs+10_1:1_cnfonf=lazy_gen:kws=inv_precedence:nwc=10.0:i=1312:si=on:rtra=on_0 on DTF2THF_3077 for (2934ds/1312Mi)
% 53.98/12.05 % (3353)Instruction limit reached!
% 53.98/12.05 % (3353)------------------------------
% 53.98/12.05 % (3353)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.98/12.05 % (3353)Termination reason: Unknown
% 53.98/12.05 % (3353)Termination phase: Saturation
% 53.98/12.05
% 53.98/12.05 % (3353)Memory used [KB]: 34541
% 53.98/12.05 % (3353)Time elapsed: 1.424 s
% 53.98/12.05 % (3353)Instructions burned: 4528 (million)
% 53.98/12.05 % (3353)------------------------------
% 53.98/12.05 % (3353)------------------------------
% 53.98/12.07 % (3375)ott+2_8:1_add=large:c=on:cnfonf=off:e2e=on:er=filter:hud=5:plsq=on:plsqr=1,1:ss=axioms:i=672:si=on:rtra=on_0 on DTF2THF_3077 for (2930ds/672Mi)
% 53.98/12.08 % (3374)Instruction limit reached!
% 53.98/12.08 % (3374)------------------------------
% 53.98/12.08 % (3374)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.98/12.08 % (3374)Termination reason: Unknown
% 53.98/12.08 % (3374)Termination phase: shuffling
% 53.98/12.08
% 53.98/12.08 % (3374)Memory used [KB]: 38506
% 53.98/12.08 % (3374)Time elapsed: 0.413 s
% 53.98/12.08 % (3374)Instructions burned: 1312 (million)
% 53.98/12.08 % (3374)------------------------------
% 53.98/12.08 % (3374)------------------------------
% 53.98/12.11 % (3376)dis+10_1:1_au=on:c=on:e2e=on:fsr=off:sos=all:sp=const_frequency:ss=axioms:i=1897:si=on:rtra=on_0 on DTF2THF_3077 for (2930ds/1897Mi)
% 53.98/12.11 % (3373)Instruction limit reached!
% 53.98/12.11 % (3373)------------------------------
% 53.98/12.11 % (3373)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 53.98/12.11 % (3373)Termination reason: Unknown
% 53.98/12.11 % (3373)Termination phase: Property scanning
% 53.98/12.11
% 53.98/12.11 % (3373)Memory used [KB]: 29551
% 53.98/12.11 % (3373)Time elapsed: 0.507 s
% 53.98/12.11 % (3373)Instructions burned: 1321 (million)
% 53.98/12.11 % (3373)------------------------------
% 53.98/12.11 % (3373)------------------------------
% 53.98/12.13 % (3377)lrs+10_1:40_av=off:fde=unused:ntd=on:plsq=on:plsqc=2:plsqr=411915,1048576:sp=unary_frequency:tgt=full:i=1202:si=on:rtra=on_0 on DTF2THF_3077 for (2930ds/1202Mi)
% 54.88/12.22 % (3370)Instruction limit reached!
% 54.88/12.22 % (3370)------------------------------
% 54.88/12.22 % (3370)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.88/12.22 % (3370)Termination reason: Unknown
% 54.88/12.22 % (3370)Termination phase: Property scanning
% 54.88/12.22
% 54.88/12.22 % (3370)Memory used [KB]: 41449
% 54.88/12.22 % (3370)Time elapsed: 0.751 s
% 54.88/12.22 % (3370)Instructions burned: 2404 (million)
% 54.88/12.22 % (3370)------------------------------
% 54.88/12.22 % (3370)------------------------------
% 54.88/12.24 % (3378)dis+21_1:1_au=on:cnfonf=lazy_pi_sigma_gen:hfsq=on:hfsqc=40:hfsqr=8,1:ntd=on:sos=on:sp=occurrence:ss=axioms:i=1493:si=on:rtra=on_0 on DTF2THF_3077 for (2929ds/1493Mi)
% 54.88/12.30 % (3375)Instruction limit reached!
% 54.88/12.30 % (3375)------------------------------
% 54.88/12.30 % (3375)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 54.88/12.30 % (3375)Termination reason: Unknown
% 54.88/12.30 % (3375)Termination phase: shuffling
% 54.88/12.30
% 54.88/12.30 % (3375)Memory used [KB]: 23539
% 54.88/12.30 % (3375)Time elapsed: 0.231 s
% 54.88/12.30 % (3375)Instructions burned: 673 (million)
% 54.88/12.30 % (3375)------------------------------
% 54.88/12.30 % (3375)------------------------------
% 55.21/12.31 % (3379)dis+1010_3:1_av=off:fde=unused:hud=10:nwc=10.0:s2a=on:ss=axioms:i=7679:si=on:rtra=on_0 on DTF2THF_3077 for (2928ds/7679Mi)
% 56.07/12.53 % (3377)Instruction limit reached!
% 56.07/12.53 % (3377)------------------------------
% 56.07/12.53 % (3377)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.07/12.53 % (3377)Termination reason: Unknown
% 56.07/12.53 % (3377)Termination phase: Property scanning
% 56.07/12.53
% 56.07/12.53 % (3377)Memory used [KB]: 29551
% 56.07/12.53 % (3377)Time elapsed: 0.405 s
% 56.07/12.53 % (3377)Instructions burned: 1204 (million)
% 56.07/12.53 % (3377)------------------------------
% 56.07/12.53 % (3377)------------------------------
% 56.07/12.54 % (3380)lrs+2_1:1024_anc=all:cnfonf=off:cs=on:e2e=on:nm=20:s2at=1.5:slsq=on:slsqr=1,2:spb=goal:i=340:si=on:rtra=on_0 on DTF2THF_3077 for (2926ds/340Mi)
% 56.77/12.66 % (3380)Instruction limit reached!
% 56.77/12.66 % (3380)------------------------------
% 56.77/12.66 % (3380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.77/12.66 % (3380)Termination reason: Unknown
% 56.77/12.66 % (3380)Termination phase: shuffling
% 56.77/12.66
% 56.77/12.66 % (3380)Memory used [KB]: 19061
% 56.77/12.66 % (3380)Time elapsed: 0.121 s
% 56.77/12.66 % (3380)Instructions burned: 340 (million)
% 56.77/12.66 % (3380)------------------------------
% 56.77/12.66 % (3380)------------------------------
% 56.77/12.67 % (3350)Instruction limit reached!
% 56.77/12.67 % (3350)------------------------------
% 56.77/12.67 % (3350)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.77/12.67 % (3350)Termination reason: Unknown
% 56.77/12.67 % (3350)Termination phase: Saturation
% 56.77/12.67
% 56.77/12.67 % (3350)Memory used [KB]: 40041
% 56.77/12.67 % (3350)Time elapsed: 2.167 s
% 56.77/12.67 % (3350)Instructions burned: 6455 (million)
% 56.77/12.67 % (3350)------------------------------
% 56.77/12.67 % (3350)------------------------------
% 56.77/12.68 % (3378)Instruction limit reached!
% 56.77/12.68 % (3378)------------------------------
% 56.77/12.68 % (3378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 56.77/12.68 % (3378)Termination reason: Unknown
% 56.77/12.68 % (3378)Termination phase: shuffling
% 56.77/12.68
% 56.77/12.68 % (3378)Memory used [KB]: 41065
% 56.77/12.68 % (3378)Time elapsed: 0.440 s
% 56.77/12.68 % (3378)Instructions burned: 1495 (million)
% 56.77/12.68 % (3378)------------------------------
% 56.77/12.68 % (3378)------------------------------
% 56.77/12.68 % (3381)dis+1002_1:1_acc=model:au=on:nwc=5.0:sd=5:ss=axioms:st=2.5:i=3672:si=on:rtra=on_0 on DTF2THF_3077 for (2924ds/3672Mi)
% 56.77/12.68 % (3382)dis-1002_3:1_acc=model:bd=off:bsd=on:bsdmm=1:cnfonf=lazy_not_be_gen:fd=off:fe=off:ntd=on:nwc=10.0:piset=equals:plsq=on:plsqc=1:plsqr=32,1:sp=const_max:uhcvi=on:i=1535:si=on:rtra=on_0 on DTF2THF_3077 for (2924ds/1535Mi)
% 56.77/12.69 % (3383)dis+10_16:1_hud=1:plsq=on:plsqc=1:plsqr=32,1:prag=on:sd=1:sgt=64:ss=axioms:i=4637:si=on:rtra=on_0 on DTF2THF_3077 for (2924ds/4637Mi)
% 57.87/12.81 % (3369)Instruction limit reached!
% 57.87/12.81 % (3369)------------------------------
% 57.87/12.81 % (3369)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.87/12.81 % (3369)Termination reason: Unknown
% 57.87/12.81 % (3369)Termination phase: Property scanning
% 57.87/12.81
% 57.87/12.81 % (3369)Memory used [KB]: 61406
% 57.87/12.81 % (3369)Time elapsed: 1.394 s
% 57.87/12.81 % (3369)Instructions burned: 5063 (million)
% 57.87/12.81 % (3369)------------------------------
% 57.87/12.81 % (3369)------------------------------
% 57.87/12.83 % (3384)lrs+10_1:1_fde=unused:prag=on:sd=2:sgt=32:ss=axioms:i=3174:si=on:rtra=on_0 on DTF2THF_3077 for (2923ds/3174Mi)
% 57.99/12.91 % (3376)Instruction limit reached!
% 57.99/12.91 % (3376)------------------------------
% 57.99/12.91 % (3376)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 57.99/12.91 % (3376)Termination reason: Unknown
% 57.99/12.91 % (3376)Termination phase: Preprocessing 3
% 57.99/12.91
% 57.99/12.91 % (3376)Memory used [KB]: 32622
% 57.99/12.91 % (3376)Time elapsed: 0.800 s
% 57.99/12.91 % (3376)Instructions burned: 1897 (million)
% 57.99/12.91 % (3376)------------------------------
% 57.99/12.91 % (3376)------------------------------
% 57.99/12.92 % (3385)lrs+1010_1:10_au=on:bd=off:cnfonf=lazy_pi_sigma_gen:sp=occurrence:i=307:si=on:rtra=on_0 on DTF2THF_3077 for (2922ds/307Mi)
% 60.86/13.02 % (3385)Instruction limit reached!
% 60.86/13.02 % (3385)------------------------------
% 60.86/13.02 % (3385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 60.86/13.02 % (3385)Termination reason: Unknown
% 60.86/13.02 % (3385)Termination phase: shuffling
% 60.86/13.02
% 60.86/13.02 % (3385)Memory used [KB]: 20852
% 60.86/13.02 % (3385)Time elapsed: 0.099 s
% 60.86/13.02 % (3385)Instructions burned: 310 (million)
% 60.86/13.02 % (3385)------------------------------
% 60.86/13.02 % (3385)------------------------------
% 60.86/13.03 % (3386)dis+1002_5:1_au=on:bd=off:e2e=on:fde=none:fs=off:fsr=off:sos=on:i=3619:si=on:rtra=on_0 on DTF2THF_3077 for (2921ds/3619Mi)
% 61.50/13.14 % (3382)Instruction limit reached!
% 61.50/13.14 % (3382)------------------------------
% 61.50/13.14 % (3382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 61.50/13.14 % (3382)Termination reason: Unknown
% 61.50/13.14 % (3382)Termination phase: shuffling
% 61.50/13.14
% 61.50/13.14 % (3382)Memory used [KB]: 41577
% 61.50/13.14 % (3382)Time elapsed: 0.453 s
% 61.50/13.14 % (3382)Instructions burned: 1537 (million)
% 61.50/13.14 % (3382)------------------------------
% 61.50/13.14 % (3382)------------------------------
% 61.50/13.15 % (3387)lrs+1002_1:1_au=on:cbe=off:fde=none:ins=1:sp=const_max:i=2325:si=on:rtra=on_0 on DTF2THF_3077 for (2920ds/2325Mi)
% 62.96/13.45 % (3383)First to succeed.
% 62.96/13.45 % (3383)Refutation found. Thanks to Tanya!
% 62.96/13.45 % SZS status Theorem for DTF2THF_3077
% 62.96/13.45 % SZS output start Proof for DTF2THF_3077
% 62.96/13.45 thf(type_def_5, type, rat: $tType).
% 62.96/13.45 thf(type_def_6, type, int: $tType).
% 62.96/13.45 thf(type_def_8, type, real: $tType).
% 62.96/13.45 thf(type_def_9, type, set_Pr1261947904930325089at_nat: $tType).
% 62.96/13.45 thf(type_def_10, type, nat: $tType).
% 62.96/13.45 thf(type_def_11, type, product_prod_nat_nat: $tType).
% 62.96/13.45 thf(type_def_12, type, product_prod_int_int: $tType).
% 62.96/13.45 thf(type_def_13, type, complex: $tType).
% 62.96/13.45 thf(type_def_14, type, code_integer: $tType).
% 62.96/13.45 thf(type_def_15, type, num: $tType).
% 62.96/13.45 thf(type_def_16, type, option_num: $tType).
% 62.96/13.45 thf(type_def_17, type, produc6271795597528267376eger_o: $tType).
% 62.96/13.45 thf(type_def_18, type, produc8923325533196201883nteger: $tType).
% 62.96/13.45 thf(type_def_19, type, set_Extended_enat: $tType).
% 62.96/13.45 thf(type_def_20, type, extended_enat: $tType).
% 62.96/13.45 thf(type_def_21, type, set_fi4554929511873752355omplex: $tType).
% 62.96/13.45 thf(type_def_22, type, filter6041513312241820739omplex: $tType).
% 62.96/13.45 thf(type_def_23, type, set_fi7789364187291644575l_real: $tType).
% 62.96/13.45 thf(type_def_24, type, filter2146258269922977983l_real: $tType).
% 62.96/13.45 thf(type_def_25, type, set_nat: $tType).
% 62.96/13.45 thf(type_def_26, type, set_set_nat: $tType).
% 62.96/13.45 thf(type_def_27, type, set_int: $tType).
% 62.96/13.45 thf(type_def_28, type, set_real: $tType).
% 62.96/13.45 thf(type_def_29, type, filter_real: $tType).
% 62.96/13.45 thf(type_def_30, type, option_nat: $tType).
% 62.96/13.45 thf(type_def_31, type, filter_nat: $tType).
% 62.96/13.45 thf(type_def_32, type, filter1242075044329608583at_nat: $tType).
% 62.96/13.45 thf(type_def_33, type, set_Pr5085853215250843933omplex: $tType).
% 62.96/13.45 thf(type_def_34, type, set_Pr6218003697084177305l_real: $tType).
% 62.96/13.45 thf(type_def_35, type, set_o: $tType).
% 62.96/13.45 thf(type_def_36, type, set_complex: $tType).
% 62.96/13.45 thf(type_def_37, type, set_list_o: $tType).
% 62.96/13.45 thf(type_def_38, type, set_list_complex: $tType).
% 62.96/13.45 thf(type_def_39, type, set_li5464603477888414924d_enat: $tType).
% 62.96/13.45 thf(type_def_40, type, set_list_int: $tType).
% 62.96/13.45 thf(type_def_41, type, set_list_list_nat: $tType).
% 62.96/13.45 thf(type_def_42, type, set_list_nat: $tType).
% 62.96/13.45 thf(type_def_43, type, set_li5450038453877631591at_nat: $tType).
% 62.96/13.45 thf(type_def_44, type, set_list_set_nat: $tType).
% 62.96/13.45 thf(type_def_45, type, set_list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_46, type, set_Pr8693737435421807431at_nat: $tType).
% 62.96/13.45 thf(type_def_47, type, set_Pr4329608150637261639at_nat: $tType).
% 62.96/13.45 thf(type_def_48, type, set_Product_unit: $tType).
% 62.96/13.45 thf(type_def_49, type, set_VEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_50, type, set_num: $tType).
% 62.96/13.45 thf(type_def_51, type, set_rat: $tType).
% 62.96/13.45 thf(type_def_52, type, set_set_complex: $tType).
% 62.96/13.45 thf(type_def_53, type, set_se7270636423289371942d_enat: $tType).
% 62.96/13.45 thf(type_def_54, type, set_set_int: $tType).
% 62.96/13.45 thf(type_def_55, type, set_se7855581050983116737at_nat: $tType).
% 62.96/13.45 thf(type_def_56, type, char: $tType).
% 62.96/13.45 thf(type_def_57, type, produc1319942482725812455at_nat: $tType).
% 62.96/13.45 thf(type_def_58, type, list_nat: $tType).
% 62.96/13.45 thf(type_def_59, type, literal: $tType).
% 62.96/13.45 thf(type_def_60, type, produc859450856879609959at_nat: $tType).
% 62.96/13.45 thf(type_def_61, type, produc3843707927480180839at_nat: $tType).
% 62.96/13.45 thf(type_def_62, type, list_o: $tType).
% 62.96/13.45 thf(type_def_63, type, list_int: $tType).
% 62.96/13.45 thf(type_def_64, type, vEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_65, type, set_Pr958786334691620121nt_int: $tType).
% 62.96/13.45 thf(type_def_66, type, set_Code_integer: $tType).
% 62.96/13.45 thf(type_def_67, type, list_list_o: $tType).
% 62.96/13.45 thf(type_def_68, type, list_list_nat: $tType).
% 62.96/13.45 thf(type_def_69, type, list_list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_70, type, list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_71, type, list_P7333126701944960589_nat_o: $tType).
% 62.96/13.45 thf(type_def_72, type, list_P3521021558325789923at_int: $tType).
% 62.96/13.45 thf(type_def_73, type, list_P6011104703257516679at_nat: $tType).
% 62.96/13.45 thf(type_def_74, type, list_P5647936690300460905T_VEBT: $tType).
% 62.96/13.45 thf(type_def_75, type, option_o: $tType).
% 62.96/13.45 thf(type_def_76, type, option_int: $tType).
% 62.96/13.45 thf(type_def_77, type, list_num: $tType).
% 62.96/13.45 thf(type_def_78, type, option4927543243414619207at_nat: $tType).
% 62.96/13.45 thf(type_def_79, type, list_real: $tType).
% 62.96/13.45 thf(type_def_80, type, option_real: $tType).
% 62.96/13.45 thf(type_def_81, type, list_set_nat: $tType).
% 62.96/13.45 thf(type_def_82, type, option_set_nat: $tType).
% 62.96/13.45 thf(type_def_83, type, option_VEBT_VEBT: $tType).
% 62.96/13.45 thf(type_def_84, type, list_complex: $tType).
% 62.96/13.45 thf(type_def_85, type, list_Extended_enat: $tType).
% 62.96/13.45 thf(type_def_86, type, list_P3795440434834930179_o_int: $tType).
% 62.96/13.45 thf(type_def_87, type, product_prod_o_int: $tType).
% 62.96/13.45 thf(type_def_88, type, list_P7495141550334521929T_VEBT: $tType).
% 62.96/13.45 thf(type_def_89, type, produc2504756804600209347T_VEBT: $tType).
% 62.96/13.45 thf(type_def_90, type, list_P5087981734274514673_int_o: $tType).
% 62.96/13.45 thf(type_def_91, type, product_prod_int_o: $tType).
% 62.96/13.45 thf(type_def_92, type, list_P5707943133018811711nt_int: $tType).
% 62.96/13.45 thf(type_def_93, type, list_P8198026277950538467nt_nat: $tType).
% 62.96/13.45 thf(type_def_94, type, product_prod_int_nat: $tType).
% 62.96/13.45 thf(type_def_95, type, list_P7524865323317820941T_VEBT: $tType).
% 62.96/13.45 thf(type_def_96, type, produc1531783533982839933T_VEBT: $tType).
% 62.96/13.45 thf(type_def_97, type, product_prod_nat_o: $tType).
% 62.96/13.45 thf(type_def_98, type, product_prod_nat_int: $tType).
% 62.96/13.45 thf(type_def_99, type, produc8025551001238799321T_VEBT: $tType).
% 62.96/13.45 thf(type_def_100, type, list_P3126845725202233233VEBT_o: $tType).
% 62.96/13.45 thf(type_def_101, type, produc334124729049499915VEBT_o: $tType).
% 62.96/13.45 thf(type_def_102, type, list_P4547456442757143711BT_int: $tType).
% 62.96/13.45 thf(type_def_103, type, produc4894624898956917775BT_int: $tType).
% 62.96/13.45 thf(type_def_104, type, list_P7037539587688870467BT_nat: $tType).
% 62.96/13.45 thf(type_def_105, type, produc9072475918466114483BT_nat: $tType).
% 62.96/13.45 thf(type_def_106, type, list_P7413028617227757229T_VEBT: $tType).
% 62.96/13.45 thf(type_def_107, type, produc8243902056947475879T_VEBT: $tType).
% 62.96/13.45 thf(type_def_108, type, set_Pr6588086440996610945on_nat: $tType).
% 62.96/13.45 thf(type_def_109, type, set_Pr7459493094073627847at_nat: $tType).
% 62.96/13.45 thf(type_def_110, type, produc4953844613479565601on_nat: $tType).
% 62.96/13.45 thf(type_def_111, type, set_char: $tType).
% 62.96/13.45 thf(type_def_112, type, produc2233624965454879586on_nat: $tType).
% 62.96/13.45 thf(type_def_113, type, produc7248412053542808358at_nat: $tType).
% 62.96/13.45 thf(type_def_114, type, produc4471711990508489141at_nat: $tType).
% 62.96/13.45 thf(type_def_115, type, produc8306885398267862888on_nat: $tType).
% 62.96/13.45 thf(type_def_116, type, produc3447558737645232053on_num: $tType).
% 62.96/13.45 thf(type_def_117, type, produc7036089656553540234on_num: $tType).
% 62.96/13.45 thf(type_def_118, type, produc1193250871479095198on_num: $tType).
% 62.96/13.45 thf(type_def_119, type, produc6121120109295599847at_nat: $tType).
% 62.96/13.45 thf(type_def_120, type, produc5491161045314408544at_nat: $tType).
% 62.96/13.45 thf(type_def_121, type, produc5542196010084753463at_nat: $tType).
% 62.96/13.45 thf(type_def_122, type, produc8064648209034914857omplex: $tType).
% 62.96/13.45 thf(type_def_123, type, produc1621487020699730983d_enat: $tType).
% 62.96/13.45 thf(type_def_124, type, produc2115011035271226405et_int: $tType).
% 62.96/13.45 thf(type_def_125, type, produc7819656566062154093et_nat: $tType).
% 62.96/13.45 thf(type_def_126, type, produc4411394909380815293omplex: $tType).
% 62.96/13.45 thf(type_def_127, type, produc2422161461964618553l_real: $tType).
% 62.96/13.45 thf(type_def_128, type, product_unit: $tType).
% 62.96/13.45 thf(type_def_129, type, set_Pr6308028481084910985omplex: $tType).
% 62.96/13.45 thf(type_def_130, type, set_Pr2112562347474612743d_enat: $tType).
% 62.96/13.45 thf(type_def_131, type, set_Pr2522554150109002629et_int: $tType).
% 62.96/13.45 thf(type_def_132, type, set_Pr5488025237498180813et_nat: $tType).
% 62.96/13.45 thf(func_def_0, type, set_Pr7459493094073627847at_nat: $tType).
% 62.96/13.45 thf(func_def_1, type, produc1319942482725812455at_nat: $tType).
% 62.96/13.45 thf(func_def_2, type, produc5542196010084753463at_nat: $tType).
% 62.96/13.45 thf(func_def_3, type, produc5491161045314408544at_nat: $tType).
% 62.96/13.45 thf(func_def_4, type, set_Pr4329608150637261639at_nat: $tType).
% 62.96/13.45 thf(func_def_5, type, produc1193250871479095198on_num: $tType).
% 62.96/13.45 thf(func_def_6, type, produc8306885398267862888on_nat: $tType).
% 62.96/13.45 thf(func_def_7, type, produc6121120109295599847at_nat: $tType).
% 62.96/13.45 thf(func_def_8, type, produc4471711990508489141at_nat: $tType).
% 62.96/13.45 thf(func_def_9, type, produc7036089656553540234on_num: $tType).
% 62.96/13.45 thf(func_def_10, type, produc2233624965454879586on_nat: $tType).
% 62.96/13.45 thf(func_def_11, type, produc3843707927480180839at_nat: $tType).
% 62.96/13.45 thf(func_def_12, type, set_Pr8693737435421807431at_nat: $tType).
% 62.96/13.45 thf(func_def_13, type, produc859450856879609959at_nat: $tType).
% 62.96/13.45 thf(func_def_14, type, set_Pr2112562347474612743d_enat: $tType).
% 62.96/13.45 thf(func_def_15, type, set_Pr6308028481084910985omplex: $tType).
% 62.96/13.45 thf(func_def_16, type, set_Pr6588086440996610945on_nat: $tType).
% 62.96/13.45 thf(func_def_17, type, produc1621487020699730983d_enat: $tType).
% 62.96/13.45 thf(func_def_18, type, set_fi4554929511873752355omplex: $tType).
% 62.96/13.45 thf(func_def_19, type, list_P7413028617227757229T_VEBT: $tType).
% 62.96/13.45 thf(func_def_20, type, set_Pr5488025237498180813et_nat: $tType).
% 62.96/13.45 thf(func_def_21, type, set_Pr2522554150109002629et_int: $tType).
% 62.96/13.45 thf(func_def_22, type, produc8064648209034914857omplex: $tType).
% 62.96/13.45 thf(func_def_23, type, produc3447558737645232053on_num: $tType).
% 62.96/13.45 thf(func_def_24, type, produc4953844613479565601on_nat: $tType).
% 62.96/13.45 thf(func_def_25, type, produc7248412053542808358at_nat: $tType).
% 62.96/13.45 thf(func_def_26, type, set_fi7789364187291644575l_real: $tType).
% 62.96/13.45 thf(func_def_27, type, filter6041513312241820739omplex: $tType).
% 62.96/13.45 thf(func_def_28, type, list_P7037539587688870467BT_nat: $tType).
% 62.96/13.45 thf(func_def_29, type, list_P4547456442757143711BT_int: $tType).
% 62.96/13.45 thf(func_def_30, type, list_P5647936690300460905T_VEBT: $tType).
% 62.96/13.45 thf(func_def_31, type, list_P7524865323317820941T_VEBT: $tType).
% 62.96/13.45 thf(func_def_32, type, set_li5450038453877631591at_nat: $tType).
% 62.96/13.45 thf(func_def_33, type, produc8243902056947475879T_VEBT: $tType).
% 62.96/13.45 thf(func_def_34, type, set_se7855581050983116737at_nat: $tType).
% 62.96/13.45 thf(func_def_35, type, set_Pr5085853215250843933omplex: $tType).
% 62.96/13.45 thf(func_def_36, type, produc7819656566062154093et_nat: $tType).
% 62.96/13.45 thf(func_def_37, type, produc2115011035271226405et_int: $tType).
% 62.96/13.45 thf(func_def_38, type, produc8923325533196201883nteger: $tType).
% 62.96/13.45 thf(func_def_39, type, list_P3126845725202233233VEBT_o: $tType).
% 62.96/13.45 thf(func_def_40, type, list_P7495141550334521929T_VEBT: $tType).
% 62.96/13.45 thf(func_def_41, type, filter2146258269922977983l_real: $tType).
% 62.96/13.45 thf(func_def_42, type, option4927543243414619207at_nat: $tType).
% 62.96/13.45 thf(func_def_43, type, filter1242075044329608583at_nat: $tType).
% 62.96/13.45 thf(func_def_44, type, set_Pr6218003697084177305l_real: $tType).
% 62.96/13.45 thf(func_def_45, type, list_P6011104703257516679at_nat: $tType).
% 62.96/13.45 thf(func_def_46, type, list_P3521021558325789923at_int: $tType).
% 62.96/13.45 thf(func_def_47, type, list_P8198026277950538467nt_nat: $tType).
% 62.96/13.45 thf(func_def_48, type, list_P5707943133018811711nt_int: $tType).
% 62.96/13.45 thf(func_def_49, type, produc9072475918466114483BT_nat: $tType).
% 62.96/13.45 thf(func_def_50, type, produc4894624898956917775BT_int: $tType).
% 62.96/13.45 thf(func_def_51, type, produc8025551001238799321T_VEBT: $tType).
% 62.96/13.45 thf(func_def_52, type, produc1531783533982839933T_VEBT: $tType).
% 62.96/13.45 thf(func_def_53, type, set_Pr1261947904930325089at_nat: $tType).
% 62.96/13.45 thf(func_def_54, type, set_Pr958786334691620121nt_int: $tType).
% 62.96/13.45 thf(func_def_55, type, produc4411394909380815293omplex: $tType).
% 62.96/13.45 thf(func_def_56, type, list_list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_57, type, set_list_list_nat: $tType).
% 62.96/13.45 thf(func_def_58, type, list_P7333126701944960589_nat_o: $tType).
% 62.96/13.45 thf(func_def_59, type, list_P5087981734274514673_int_o: $tType).
% 62.96/13.45 thf(func_def_60, type, list_P3795440434834930179_o_int: $tType).
% 62.96/13.45 thf(func_def_61, type, set_list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_62, type, produc334124729049499915VEBT_o: $tType).
% 62.96/13.45 thf(func_def_63, type, produc2504756804600209347T_VEBT: $tType).
% 62.96/13.45 thf(func_def_64, type, set_list_set_nat: $tType).
% 62.96/13.45 thf(func_def_65, type, produc6271795597528267376eger_o: $tType).
% 62.96/13.45 thf(func_def_66, type, set_li5464603477888414924d_enat: $tType).
% 62.96/13.45 thf(func_def_67, type, produc2422161461964618553l_real: $tType).
% 62.96/13.45 thf(func_def_68, type, set_se7270636423289371942d_enat: $tType).
% 62.96/13.45 thf(func_def_69, type, product_prod_nat_nat: $tType).
% 62.96/13.45 thf(func_def_70, type, product_prod_nat_int: $tType).
% 62.96/13.45 thf(func_def_71, type, product_prod_int_nat: $tType).
% 62.96/13.45 thf(func_def_72, type, product_prod_int_int: $tType).
% 62.96/13.45 thf(func_def_73, type, set_list_complex: $tType).
% 62.96/13.45 thf(func_def_74, type, set_set_complex: $tType).
% 62.96/13.45 thf(func_def_75, type, option_VEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_76, type, option_set_nat: $tType).
% 62.96/13.45 thf(func_def_77, type, list_list_nat: $tType).
% 62.96/13.45 thf(func_def_78, type, list_VEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_79, type, set_list_nat: $tType).
% 62.96/13.45 thf(func_def_80, type, set_list_int: $tType).
% 62.96/13.45 thf(func_def_81, type, product_prod_nat_o: $tType).
% 62.96/13.45 thf(func_def_82, type, product_prod_int_o: $tType).
% 62.96/13.45 thf(func_def_83, type, product_prod_o_int: $tType).
% 62.96/13.45 thf(func_def_84, type, list_set_nat: $tType).
% 62.96/13.45 thf(func_def_85, type, set_VEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_86, type, set_set_nat: $tType).
% 62.96/13.45 thf(func_def_87, type, set_set_int: $tType).
% 62.96/13.45 thf(func_def_88, type, set_Code_integer: $tType).
% 62.96/13.45 thf(func_def_89, type, list_Extended_enat: $tType).
% 62.96/13.45 thf(func_def_90, type, set_Product_unit: $tType).
% 62.96/13.45 thf(func_def_91, type, set_Extended_enat: $tType).
% 62.96/13.45 thf(func_def_92, type, list_list_o: $tType).
% 62.96/13.45 thf(func_def_93, type, list_complex: $tType).
% 62.96/13.45 thf(func_def_94, type, set_list_o: $tType).
% 62.96/13.45 thf(func_def_95, type, set_complex: $tType).
% 62.96/13.45 thf(func_def_96, type, option_real: $tType).
% 62.96/13.45 thf(func_def_97, type, filter_real: $tType).
% 62.96/13.45 thf(func_def_98, type, option_num: $tType).
% 62.96/13.45 thf(func_def_99, type, option_nat: $tType).
% 62.96/13.45 thf(func_def_100, type, option_int: $tType).
% 62.96/13.45 thf(func_def_101, type, filter_nat: $tType).
% 62.96/13.45 thf(func_def_102, type, set_char: $tType).
% 62.96/13.45 thf(func_def_103, type, list_real: $tType).
% 62.96/13.45 thf(func_def_104, type, set_real: $tType).
% 62.96/13.45 thf(func_def_105, type, list_num: $tType).
% 62.96/13.45 thf(func_def_106, type, list_nat: $tType).
% 62.96/13.45 thf(func_def_107, type, list_int: $tType).
% 62.96/13.45 thf(func_def_108, type, vEBT_VEBT: $tType).
% 62.96/13.45 thf(func_def_109, type, set_rat: $tType).
% 62.96/13.45 thf(func_def_110, type, set_num: $tType).
% 62.96/13.45 thf(func_def_111, type, set_nat: $tType).
% 62.96/13.45 thf(func_def_112, type, set_int: $tType).
% 62.96/13.45 thf(func_def_113, type, code_integer: $tType).
% 62.96/13.45 thf(func_def_114, type, product_unit: $tType).
% 62.96/13.45 thf(func_def_115, type, option_o: $tType).
% 62.96/13.45 thf(func_def_116, type, extended_enat: $tType).
% 62.96/13.45 thf(func_def_117, type, list_o: $tType).
% 62.96/13.45 thf(func_def_118, type, complex: $tType).
% 62.96/13.45 thf(func_def_119, type, literal: $tType).
% 62.96/13.45 thf(func_def_120, type, set_o: $tType).
% 62.96/13.45 thf(func_def_121, type, char: $tType).
% 62.96/13.45 thf(func_def_122, type, real: $tType).
% 62.96/13.45 thf(func_def_123, type, rat: $tType).
% 62.96/13.45 thf(func_def_124, type, num: $tType).
% 62.96/13.45 thf(func_def_125, type, nat: $tType).
% 62.96/13.45 thf(func_def_126, type, int: $tType).
% 62.96/13.45 thf(func_def_127, type, archim2889992004027027881ng_rat: rat > int).
% 62.96/13.45 thf(func_def_128, type, archim7802044766580827645g_real: real > int).
% 62.96/13.45 thf(func_def_129, type, archim3151403230148437115or_rat: rat > int).
% 62.96/13.45 thf(func_def_130, type, archim6058952711729229775r_real: real > int).
% 62.96/13.45 thf(func_def_131, type, archimedean_frac_rat: rat > rat).
% 62.96/13.45 thf(func_def_132, type, archim2898591450579166408c_real: real > real).
% 62.96/13.45 thf(func_def_133, type, archim7778729529865785530nd_rat: rat > int).
% 62.96/13.45 thf(func_def_134, type, archim8280529875227126926d_real: real > int).
% 62.96/13.45 thf(func_def_135, type, bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat).
% 62.96/13.45 thf(func_def_136, type, bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat).
% 62.96/13.45 thf(func_def_137, type, bNF_re728719798268516973at_o_o: ((nat > rat) > (nat > rat) > $o) > ($o > $o > $o) > ((nat > rat) > $o) > ((nat > rat) > $o) > $o).
% 62.96/13.45 thf(func_def_138, type, bNF_re4297313714947099218al_o_o: ((nat > rat) > real > $o) > ($o > $o > $o) > ((nat > rat) > $o) > (real > $o) > $o).
% 62.96/13.45 thf(func_def_139, type, bNF_re3403563459893282935_int_o: (int > int > $o) > ((int > $o) > (int > $o) > $o) > (int > int > $o) > (int > int > $o) > $o).
% 62.96/13.45 thf(func_def_140, type, bNF_re5089333283451836215nt_o_o: (int > int > $o) > ($o > $o > $o) > (int > $o) > (int > $o) > $o).
% 62.96/13.45 thf(func_def_141, type, bNF_re578469030762574527_nat_o: (nat > nat > $o) > ((nat > $o) > (nat > $o) > $o) > (nat > nat > $o) > (nat > nat > $o) > $o).
% 62.96/13.45 thf(func_def_142, type, bNF_re4705727531993890431at_o_o: (nat > nat > $o) > ($o > $o > $o) > (nat > $o) > (nat > $o) > $o).
% 62.96/13.45 thf(func_def_143, type, bNF_re6830278522597306478at_int: (nat > nat > $o) > (product_prod_nat_nat > int > $o) > (nat > product_prod_nat_nat) > (nat > int) > $o).
% 62.96/13.45 thf(func_def_144, type, bNF_re8699439704749558557nt_o_o: (product_prod_int_int > product_prod_int_int > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 62.96/13.45 thf(func_def_145, type, bNF_re1494630372529172596at_o_o: (product_prod_int_int > rat > $o) > ($o > $o > $o) > (product_prod_int_int > $o) > (rat > $o) > $o).
% 62.96/13.45 thf(func_def_146, type, bNF_re717283939379294677_int_o: (product_prod_nat_nat > int > $o) > ((product_prod_nat_nat > $o) > (int > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (int > int > $o) > $o).
% 62.96/13.45 thf(func_def_147, type, bNF_re6644619430987730960nt_o_o: (product_prod_nat_nat > int > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (int > $o) > $o).
% 62.96/13.45 thf(func_def_148, type, bNF_re4202695980764964119_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ((product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 62.96/13.45 thf(func_def_149, type, bNF_re3666534408544137501at_o_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > ($o > $o > $o) > (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 62.96/13.45 thf(func_def_150, type, bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_nat > $o).
% 62.96/13.45 thf(func_def_151, type, binomial: nat > nat > nat).
% 62.96/13.45 thf(func_def_152, type, gbinomial_complex: complex > nat > complex).
% 62.96/13.45 thf(func_def_153, type, gbinomial_int: int > nat > int).
% 62.96/13.45 thf(func_def_154, type, gbinomial_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_155, type, gbinomial_rat: rat > nat > rat).
% 62.96/13.45 thf(func_def_156, type, gbinomial_real: real > nat > real).
% 62.96/13.45 thf(func_def_157, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 62.96/13.45 thf(func_def_158, type, bit_concat_bit: nat > int > int > int).
% 62.96/13.45 thf(func_def_159, type, bit_ri7919022796975470100ot_int: int > int).
% 62.96/13.45 thf(func_def_160, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 62.96/13.45 thf(func_def_161, type, bit_ri631733984087533419it_int: nat > int > int).
% 62.96/13.45 thf(func_def_162, type, bit_se725231765392027082nd_int: int > int > int).
% 62.96/13.45 thf(func_def_163, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_164, type, bit_se8568078237143864401it_int: nat > int > int).
% 62.96/13.45 thf(func_def_165, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_166, type, bit_se2159334234014336723it_int: nat > int > int).
% 62.96/13.45 thf(func_def_167, type, bit_se2000444600071755411sk_int: nat > int).
% 62.96/13.45 thf(func_def_168, type, bit_se2002935070580805687sk_nat: nat > nat).
% 62.96/13.45 thf(func_def_169, type, bit_se1409905431419307370or_int: int > int > int).
% 62.96/13.45 thf(func_def_170, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_171, type, bit_se545348938243370406it_int: nat > int > int).
% 62.96/13.45 thf(func_def_172, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_173, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 62.96/13.45 thf(func_def_174, type, bit_se7879613467334960850it_int: nat > int > int).
% 62.96/13.45 thf(func_def_175, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_176, type, bit_se2923211474154528505it_int: nat > int > int).
% 62.96/13.45 thf(func_def_177, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_178, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 62.96/13.45 thf(func_def_179, type, bit_se4203085406695923979it_int: nat > int > int).
% 62.96/13.45 thf(func_def_180, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_181, type, bit_se6526347334894502574or_int: int > int > int).
% 62.96/13.45 thf(func_def_182, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 62.96/13.45 thf(func_def_183, type, bit_se1146084159140164899it_int: int > nat > $o).
% 62.96/13.45 thf(func_def_184, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 62.96/13.45 thf(func_def_185, type, bit_take_bit_num: nat > num > option_num).
% 62.96/13.45 thf(func_def_186, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 62.96/13.45 thf(func_def_187, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 62.96/13.45 thf(func_def_188, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 62.96/13.45 thf(func_def_189, type, code_int_of_integer: code_integer > int).
% 62.96/13.45 thf(func_def_190, type, code_integer_of_int: int > code_integer).
% 62.96/13.45 thf(func_def_191, type, code_nat_of_integer: code_integer > nat).
% 62.96/13.45 thf(func_def_192, type, code_num_of_integer: code_integer > num).
% 62.96/13.45 thf(func_def_193, type, comple2295165028678016749d_enat: set_Extended_enat > extended_enat).
% 62.96/13.45 thf(func_def_194, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 62.96/13.45 thf(func_def_195, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 62.96/13.45 thf(func_def_196, type, complete_Inf_Inf_nat: set_nat > nat).
% 62.96/13.45 thf(func_def_197, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 62.96/13.45 thf(func_def_198, type, comple4398354569131411667d_enat: set_Extended_enat > extended_enat).
% 62.96/13.45 thf(func_def_199, type, complete_Sup_Sup_int: set_int > int).
% 62.96/13.45 thf(func_def_200, type, complete_Sup_Sup_nat: set_nat > nat).
% 62.96/13.45 thf(func_def_201, type, comple1385675409528146559p_real: set_real > real).
% 62.96/13.45 thf(func_def_202, type, comple7399068483239264473et_nat: set_set_nat > set_nat).
% 62.96/13.45 thf(func_def_203, type, arg: complex > real).
% 62.96/13.45 thf(func_def_204, type, cis: real > complex).
% 62.96/13.45 thf(func_def_205, type, cnj: complex > complex).
% 62.96/13.45 thf(func_def_206, type, complex2: real > real > complex).
% 62.96/13.45 thf(func_def_207, type, im: complex > real).
% 62.96/13.45 thf(func_def_208, type, re: complex > real).
% 62.96/13.45 thf(func_def_209, type, csqrt: complex > complex).
% 62.96/13.45 thf(func_def_210, type, imaginary_unit: complex).
% 62.96/13.45 thf(func_def_211, type, condit2214826472909112428ve_nat: set_nat > $o).
% 62.96/13.45 thf(func_def_212, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 62.96/13.45 thf(func_def_213, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 62.96/13.45 thf(func_def_214, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 62.96/13.45 thf(func_def_215, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 62.96/13.45 thf(func_def_216, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 62.96/13.45 thf(func_def_217, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 62.96/13.45 thf(func_def_218, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 62.96/13.45 thf(func_def_219, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 62.96/13.45 thf(func_def_220, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 62.96/13.45 thf(func_def_221, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 62.96/13.45 thf(func_def_222, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.96/13.45 thf(func_def_223, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 62.96/13.45 thf(func_def_224, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 62.96/13.45 thf(func_def_225, type, euclid3395696857347342551nt_int: int > int).
% 62.96/13.45 thf(func_def_226, type, extended_eSuc: extended_enat > extended_enat).
% 62.96/13.45 thf(func_def_227, type, extended_enat2: nat > extended_enat).
% 62.96/13.45 thf(func_def_228, type, extended_Abs_enat: option_nat > extended_enat).
% 62.96/13.45 thf(func_def_229, type, extended_case_enat_o: (nat > $o) > $o > extended_enat > $o).
% 62.96/13.45 thf(func_def_230, type, extend3600170679010898289d_enat: (nat > extended_enat) > extended_enat > extended_enat > extended_enat).
% 62.96/13.45 thf(func_def_231, type, extend5688581933313929465d_enat: extended_enat).
% 62.96/13.45 thf(func_def_232, type, semiri1408675320244567234ct_nat: nat > nat).
% 62.96/13.45 thf(func_def_233, type, semiri2265585572941072030t_real: nat > real).
% 62.96/13.45 thf(func_def_234, type, inverse_inverse_real: real > real).
% 62.96/13.45 thf(func_def_235, type, at_bot_real: filter_real).
% 62.96/13.45 thf(func_def_236, type, at_top_nat: filter_nat).
% 62.96/13.45 thf(func_def_237, type, at_top_real: filter_real).
% 62.96/13.45 thf(func_def_238, type, cofinite_nat: filter_nat).
% 62.96/13.45 thf(func_def_239, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 62.96/13.45 thf(func_def_240, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 62.96/13.45 thf(func_def_241, type, eventually_real: (real > $o) > filter_real > $o).
% 62.96/13.45 thf(func_def_242, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 62.96/13.45 thf(func_def_243, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 62.96/13.45 thf(func_def_244, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 62.96/13.45 thf(func_def_245, type, frequently_nat: (nat > $o) > filter_nat > $o).
% 62.96/13.45 thf(func_def_246, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 62.96/13.46 thf(func_def_247, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 62.96/13.46 thf(func_def_248, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 62.96/13.46 thf(func_def_249, type, finite_card_o: set_o > nat).
% 62.96/13.46 thf(func_def_250, type, finite_card_complex: set_complex > nat).
% 62.96/13.46 thf(func_def_251, type, finite121521170596916366d_enat: set_Extended_enat > nat).
% 62.96/13.46 thf(func_def_252, type, finite_card_int: set_int > nat).
% 62.96/13.46 thf(func_def_253, type, finite_card_list_o: set_list_o > nat).
% 62.96/13.46 thf(func_def_254, type, finite5120063068150530198omplex: set_list_complex > nat).
% 62.96/13.46 thf(func_def_255, type, finite7441382602597825044d_enat: set_li5464603477888414924d_enat > nat).
% 62.96/13.46 thf(func_def_256, type, finite_card_list_int: set_list_int > nat).
% 62.96/13.46 thf(func_def_257, type, finite7325466520557071688st_nat: set_list_list_nat > nat).
% 62.96/13.46 thf(func_def_258, type, finite_card_list_nat: set_list_nat > nat).
% 62.96/13.46 thf(func_def_259, type, finite249151656366948015at_nat: set_li5450038453877631591at_nat > nat).
% 62.96/13.46 thf(func_def_260, type, finite5631907774883551598et_nat: set_list_set_nat > nat).
% 62.96/13.46 thf(func_def_261, type, finite5915292604075114978T_VEBT: set_list_VEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_262, type, finite_card_nat: set_nat > nat).
% 62.96/13.46 thf(func_def_263, type, finite711546835091564841at_nat: set_Pr1261947904930325089at_nat > nat).
% 62.96/13.46 thf(func_def_264, type, finite1207074278014112911at_nat: set_Pr8693737435421807431at_nat > nat).
% 62.96/13.46 thf(func_def_265, type, finite3771342082235030671at_nat: set_Pr4329608150637261639at_nat > nat).
% 62.96/13.46 thf(func_def_266, type, finite410649719033368117t_unit: set_Product_unit > nat).
% 62.96/13.46 thf(func_def_267, type, finite_card_real: set_real > nat).
% 62.96/13.46 thf(func_def_268, type, finite_card_set_nat: set_set_nat > nat).
% 62.96/13.46 thf(func_def_269, type, finite7802652506058667612T_VEBT: set_VEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_270, type, finite_finite_o: set_o > $o).
% 62.96/13.46 thf(func_def_271, type, finite3207457112153483333omplex: set_complex > $o).
% 62.96/13.46 thf(func_def_272, type, finite4001608067531595151d_enat: set_Extended_enat > $o).
% 62.96/13.46 thf(func_def_273, type, finite_finite_int: set_int > $o).
% 62.96/13.46 thf(func_def_274, type, finite_finite_list_o: set_list_o > $o).
% 62.96/13.46 thf(func_def_275, type, finite8712137658972009173omplex: set_list_complex > $o).
% 62.96/13.46 thf(func_def_276, type, finite1862508098717546133d_enat: set_li5464603477888414924d_enat > $o).
% 62.96/13.46 thf(func_def_277, type, finite3922522038869484883st_int: set_list_int > $o).
% 62.96/13.46 thf(func_def_278, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 62.96/13.46 thf(func_def_279, type, finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o).
% 62.96/13.46 thf(func_def_280, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_281, type, finite_finite_nat: set_nat > $o).
% 62.96/13.46 thf(func_def_282, type, finite_finite_num: set_num > $o).
% 62.96/13.46 thf(func_def_283, type, finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_284, type, finite4392333629123659920at_nat: set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_285, type, finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o).
% 62.96/13.46 thf(func_def_286, type, finite_finite_rat: set_rat > $o).
% 62.96/13.46 thf(func_def_287, type, finite_finite_real: set_real > $o).
% 62.96/13.46 thf(func_def_288, type, finite6551019134538273531omplex: set_set_complex > $o).
% 62.96/13.46 thf(func_def_289, type, finite5468666774076196335d_enat: set_se7270636423289371942d_enat > $o).
% 62.96/13.46 thf(func_def_290, type, finite6197958912794628473et_int: set_set_int > $o).
% 62.96/13.46 thf(func_def_291, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 62.96/13.46 thf(func_def_292, type, finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o).
% 62.96/13.46 thf(func_def_293, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_294, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 62.96/13.46 thf(func_def_295, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 62.96/13.46 thf(func_def_296, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 62.96/13.46 thf(func_def_297, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 62.96/13.46 thf(func_def_298, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 62.96/13.46 thf(func_def_299, type, comp_nat_real_nat: (nat > real) > (nat > nat) > nat > real).
% 62.96/13.46 thf(func_def_300, type, id_o: $o > $o).
% 62.96/13.46 thf(func_def_301, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 62.96/13.46 thf(func_def_302, type, inj_on_nat_char: (nat > char) > set_nat > $o).
% 62.96/13.46 thf(func_def_303, type, inj_on_real_real: (real > real) > set_real > $o).
% 62.96/13.46 thf(func_def_304, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 62.96/13.46 thf(func_def_305, type, map_fu434086159418415080_int_o: (int > product_prod_nat_nat) > ((product_prod_nat_nat > $o) > int > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > int > int > $o).
% 62.96/13.46 thf(func_def_306, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 62.96/13.46 thf(func_def_307, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 62.96/13.46 thf(func_def_308, type, map_fu1856342031159181835at_o_o: (real > nat > rat) > ($o > $o) > ((nat > rat) > $o) > real > $o).
% 62.96/13.46 thf(func_def_309, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 62.96/13.46 thf(func_def_310, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 62.96/13.46 thf(func_def_311, type, fun_max_strict: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_312, type, fun_max_weak: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_313, type, fun_min_strict: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_314, type, fun_min_weak: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_315, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_316, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_317, type, fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o).
% 62.96/13.46 thf(func_def_318, type, gcd_Gcd_int: set_int > int).
% 62.96/13.46 thf(func_def_319, type, gcd_Gcd_nat: set_nat > nat).
% 62.96/13.46 thf(func_def_320, type, gcd_Lcm_int: set_int > int).
% 62.96/13.46 thf(func_def_321, type, gcd_Lcm_nat: set_nat > nat).
% 62.96/13.46 thf(func_def_322, type, bezw: nat > nat > product_prod_int_int).
% 62.96/13.46 thf(func_def_323, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_324, type, gcd_gcd_int: int > int > int).
% 62.96/13.46 thf(func_def_325, type, gcd_gcd_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_326, type, gcd_lcm_int: int > int > int).
% 62.96/13.46 thf(func_def_327, type, gcd_lcm_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_328, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_329, type, abs_abs_Code_integer: code_integer > code_integer).
% 62.96/13.46 thf(func_def_330, type, abs_abs_int: int > int).
% 62.96/13.46 thf(func_def_331, type, abs_abs_rat: rat > rat).
% 62.96/13.46 thf(func_def_332, type, abs_abs_real: real > real).
% 62.96/13.46 thf(func_def_333, type, minus_minus_o_o: ($o > $o) > ($o > $o) > $o > $o).
% 62.96/13.46 thf(func_def_334, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 62.96/13.46 thf(func_def_335, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_336, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_337, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 62.96/13.46 thf(func_def_338, type, minus_6910147592129066416_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 62.96/13.46 thf(func_def_339, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_340, type, minus_minus_complex: complex > complex > complex).
% 62.96/13.46 thf(func_def_341, type, minus_3235023915231533773d_enat: extended_enat > extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_342, type, minus_minus_int: int > int > int).
% 62.96/13.46 thf(func_def_343, type, minus_minus_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_344, type, minus_minus_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_345, type, minus_minus_real: real > real > real).
% 62.96/13.46 thf(func_def_346, type, minus_minus_set_o: set_o > set_o > set_o).
% 62.96/13.46 thf(func_def_347, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 62.96/13.46 thf(func_def_348, type, minus_925952699566721837d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_349, type, minus_minus_set_int: set_int > set_int > set_int).
% 62.96/13.46 thf(func_def_350, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_351, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_352, type, minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_353, type, minus_8321449233255521966at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_354, type, minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_355, type, minus_minus_set_real: set_real > set_real > set_real).
% 62.96/13.46 thf(func_def_356, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_357, type, minus_5127226145743854075T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 62.96/13.46 thf(func_def_358, type, one_one_Code_integer: code_integer).
% 62.96/13.46 thf(func_def_359, type, one_one_complex: complex).
% 62.96/13.46 thf(func_def_360, type, one_on7984719198319812577d_enat: extended_enat).
% 62.96/13.46 thf(func_def_361, type, one_one_int: int).
% 62.96/13.46 thf(func_def_362, type, one_one_nat: nat).
% 62.96/13.46 thf(func_def_363, type, one_one_rat: rat).
% 62.96/13.46 thf(func_def_364, type, one_one_real: real).
% 62.96/13.46 thf(func_def_365, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_366, type, plus_plus_complex: complex > complex > complex).
% 62.96/13.46 thf(func_def_367, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_368, type, plus_plus_int: int > int > int).
% 62.96/13.46 thf(func_def_369, type, plus_plus_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_370, type, plus_plus_num: num > num > num).
% 62.96/13.46 thf(func_def_371, type, plus_plus_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_372, type, plus_plus_real: real > real > real).
% 62.96/13.46 thf(func_def_373, type, plus_plus_literal: literal > literal > literal).
% 62.96/13.46 thf(func_def_374, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 62.96/13.46 thf(func_def_375, type, sgn_sgn_complex: complex > complex).
% 62.96/13.46 thf(func_def_376, type, sgn_sgn_int: int > int).
% 62.96/13.46 thf(func_def_377, type, sgn_sgn_rat: rat > rat).
% 62.96/13.46 thf(func_def_378, type, sgn_sgn_real: real > real).
% 62.96/13.46 thf(func_def_379, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_380, type, times_times_complex: complex > complex > complex).
% 62.96/13.46 thf(func_def_381, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_382, type, times_times_int: int > int > int).
% 62.96/13.46 thf(func_def_383, type, times_times_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_384, type, times_times_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_385, type, times_times_real: real > real > real).
% 62.96/13.46 thf(func_def_386, type, uminus_uminus_o_o: ($o > $o) > $o > $o).
% 62.96/13.46 thf(func_def_387, type, uminus_uminus_int_o: (int > $o) > int > $o).
% 62.96/13.46 thf(func_def_388, type, uminus5770388063884162150_nat_o: (list_nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_389, type, uminus_uminus_nat_o: (nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_390, type, uminus_uminus_real_o: (real > $o) > real > $o).
% 62.96/13.46 thf(func_def_391, type, uminus6401447641752708672_nat_o: (set_nat > $o) > set_nat > $o).
% 62.96/13.46 thf(func_def_392, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 62.96/13.46 thf(func_def_393, type, uminus1482373934393186551omplex: complex > complex).
% 62.96/13.46 thf(func_def_394, type, uminus_uminus_int: int > int).
% 62.96/13.46 thf(func_def_395, type, uminus_uminus_rat: rat > rat).
% 62.96/13.46 thf(func_def_396, type, uminus_uminus_real: real > real).
% 62.96/13.46 thf(func_def_397, type, uminus_uminus_set_o: set_o > set_o).
% 62.96/13.46 thf(func_def_398, type, uminus8566677241136511917omplex: set_complex > set_complex).
% 62.96/13.46 thf(func_def_399, type, uminus417252749190364093d_enat: set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_400, type, uminus1532241313380277803et_int: set_int > set_int).
% 62.96/13.46 thf(func_def_401, type, uminus3195874150345416415st_nat: set_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_402, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 62.96/13.46 thf(func_def_403, type, uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_404, type, uminus4384627049435823934at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_405, type, uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_406, type, uminus612125837232591019t_real: set_real > set_real).
% 62.96/13.46 thf(func_def_407, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_408, type, zero_z3403309356797280102nteger: code_integer).
% 62.96/13.46 thf(func_def_409, type, zero_zero_complex: complex).
% 62.96/13.46 thf(func_def_410, type, zero_z5237406670263579293d_enat: extended_enat).
% 62.96/13.46 thf(func_def_411, type, zero_zero_int: int).
% 62.96/13.46 thf(func_def_412, type, zero_zero_nat: nat).
% 62.96/13.46 thf(func_def_413, type, zero_zero_rat: rat).
% 62.96/13.46 thf(func_def_414, type, zero_zero_real: real).
% 62.96/13.46 thf(func_def_415, type, zero_zero_literal: literal).
% 62.96/13.46 thf(func_def_416, type, groups4406642042086082107nteger: ($o > code_integer) > set_o > code_integer).
% 62.96/13.46 thf(func_def_417, type, groups5328290441151304332omplex: ($o > complex) > set_o > complex).
% 62.96/13.46 thf(func_def_418, type, groups8505340233167759370_o_int: ($o > int) > set_o > int).
% 62.96/13.46 thf(func_def_419, type, groups8507830703676809646_o_nat: ($o > nat) > set_o > nat).
% 62.96/13.46 thf(func_def_420, type, groups7872700643590313910_o_rat: ($o > rat) > set_o > rat).
% 62.96/13.46 thf(func_def_421, type, groups8691415230153176458o_real: ($o > real) > set_o > real).
% 62.96/13.46 thf(func_def_422, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 62.96/13.46 thf(func_def_423, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 62.96/13.46 thf(func_def_424, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 62.96/13.46 thf(func_def_425, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 62.96/13.46 thf(func_def_426, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 62.96/13.46 thf(func_def_427, type, groups6818542070133387226omplex: (extended_enat > complex) > set_Extended_enat > complex).
% 62.96/13.46 thf(func_def_428, type, groups2025484359314973016at_int: (extended_enat > int) > set_Extended_enat > int).
% 62.96/13.46 thf(func_def_429, type, groups2027974829824023292at_nat: (extended_enat > nat) > set_Extended_enat > nat).
% 62.96/13.46 thf(func_def_430, type, groups1392844769737527556at_rat: (extended_enat > rat) > set_Extended_enat > rat).
% 62.96/13.46 thf(func_def_431, type, groups4148127829035722712t_real: (extended_enat > real) > set_Extended_enat > real).
% 62.96/13.46 thf(func_def_432, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 62.96/13.46 thf(func_def_433, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 62.96/13.46 thf(func_def_434, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 62.96/13.46 thf(func_def_435, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 62.96/13.46 thf(func_def_436, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 62.96/13.46 thf(func_def_437, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 62.96/13.46 thf(func_def_438, type, groups6529277132148336714omplex: (list_nat > complex) > set_list_nat > complex).
% 62.96/13.46 thf(func_def_439, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 62.96/13.46 thf(func_def_440, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 62.96/13.46 thf(func_def_441, type, groups7108830773950497114d_enat: (nat > extended_enat) > set_nat > extended_enat).
% 62.96/13.46 thf(func_def_442, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 62.96/13.46 thf(func_def_443, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 62.96/13.46 thf(func_def_444, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 62.96/13.46 thf(func_def_445, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 62.96/13.46 thf(func_def_446, type, groups977919841031483927at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 62.96/13.46 thf(func_def_447, type, groups1900718384385340925at_nat: (produc859450856879609959at_nat > nat) > set_Pr8693737435421807431at_nat > nat).
% 62.96/13.46 thf(func_def_448, type, groups3860910324918113789at_nat: (produc3843707927480180839at_nat > nat) > set_Pr4329608150637261639at_nat > nat).
% 62.96/13.46 thf(func_def_449, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 62.96/13.46 thf(func_def_450, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 62.96/13.46 thf(func_def_451, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 62.96/13.46 thf(func_def_452, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 62.96/13.46 thf(func_def_453, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 62.96/13.46 thf(func_def_454, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 62.96/13.46 thf(func_def_455, type, groups8255218700646806128omplex: (set_nat > complex) > set_set_nat > complex).
% 62.96/13.46 thf(func_def_456, type, groups8294997508430121362at_nat: (set_nat > nat) > set_set_nat > nat).
% 62.96/13.46 thf(func_def_457, type, groups5107569545109728110t_real: (set_nat > real) > set_set_nat > real).
% 62.96/13.46 thf(func_def_458, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 62.96/13.46 thf(func_def_459, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 62.96/13.46 thf(func_def_460, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 62.96/13.46 thf(func_def_461, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 62.96/13.46 thf(func_def_462, type, groups4561878855575611511st_nat: list_nat > nat).
% 62.96/13.46 thf(func_def_463, type, the_int: (int > $o) > int).
% 62.96/13.46 thf(func_def_464, type, the_Pr4378521158711661632nt_int: (product_prod_int_int > $o) > product_prod_int_int).
% 62.96/13.46 thf(func_def_465, type, the_real: (real > $o) > real).
% 62.96/13.46 thf(func_def_466, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_467, type, if_complex: $o > complex > complex > complex).
% 62.96/13.46 thf(func_def_468, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_469, type, if_int: $o > int > int > int).
% 62.96/13.46 thf(func_def_470, type, if_list_int: $o > list_int > list_int > list_int).
% 62.96/13.46 thf(func_def_471, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 62.96/13.46 thf(func_def_472, type, if_nat: $o > nat > nat > nat).
% 62.96/13.46 thf(func_def_473, type, if_num: $o > num > num > num).
% 62.96/13.46 thf(func_def_474, type, if_option_nat: $o > option_nat > option_nat > option_nat).
% 62.96/13.46 thf(func_def_475, type, if_option_num: $o > option_num > option_num > option_num).
% 62.96/13.46 thf(func_def_476, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 62.96/13.46 thf(func_def_477, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.96/13.46 thf(func_def_478, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 62.96/13.46 thf(func_def_479, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_480, type, if_rat: $o > rat > rat > rat).
% 62.96/13.46 thf(func_def_481, type, if_real: $o > real > real > real).
% 62.96/13.46 thf(func_def_482, type, if_set_int: $o > set_int > set_int > set_int).
% 62.96/13.46 thf(func_def_483, type, if_VEBT_VEBT: $o > vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_484, type, infini7641415182203889163d_enat: set_Extended_enat > nat > extended_enat).
% 62.96/13.46 thf(func_def_485, type, infini8530281810654367211te_nat: set_nat > nat > nat).
% 62.96/13.46 thf(func_def_486, type, abs_Integ: product_prod_nat_nat > int).
% 62.96/13.46 thf(func_def_487, type, rep_Integ: int > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_488, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 62.96/13.46 thf(func_def_489, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 62.96/13.46 thf(func_def_490, type, intrel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_491, type, nat2: int > nat).
% 62.96/13.46 thf(func_def_492, type, pcr_int: product_prod_nat_nat > int > $o).
% 62.96/13.46 thf(func_def_493, type, power_int_real: real > int > real).
% 62.96/13.46 thf(func_def_494, type, ring_11222124179247155820nteger: set_Code_integer).
% 62.96/13.46 thf(func_def_495, type, ring_1_Ints_complex: set_complex).
% 62.96/13.46 thf(func_def_496, type, ring_1_Ints_int: set_int).
% 62.96/13.46 thf(func_def_497, type, ring_1_Ints_rat: set_rat).
% 62.96/13.46 thf(func_def_498, type, ring_1_Ints_real: set_real).
% 62.96/13.46 thf(func_def_499, type, ring_18347121197199848620nteger: int > code_integer).
% 62.96/13.46 thf(func_def_500, type, ring_17405671764205052669omplex: int > complex).
% 62.96/13.46 thf(func_def_501, type, ring_1_of_int_int: int > int).
% 62.96/13.46 thf(func_def_502, type, ring_1_of_int_rat: int > rat).
% 62.96/13.46 thf(func_def_503, type, ring_1_of_int_real: int > real).
% 62.96/13.46 thf(func_def_504, type, inf_inf_o_o: ($o > $o) > ($o > $o) > $o > $o).
% 62.96/13.46 thf(func_def_505, type, inf_inf_int_o: (int > $o) > (int > $o) > int > $o).
% 62.96/13.46 thf(func_def_506, type, inf_inf_list_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_507, type, inf_inf_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_508, type, inf_in5163264567034779214_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_509, type, inf_inf_real_o: (real > $o) > (real > $o) > real > $o).
% 62.96/13.46 thf(func_def_510, type, inf_inf_set_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 62.96/13.46 thf(func_def_511, type, inf_inf_int: int > int > int).
% 62.96/13.46 thf(func_def_512, type, inf_inf_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_513, type, inf_inf_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_514, type, inf_inf_real: real > real > real).
% 62.96/13.46 thf(func_def_515, type, inf_inf_set_o: set_o > set_o > set_o).
% 62.96/13.46 thf(func_def_516, type, inf_inf_set_complex: set_complex > set_complex > set_complex).
% 62.96/13.46 thf(func_def_517, type, inf_in8357106775501769908d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_518, type, inf_inf_set_int: set_int > set_int > set_int).
% 62.96/13.46 thf(func_def_519, type, inf_inf_set_list_nat: set_list_nat > set_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_520, type, inf_inf_set_nat: set_nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_521, type, inf_inf_set_num: set_num > set_num > set_num).
% 62.96/13.46 thf(func_def_522, type, inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_523, type, inf_in4302113700860409141at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_524, type, inf_in7913087082777306421at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_525, type, inf_inf_set_rat: set_rat > set_rat > set_rat).
% 62.96/13.46 thf(func_def_526, type, inf_inf_set_real: set_real > set_real > set_real).
% 62.96/13.46 thf(func_def_527, type, inf_inf_set_set_nat: set_set_nat > set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_528, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 62.96/13.46 thf(func_def_529, type, sup_sup_o_o: ($o > $o) > ($o > $o) > $o > $o).
% 62.96/13.46 thf(func_def_530, type, sup_sup_int_o: (int > $o) > (int > $o) > int > $o).
% 62.96/13.46 thf(func_def_531, type, sup_sup_list_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_532, type, sup_sup_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_533, type, sup_su8986005896011022210_nat_o: (produc859450856879609959at_nat > $o) > (produc859450856879609959at_nat > $o) > produc859450856879609959at_nat > $o).
% 62.96/13.46 thf(func_def_534, type, sup_su2080679670758317954_nat_o: (produc3843707927480180839at_nat > $o) > (produc3843707927480180839at_nat > $o) > produc3843707927480180839at_nat > $o).
% 62.96/13.46 thf(func_def_535, type, sup_sup_real_o: (real > $o) > (real > $o) > real > $o).
% 62.96/13.46 thf(func_def_536, type, sup_sup_set_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 62.96/13.46 thf(func_def_537, type, sup_sup_int: int > int > int).
% 62.96/13.46 thf(func_def_538, type, sup_sup_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_539, type, sup_sup_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_540, type, sup_sup_real: real > real > real).
% 62.96/13.46 thf(func_def_541, type, sup_sup_set_o: set_o > set_o > set_o).
% 62.96/13.46 thf(func_def_542, type, sup_sup_set_complex: set_complex > set_complex > set_complex).
% 62.96/13.46 thf(func_def_543, type, sup_su4489774667511045786d_enat: set_Extended_enat > set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_544, type, sup_sup_set_int: set_int > set_int > set_int).
% 62.96/13.46 thf(func_def_545, type, sup_sup_set_list_nat: set_list_nat > set_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_546, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_547, type, sup_sup_set_num: set_num > set_num > set_num).
% 62.96/13.46 thf(func_def_548, type, sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_549, type, sup_su718114333110466843at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_550, type, sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_551, type, sup_sup_set_rat: set_rat > set_rat > set_rat).
% 62.96/13.46 thf(func_def_552, type, sup_sup_set_real: set_real > set_real > set_real).
% 62.96/13.46 thf(func_def_553, type, sup_sup_set_set_nat: set_set_nat > set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_554, type, lattic921264341876707157d_enat: set_Extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_555, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 62.96/13.46 thf(func_def_556, type, lattic8556559851467007577_o_num: ($o > num) > set_o > $o).
% 62.96/13.46 thf(func_def_557, type, lattic2140725968369957399_o_rat: ($o > rat) > set_o > $o).
% 62.96/13.46 thf(func_def_558, type, lattic8697145971487455083o_real: ($o > real) > set_o > $o).
% 62.96/13.46 thf(func_def_559, type, lattic1922116423962787043ex_num: (complex > num) > set_complex > complex).
% 62.96/13.46 thf(func_def_560, type, lattic4729654577720512673ex_rat: (complex > rat) > set_complex > complex).
% 62.96/13.46 thf(func_def_561, type, lattic8794016678065449205x_real: (complex > real) > set_complex > complex).
% 62.96/13.46 thf(func_def_562, type, lattic402713867396545063at_num: (extended_enat > num) > set_Extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_563, type, lattic3210252021154270693at_rat: (extended_enat > rat) > set_Extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_564, type, lattic1189837152898106425t_real: (extended_enat > real) > set_Extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_565, type, lattic7811156612396918303nt_rat: (int > rat) > set_int > int).
% 62.96/13.46 thf(func_def_566, type, lattic2675449441010098035t_real: (int > real) > set_int > int).
% 62.96/13.46 thf(func_def_567, type, lattic6811802900495863747at_rat: (nat > rat) > set_nat > nat).
% 62.96/13.46 thf(func_def_568, type, lattic488527866317076247t_real: (nat > real) > set_nat > nat).
% 62.96/13.46 thf(func_def_569, type, lattic1613168225601753569al_num: (real > num) > set_real > real).
% 62.96/13.46 thf(func_def_570, type, lattic4420706379359479199al_rat: (real > rat) > set_real > real).
% 62.96/13.46 thf(func_def_571, type, lattic8440615504127631091l_real: (real > real) > set_real > real).
% 62.96/13.46 thf(func_def_572, type, lattic7826324295020591184_F_nat: (nat > nat > nat) > nat > set_nat > nat).
% 62.96/13.46 thf(func_def_573, type, bfun_nat_real: (nat > real) > filter_nat > $o).
% 62.96/13.46 thf(func_def_574, type, at_infinity_real: filter_real).
% 62.96/13.46 thf(func_def_575, type, append_int: list_int > list_int > list_int).
% 62.96/13.46 thf(func_def_576, type, append_nat: list_nat > list_nat > list_nat).
% 62.96/13.46 thf(func_def_577, type, concat_o: list_list_o > list_o).
% 62.96/13.46 thf(func_def_578, type, concat_nat: list_list_nat > list_nat).
% 62.96/13.46 thf(func_def_579, type, concat_VEBT_VEBT: list_list_VEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_580, type, enumerate_o: nat > list_o > list_P7333126701944960589_nat_o).
% 62.96/13.46 thf(func_def_581, type, enumerate_int: nat > list_int > list_P3521021558325789923at_int).
% 62.96/13.46 thf(func_def_582, type, enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat).
% 62.96/13.46 thf(func_def_583, type, enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 62.96/13.46 thf(func_def_584, type, find_o: ($o > $o) > list_o > option_o).
% 62.96/13.46 thf(func_def_585, type, find_int: (int > $o) > list_int > option_int).
% 62.96/13.46 thf(func_def_586, type, find_nat: (nat > $o) > list_nat > option_nat).
% 62.96/13.46 thf(func_def_587, type, find_num: (num > $o) > list_num > option_num).
% 62.96/13.46 thf(func_def_588, type, find_P8199882355184865565at_nat: (product_prod_nat_nat > $o) > list_P6011104703257516679at_nat > option4927543243414619207at_nat).
% 62.96/13.46 thf(func_def_589, type, find_real: (real > $o) > list_real > option_real).
% 62.96/13.46 thf(func_def_590, type, find_set_nat: (set_nat > $o) > list_set_nat > option_set_nat).
% 62.96/13.46 thf(func_def_591, type, find_VEBT_VEBT: (vEBT_VEBT > $o) > list_VEBT_VEBT > option_VEBT_VEBT).
% 62.96/13.46 thf(func_def_592, type, fold_nat_nat: (nat > nat > nat) > list_nat > nat > nat).
% 62.96/13.46 thf(func_def_593, type, last_nat: list_nat > nat).
% 62.96/13.46 thf(func_def_594, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 62.96/13.46 thf(func_def_595, type, cons_o: $o > list_o > list_o).
% 62.96/13.46 thf(func_def_596, type, cons_int: int > list_int > list_int).
% 62.96/13.46 thf(func_def_597, type, cons_nat: nat > list_nat > list_nat).
% 62.96/13.46 thf(func_def_598, type, cons_num: num > list_num > list_num).
% 62.96/13.46 thf(func_def_599, type, cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat).
% 62.96/13.46 thf(func_def_600, type, cons_real: real > list_real > list_real).
% 62.96/13.46 thf(func_def_601, type, cons_set_nat: set_nat > list_set_nat > list_set_nat).
% 62.96/13.46 thf(func_def_602, type, cons_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_603, type, nil_int: list_int).
% 62.96/13.46 thf(func_def_604, type, nil_nat: list_nat).
% 62.96/13.46 thf(func_def_605, type, hd_nat: list_nat > nat).
% 62.96/13.46 thf(func_def_606, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 62.96/13.46 thf(func_def_607, type, set_o2: list_o > set_o).
% 62.96/13.46 thf(func_def_608, type, set_complex2: list_complex > set_complex).
% 62.96/13.46 thf(func_def_609, type, set_Extended_enat2: list_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_610, type, set_int2: list_int > set_int).
% 62.96/13.46 thf(func_def_611, type, set_list_o2: list_list_o > set_list_o).
% 62.96/13.46 thf(func_def_612, type, set_list_nat2: list_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_613, type, set_list_VEBT_VEBT2: list_list_VEBT_VEBT > set_list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_614, type, set_nat2: list_nat > set_nat).
% 62.96/13.46 thf(func_def_615, type, set_num2: list_num > set_num).
% 62.96/13.46 thf(func_def_616, type, set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_617, type, set_real2: list_real > set_real).
% 62.96/13.46 thf(func_def_618, type, set_set_nat2: list_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_619, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 62.96/13.46 thf(func_def_620, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_621, type, list_update_o: list_o > nat > $o > list_o).
% 62.96/13.46 thf(func_def_622, type, list_update_int: list_int > nat > int > list_int).
% 62.96/13.46 thf(func_def_623, type, list_update_nat: list_nat > nat > nat > list_nat).
% 62.96/13.46 thf(func_def_624, type, list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 62.96/13.46 thf(func_def_625, type, list_update_real: list_real > nat > real > list_real).
% 62.96/13.46 thf(func_def_626, type, list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat).
% 62.96/13.46 thf(func_def_627, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_628, type, nth_o: list_o > nat > $o).
% 62.96/13.46 thf(func_def_629, type, nth_int: list_int > nat > int).
% 62.96/13.46 thf(func_def_630, type, nth_nat: list_nat > nat > nat).
% 62.96/13.46 thf(func_def_631, type, nth_num: list_num > nat > num).
% 62.96/13.46 thf(func_def_632, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 62.96/13.46 thf(func_def_633, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 62.96/13.46 thf(func_def_634, type, nth_Pr7514405829937366042_int_o: list_P5087981734274514673_int_o > nat > product_prod_int_o).
% 62.96/13.46 thf(func_def_635, type, nth_Pr4439495888332055232nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int).
% 62.96/13.46 thf(func_def_636, type, nth_Pr8617346907841251940nt_nat: list_P8198026277950538467nt_nat > nat > product_prod_int_nat).
% 62.96/13.46 thf(func_def_637, type, nth_Pr3474266648193625910T_VEBT: list_P7524865323317820941T_VEBT > nat > produc1531783533982839933T_VEBT).
% 62.96/13.46 thf(func_def_638, type, nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o).
% 62.96/13.46 thf(func_def_639, type, nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int).
% 62.96/13.46 thf(func_def_640, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_641, type, nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT).
% 62.96/13.46 thf(func_def_642, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 62.96/13.46 thf(func_def_643, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 62.96/13.46 thf(func_def_644, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 62.96/13.46 thf(func_def_645, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 62.96/13.46 thf(func_def_646, type, nth_real: list_real > nat > real).
% 62.96/13.46 thf(func_def_647, type, nth_set_nat: list_set_nat > nat > set_nat).
% 62.96/13.46 thf(func_def_648, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_649, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 62.96/13.46 thf(func_def_650, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 62.96/13.46 thf(func_def_651, type, product_int_o: list_int > list_o > list_P5087981734274514673_int_o).
% 62.96/13.46 thf(func_def_652, type, product_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 62.96/13.46 thf(func_def_653, type, product_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat).
% 62.96/13.46 thf(func_def_654, type, produc662631939642741121T_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 62.96/13.46 thf(func_def_655, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 62.96/13.46 thf(func_def_656, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 62.96/13.46 thf(func_def_657, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 62.96/13.46 thf(func_def_658, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 62.96/13.46 thf(func_def_659, type, removeAll_o: $o > list_o > list_o).
% 62.96/13.46 thf(func_def_660, type, removeAll_int: int > list_int > list_int).
% 62.96/13.46 thf(func_def_661, type, removeAll_nat: nat > list_nat > list_nat).
% 62.96/13.46 thf(func_def_662, type, remove3673390508374433037at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat).
% 62.96/13.46 thf(func_def_663, type, removeAll_real: real > list_real > list_real).
% 62.96/13.46 thf(func_def_664, type, removeAll_set_nat: set_nat > list_set_nat > list_set_nat).
% 62.96/13.46 thf(func_def_665, type, removeAll_VEBT_VEBT: vEBT_VEBT > list_VEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_666, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_667, type, rotate1_o: list_o > list_o).
% 62.96/13.46 thf(func_def_668, type, rotate1_int: list_int > list_int).
% 62.96/13.46 thf(func_def_669, type, rotate1_nat: list_nat > list_nat).
% 62.96/13.46 thf(func_def_670, type, rotate1_VEBT_VEBT: list_VEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_671, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 62.96/13.46 thf(func_def_672, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_673, type, take_nat: nat > list_nat > list_nat).
% 62.96/13.46 thf(func_def_674, type, upt: nat > nat > list_nat).
% 62.96/13.46 thf(func_def_675, type, upto: int > int > list_int).
% 62.96/13.46 thf(func_def_676, type, upto_aux: int > int > list_int > list_int).
% 62.96/13.46 thf(func_def_677, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 62.96/13.46 thf(func_def_678, type, zip_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 62.96/13.46 thf(func_def_679, type, zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 62.96/13.46 thf(func_def_680, type, zip_int_o: list_int > list_o > list_P5087981734274514673_int_o).
% 62.96/13.46 thf(func_def_681, type, zip_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 62.96/13.46 thf(func_def_682, type, zip_int_nat: list_int > list_nat > list_P8198026277950538467nt_nat).
% 62.96/13.46 thf(func_def_683, type, zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 62.96/13.46 thf(func_def_684, type, zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 62.96/13.46 thf(func_def_685, type, zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 62.96/13.46 thf(func_def_686, type, zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 62.96/13.46 thf(func_def_687, type, zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 62.96/13.46 thf(func_def_688, type, suc: nat > nat).
% 62.96/13.46 thf(func_def_689, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 62.96/13.46 thf(func_def_690, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_691, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 62.96/13.46 thf(func_def_692, type, pred: nat > nat).
% 62.96/13.46 thf(func_def_693, type, semiri4939895301339042750nteger: nat > code_integer).
% 62.96/13.46 thf(func_def_694, type, semiri8010041392384452111omplex: nat > complex).
% 62.96/13.46 thf(func_def_695, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 62.96/13.46 thf(func_def_696, type, semiri1314217659103216013at_int: nat > int).
% 62.96/13.46 thf(func_def_697, type, semiri1316708129612266289at_nat: nat > nat).
% 62.96/13.46 thf(func_def_698, type, semiri681578069525770553at_rat: nat > rat).
% 62.96/13.46 thf(func_def_699, type, semiri5074537144036343181t_real: nat > real).
% 62.96/13.46 thf(func_def_700, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 62.96/13.46 thf(func_def_701, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 62.96/13.46 thf(func_def_702, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 62.96/13.46 thf(func_def_703, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 62.96/13.46 thf(func_def_704, type, size_size_list_o: list_o > nat).
% 62.96/13.46 thf(func_def_705, type, size_s3451745648224563538omplex: list_complex > nat).
% 62.96/13.46 thf(func_def_706, type, size_s3941691890525107288d_enat: list_Extended_enat > nat).
% 62.96/13.46 thf(func_def_707, type, size_size_list_int: list_int > nat).
% 62.96/13.46 thf(func_def_708, type, size_s2710708370519433104list_o: list_list_o > nat).
% 62.96/13.46 thf(func_def_709, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 62.96/13.46 thf(func_def_710, type, size_s8217280938318005548T_VEBT: list_list_VEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_711, type, size_size_list_nat: list_nat > nat).
% 62.96/13.46 thf(func_def_712, type, size_size_list_num: list_num > nat).
% 62.96/13.46 thf(func_def_713, type, size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat).
% 62.96/13.46 thf(func_def_714, type, size_size_list_real: list_real > nat).
% 62.96/13.46 thf(func_def_715, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 62.96/13.46 thf(func_def_716, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_717, type, size_size_num: num > nat).
% 62.96/13.46 thf(func_def_718, type, size_size_option_nat: option_nat > nat).
% 62.96/13.46 thf(func_def_719, type, size_size_option_num: option_num > nat).
% 62.96/13.46 thf(func_def_720, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 62.96/13.46 thf(func_def_721, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_722, type, nat_list_decode: nat > list_nat).
% 62.96/13.46 thf(func_def_723, type, nat_list_decode_rel: nat > nat > $o).
% 62.96/13.46 thf(func_def_724, type, nat_list_encode: list_nat > nat).
% 62.96/13.46 thf(func_def_725, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 62.96/13.46 thf(func_def_726, type, nat_prod_decode: nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_727, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_728, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_729, type, nat_prod_encode: product_prod_nat_nat > nat).
% 62.96/13.46 thf(func_def_730, type, nat_set_decode: nat > set_nat).
% 62.96/13.46 thf(func_def_731, type, nat_set_encode: set_nat > nat).
% 62.96/13.46 thf(func_def_732, type, root: nat > real > real).
% 62.96/13.46 thf(func_def_733, type, sqrt: real > real).
% 62.96/13.46 thf(func_def_734, type, inc: num > num).
% 62.96/13.46 thf(func_def_735, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 62.96/13.46 thf(func_def_736, type, neg_nu6511756317524482435omplex: complex > complex).
% 62.96/13.46 thf(func_def_737, type, neg_nu3811975205180677377ec_int: int > int).
% 62.96/13.46 thf(func_def_738, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 62.96/13.46 thf(func_def_739, type, neg_nu6075765906172075777c_real: real > real).
% 62.96/13.46 thf(func_def_740, type, neg_nu8557863876264182079omplex: complex > complex).
% 62.96/13.46 thf(func_def_741, type, neg_nu5851722552734809277nc_int: int > int).
% 62.96/13.46 thf(func_def_742, type, neg_nu5219082963157363817nc_rat: rat > rat).
% 62.96/13.46 thf(func_def_743, type, neg_nu8295874005876285629c_real: real > real).
% 62.96/13.46 thf(func_def_744, type, bit0: num > num).
% 62.96/13.46 thf(func_def_745, type, bit1: num > num).
% 62.96/13.46 thf(func_def_746, type, one: num).
% 62.96/13.46 thf(func_def_747, type, size_num: num > nat).
% 62.96/13.46 thf(func_def_748, type, num_of_nat: nat > num).
% 62.96/13.46 thf(func_def_749, type, numera6620942414471956472nteger: num > code_integer).
% 62.96/13.46 thf(func_def_750, type, numera6690914467698888265omplex: num > complex).
% 62.96/13.46 thf(func_def_751, type, numera1916890842035813515d_enat: num > extended_enat).
% 62.96/13.46 thf(func_def_752, type, numeral_numeral_int: num > int).
% 62.96/13.46 thf(func_def_753, type, numeral_numeral_nat: num > nat).
% 62.96/13.46 thf(func_def_754, type, numeral_numeral_rat: num > rat).
% 62.96/13.46 thf(func_def_755, type, numeral_numeral_real: num > real).
% 62.96/13.46 thf(func_def_756, type, pred_numeral: num > nat).
% 62.96/13.46 thf(func_def_757, type, none_o: option_o).
% 62.96/13.46 thf(func_def_758, type, none_int: option_int).
% 62.96/13.46 thf(func_def_759, type, none_nat: option_nat).
% 62.96/13.46 thf(func_def_760, type, none_num: option_num).
% 62.96/13.46 thf(func_def_761, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 62.96/13.46 thf(func_def_762, type, none_real: option_real).
% 62.96/13.46 thf(func_def_763, type, none_set_nat: option_set_nat).
% 62.96/13.46 thf(func_def_764, type, none_VEBT_VEBT: option_VEBT_VEBT).
% 62.96/13.46 thf(func_def_765, type, some_o: $o > option_o).
% 62.96/13.46 thf(func_def_766, type, some_int: int > option_int).
% 62.96/13.46 thf(func_def_767, type, some_nat: nat > option_nat).
% 62.96/13.46 thf(func_def_768, type, some_num: num > option_num).
% 62.96/13.46 thf(func_def_769, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 62.96/13.46 thf(func_def_770, type, some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT).
% 62.96/13.46 thf(func_def_771, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 62.96/13.46 thf(func_def_772, type, size_option_nat: (nat > nat) > option_nat > nat).
% 62.96/13.46 thf(func_def_773, type, size_option_num: (num > nat) > option_num > nat).
% 62.96/13.46 thf(func_def_774, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 62.96/13.46 thf(func_def_775, type, the_nat: option_nat > nat).
% 62.96/13.46 thf(func_def_776, type, the_num: option_num > num).
% 62.96/13.46 thf(func_def_777, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_778, type, order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 62.96/13.46 thf(func_def_779, type, order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_780, type, bot_bot_o_o: $o > $o).
% 62.96/13.46 thf(func_def_781, type, bot_bot_int_int_o: int > int > $o).
% 62.96/13.46 thf(func_def_782, type, bot_bot_int_o: int > $o).
% 62.96/13.46 thf(func_def_783, type, bot_bot_list_nat_o: list_nat > $o).
% 62.96/13.46 thf(func_def_784, type, bot_bot_nat_nat_o: nat > nat > $o).
% 62.96/13.46 thf(func_def_785, type, bot_bot_nat_o: nat > $o).
% 62.96/13.46 thf(func_def_786, type, bot_bo5043116465536727218_nat_o: option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_787, type, bot_bot_real_o: real > $o).
% 62.96/13.46 thf(func_def_788, type, bot_bot_set_nat_o: set_nat > $o).
% 62.96/13.46 thf(func_def_789, type, bot_bo394778441745866138_nat_o: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_790, type, bot_bo3364206721330744218_nat_o: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 62.96/13.46 thf(func_def_792, type, bot_bo4199563552545308370d_enat: extended_enat).
% 62.96/13.46 thf(func_def_793, type, bot_bot_filter_nat: filter_nat).
% 62.96/13.46 thf(func_def_794, type, bot_bot_nat: nat).
% 62.96/13.46 thf(func_def_795, type, bot_bot_set_o: set_o).
% 62.96/13.46 thf(func_def_796, type, bot_bot_set_complex: set_complex).
% 62.96/13.46 thf(func_def_797, type, bot_bo7653980558646680370d_enat: set_Extended_enat).
% 62.96/13.46 thf(func_def_798, type, bot_bot_set_int: set_int).
% 62.96/13.46 thf(func_def_799, type, bot_bot_set_list_nat: set_list_nat).
% 62.96/13.46 thf(func_def_800, type, bot_bot_set_nat: set_nat).
% 62.96/13.46 thf(func_def_801, type, bot_bot_set_num: set_num).
% 62.96/13.46 thf(func_def_802, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 62.96/13.46 thf(func_def_803, type, bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_804, type, bot_bo232370072503712749on_nat: set_Pr6588086440996610945on_nat).
% 62.96/13.46 thf(func_def_805, type, bot_bo5327735625951526323at_nat: set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_806, type, bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_807, type, bot_bo4948859079157340979at_nat: set_Pr7459493094073627847at_nat).
% 62.96/13.46 thf(func_def_808, type, bot_bot_set_rat: set_rat).
% 62.96/13.46 thf(func_def_809, type, bot_bot_set_real: set_real).
% 62.96/13.46 thf(func_def_810, type, bot_bot_set_set_int: set_set_int).
% 62.96/13.46 thf(func_def_811, type, bot_bot_set_set_nat: set_set_nat).
% 62.96/13.46 thf(func_def_812, type, bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT).
% 62.96/13.46 thf(func_def_813, type, ord_Le1955565732374568822d_enat: (extended_enat > $o) > extended_enat).
% 62.96/13.46 thf(func_def_814, type, ord_Least_nat: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_815, type, ord_Least_real: (real > $o) > real).
% 62.96/13.46 thf(func_def_816, type, ord_less_o_o: ($o > $o) > ($o > $o) > $o).
% 62.96/13.46 thf(func_def_817, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 62.96/13.46 thf(func_def_818, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 62.96/13.46 thf(func_def_819, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 62.96/13.46 thf(func_def_820, type, ord_less_set_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 62.96/13.46 thf(func_def_821, type, ord_less_o: $o > $o > $o).
% 62.96/13.46 thf(func_def_822, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 62.96/13.46 thf(func_def_823, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 62.96/13.46 thf(func_def_824, type, ord_less_int: int > int > $o).
% 62.96/13.46 thf(func_def_825, type, ord_less_nat: nat > nat > $o).
% 62.96/13.46 thf(func_def_826, type, ord_less_num: num > num > $o).
% 62.96/13.46 thf(func_def_827, type, ord_less_rat: rat > rat > $o).
% 62.96/13.46 thf(func_def_828, type, ord_less_real: real > real > $o).
% 62.96/13.46 thf(func_def_829, type, ord_less_set_o: set_o > set_o > $o).
% 62.96/13.46 thf(func_def_830, type, ord_less_set_complex: set_complex > set_complex > $o).
% 62.96/13.46 thf(func_def_831, type, ord_le2529575680413868914d_enat: set_Extended_enat > set_Extended_enat > $o).
% 62.96/13.46 thf(func_def_832, type, ord_less_set_int: set_int > set_int > $o).
% 62.96/13.46 thf(func_def_833, type, ord_le1190675801316882794st_nat: set_list_nat > set_list_nat > $o).
% 62.96/13.46 thf(func_def_834, type, ord_less_set_nat: set_nat > set_nat > $o).
% 62.96/13.46 thf(func_def_835, type, ord_less_set_num: set_num > set_num > $o).
% 62.96/13.46 thf(func_def_836, type, ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_837, type, ord_le6428140832669894131at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_838, type, ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 62.96/13.46 thf(func_def_839, type, ord_less_set_rat: set_rat > set_rat > $o).
% 62.96/13.46 thf(func_def_840, type, ord_less_set_real: set_real > set_real > $o).
% 62.96/13.46 thf(func_def_841, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 62.96/13.46 thf(func_def_842, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 62.96/13.46 thf(func_def_843, type, ord_less_eq_o_o: ($o > $o) > ($o > $o) > $o).
% 62.96/13.46 thf(func_def_844, type, ord_le6741204236512500942_int_o: (int > int > $o) > (int > int > $o) > $o).
% 62.96/13.46 thf(func_def_845, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 62.96/13.46 thf(func_def_846, type, ord_le6558929396352911974_nat_o: (list_nat > list_nat > $o) > (list_nat > list_nat > $o) > $o).
% 62.96/13.46 thf(func_def_847, type, ord_le1520216061033275535_nat_o: (list_nat > $o) > (list_nat > $o) > $o).
% 62.96/13.46 thf(func_def_848, type, ord_le2646555220125990790_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 62.96/13.46 thf(func_def_849, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 62.96/13.46 thf(func_def_850, type, ord_le8905833333647802342_nat_o: (option_nat > option_nat > $o) > (option_nat > option_nat > $o) > $o).
% 62.96/13.46 thf(func_def_851, type, ord_le1598226405681992910_int_o: (product_prod_int_int > product_prod_int_int > $o) > (product_prod_int_int > product_prod_int_int > $o) > $o).
% 62.96/13.46 thf(func_def_852, type, ord_le8369615600986905444_int_o: (product_prod_int_int > $o) > (product_prod_int_int > $o) > $o).
% 62.96/13.46 thf(func_def_853, type, ord_le5604493270027003598_nat_o: (product_prod_nat_nat > product_prod_nat_nat > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > $o).
% 62.96/13.46 thf(func_def_854, type, ord_le704812498762024988_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 62.96/13.46 thf(func_def_855, type, ord_le7862513914298786254_nat_o: (produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o) > (produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o) > $o).
% 62.96/13.46 thf(func_def_856, type, ord_le8126618931240741628_nat_o: (produc4953844613479565601on_nat > $o) > (produc4953844613479565601on_nat > $o) > $o).
% 62.96/13.46 thf(func_def_857, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 62.96/13.46 thf(func_def_858, type, ord_le3964352015994296041_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 62.96/13.46 thf(func_def_859, type, ord_le3935385432712749774_nat_o: (set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o) > (set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o) > $o).
% 62.96/13.46 thf(func_def_860, type, ord_le3072208448688395470_nat_o: (set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o) > (set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o) > $o).
% 62.96/13.46 thf(func_def_861, type, ord_less_eq_o: $o > $o > $o).
% 62.96/13.46 thf(func_def_862, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 62.96/13.46 thf(func_def_863, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 62.96/13.46 thf(func_def_864, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 62.96/13.46 thf(func_def_865, type, ord_le4104064031414453916r_real: filter_real > filter_real > $o).
% 62.96/13.46 thf(func_def_866, type, ord_less_eq_int: int > int > $o).
% 62.96/13.46 thf(func_def_867, type, ord_less_eq_nat: nat > nat > $o).
% 62.96/13.46 thf(func_def_868, type, ord_less_eq_num: num > num > $o).
% 62.96/13.46 thf(func_def_869, type, ord_less_eq_rat: rat > rat > $o).
% 62.96/13.46 thf(func_def_870, type, ord_less_eq_real: real > real > $o).
% 62.96/13.46 thf(func_def_871, type, ord_less_eq_set_o: set_o > set_o > $o).
% 62.96/13.46 thf(func_def_872, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 62.96/13.46 thf(func_def_873, type, ord_le7203529160286727270d_enat: set_Extended_enat > set_Extended_enat > $o).
% 62.96/13.46 thf(func_def_874, type, ord_less_eq_set_int: set_int > set_int > $o).
% 62.96/13.46 thf(func_def_875, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 62.96/13.46 thf(func_def_876, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 62.96/13.46 thf(func_def_877, type, ord_less_eq_set_num: set_num > set_num > $o).
% 62.96/13.46 thf(func_def_878, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 62.96/13.46 thf(func_def_879, type, ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_880, type, ord_le6406482658798684961on_nat: set_Pr6588086440996610945on_nat > set_Pr6588086440996610945on_nat > $o).
% 62.96/13.46 thf(func_def_881, type, ord_le3000389064537975527at_nat: set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_882, type, ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 62.96/13.46 thf(func_def_883, type, ord_le5997549366648089703at_nat: set_Pr7459493094073627847at_nat > set_Pr7459493094073627847at_nat > $o).
% 62.96/13.46 thf(func_def_884, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 62.96/13.46 thf(func_def_885, type, ord_less_eq_set_real: set_real > set_real > $o).
% 62.96/13.46 thf(func_def_886, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 62.96/13.46 thf(func_def_887, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 62.96/13.46 thf(func_def_888, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_889, type, ord_max_Code_integer: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_890, type, ord_ma741700101516333627d_enat: extended_enat > extended_enat > extended_enat).
% 62.96/13.46 thf(func_def_891, type, ord_max_int: int > int > int).
% 62.96/13.46 thf(func_def_892, type, ord_max_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_893, type, ord_max_num: num > num > num).
% 62.96/13.46 thf(func_def_894, type, ord_max_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_895, type, ord_max_real: real > real > real).
% 62.96/13.46 thf(func_def_896, type, ord_max_set_o: set_o > set_o > set_o).
% 62.96/13.46 thf(func_def_897, type, ord_max_set_int: set_int > set_int > set_int).
% 62.96/13.46 thf(func_def_898, type, ord_max_set_nat: set_nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_899, type, ord_max_set_real: set_real > set_real > set_real).
% 62.96/13.46 thf(func_def_900, type, ord_min_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_901, type, order_Greatest_nat: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_902, type, order_9091379641038594480t_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_903, type, order_mono_nat_nat: (nat > nat) > $o).
% 62.96/13.46 thf(func_def_904, type, order_mono_nat_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_905, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 62.96/13.46 thf(func_def_906, type, order_7092887310737990675l_real: (real > real) > $o).
% 62.96/13.46 thf(func_def_907, type, ordering_top_nat: (nat > nat > $o) > (nat > nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_908, type, top_top_set_o: set_o).
% 62.96/13.46 thf(func_def_909, type, top_top_set_nat: set_nat).
% 62.96/13.46 thf(func_def_910, type, top_to1996260823553986621t_unit: set_Product_unit).
% 62.96/13.46 thf(func_def_911, type, top_top_set_real: set_real).
% 62.96/13.46 thf(func_def_912, type, top_top_set_char: set_char).
% 62.96/13.46 thf(func_def_913, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 62.96/13.46 thf(func_def_914, type, power_power_complex: complex > nat > complex).
% 62.96/13.46 thf(func_def_915, type, power_power_int: int > nat > int).
% 62.96/13.46 thf(func_def_916, type, power_power_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_917, type, power_power_rat: rat > nat > rat).
% 62.96/13.46 thf(func_def_918, type, power_power_real: real > nat > real).
% 62.96/13.46 thf(func_def_919, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 62.96/13.46 thf(func_def_920, type, produc3209952032786966637at_nat: (nat > nat > nat) > produc7248412053542808358at_nat > produc4471711990508489141at_nat).
% 62.96/13.46 thf(func_def_921, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 62.96/13.46 thf(func_def_922, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 62.96/13.46 thf(func_def_923, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 62.96/13.46 thf(func_def_924, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 62.96/13.46 thf(func_def_925, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 62.96/13.46 thf(func_def_926, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 62.96/13.46 thf(func_def_927, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 62.96/13.46 thf(func_def_928, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 62.96/13.46 thf(func_def_929, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 62.96/13.46 thf(func_def_930, type, product_Pair_int_o: int > $o > product_prod_int_o).
% 62.96/13.46 thf(func_def_931, type, product_Pair_int_int: int > int > product_prod_int_int).
% 62.96/13.46 thf(func_def_932, type, product_Pair_int_nat: int > nat > product_prod_int_nat).
% 62.96/13.46 thf(func_def_933, type, produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT).
% 62.96/13.46 thf(func_def_934, type, product_Pair_nat_o: nat > $o > product_prod_nat_o).
% 62.96/13.46 thf(func_def_935, type, product_Pair_nat_int: nat > int > product_prod_nat_int).
% 62.96/13.46 thf(func_def_936, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_937, type, produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat).
% 62.96/13.46 thf(func_def_938, type, produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT).
% 62.96/13.46 thf(func_def_939, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 62.96/13.46 thf(func_def_940, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 62.96/13.46 thf(func_def_941, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 62.96/13.46 thf(func_def_942, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 62.96/13.46 thf(func_def_943, type, produc3790773574474814305omplex: set_complex > set_complex > produc8064648209034914857omplex).
% 62.96/13.46 thf(func_def_944, type, produc6639060556116774935d_enat: set_Extended_enat > set_Extended_enat > produc1621487020699730983d_enat).
% 62.96/13.46 thf(func_def_945, type, produc6363374080413544029et_int: set_int > set_int > produc2115011035271226405et_int).
% 62.96/13.46 thf(func_def_946, type, produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat).
% 62.96/13.46 thf(func_def_947, type, produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat).
% 62.96/13.46 thf(func_def_948, type, produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat).
% 62.96/13.46 thf(func_def_949, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 62.96/13.46 thf(func_def_950, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 62.96/13.46 thf(func_def_951, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 62.96/13.46 thf(func_def_952, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 62.96/13.46 thf(func_def_953, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_954, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.96/13.46 thf(func_def_955, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 62.96/13.46 thf(func_def_956, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 62.96/13.46 thf(func_def_957, type, produc7336495610019696514er_num: (code_integer > code_integer > num) > produc8923325533196201883nteger > num).
% 62.96/13.46 thf(func_def_958, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 62.96/13.46 thf(func_def_959, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 62.96/13.46 thf(func_def_960, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 62.96/13.46 thf(func_def_961, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 62.96/13.46 thf(func_def_962, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 62.96/13.46 thf(func_def_963, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_964, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_965, type, produc2761476792215241774st_nat: (nat > nat > list_nat) > product_prod_nat_nat > list_nat).
% 62.96/13.46 thf(func_def_966, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_967, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 62.96/13.46 thf(func_def_968, type, product_fst_int_int: product_prod_int_int > int).
% 62.96/13.46 thf(func_def_969, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 62.96/13.46 thf(func_def_970, type, product_snd_int_int: product_prod_int_int > int).
% 62.96/13.46 thf(func_def_971, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 62.96/13.46 thf(func_def_972, type, product_Abs_unit: $o > product_unit).
% 62.96/13.46 thf(func_def_973, type, product_Rep_unit: product_unit > $o).
% 62.96/13.46 thf(func_def_974, type, abs_Rat: product_prod_int_int > rat).
% 62.96/13.46 thf(func_def_975, type, fract: int > int > rat).
% 62.96/13.46 thf(func_def_976, type, rep_Rat: rat > product_prod_int_int).
% 62.96/13.46 thf(func_def_977, type, field_5140801741446780682s_real: set_real).
% 62.96/13.46 thf(func_def_978, type, field_7254667332652039916t_real: rat > real).
% 62.96/13.46 thf(func_def_979, type, normalize: product_prod_int_int > product_prod_int_int).
% 62.96/13.46 thf(func_def_980, type, pcr_rat: product_prod_int_int > rat > $o).
% 62.96/13.46 thf(func_def_981, type, positive: rat > $o).
% 62.96/13.46 thf(func_def_982, type, quotient_of: rat > product_prod_int_int).
% 62.96/13.46 thf(func_def_983, type, ratrel: product_prod_int_int > product_prod_int_int > $o).
% 62.96/13.46 thf(func_def_984, type, real2: (nat > rat) > real).
% 62.96/13.46 thf(func_def_985, type, cauchy: (nat > rat) > $o).
% 62.96/13.46 thf(func_def_986, type, pcr_real: (nat > rat) > real > $o).
% 62.96/13.46 thf(func_def_987, type, positive2: real > $o).
% 62.96/13.46 thf(func_def_988, type, realrel: (nat > rat) > (nat > rat) > $o).
% 62.96/13.46 thf(func_def_989, type, rep_real: real > nat > rat).
% 62.96/13.46 thf(func_def_990, type, vanishes: (nat > rat) > $o).
% 62.96/13.46 thf(func_def_991, type, real_V3694042436643373181omplex: complex > complex > real).
% 62.96/13.46 thf(func_def_992, type, real_V975177566351809787t_real: real > real > real).
% 62.96/13.46 thf(func_def_993, type, real_V1022390504157884413omplex: complex > real).
% 62.96/13.46 thf(func_def_994, type, real_V7735802525324610683m_real: real > real).
% 62.96/13.46 thf(func_def_995, type, real_V4546457046886955230omplex: real > complex).
% 62.96/13.46 thf(func_def_996, type, field_nat: set_Pr1261947904930325089at_nat > set_nat).
% 62.96/13.46 thf(func_def_997, type, id_Pro2258643101195443293at_nat: set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_998, type, total_3592101749530773125at_nat: set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_999, type, trans_4347625901269045472at_nat: set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_1000, type, algebr932160517623751201me_int: int > int > $o).
% 62.96/13.46 thf(func_def_1001, type, algebr934650988132801477me_nat: nat > nat > $o).
% 62.96/13.46 thf(func_def_1002, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_1003, type, divide1717551699836669952omplex: complex > complex > complex).
% 62.96/13.46 thf(func_def_1004, type, divide_divide_int: int > int > int).
% 62.96/13.46 thf(func_def_1005, type, divide_divide_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_1006, type, divide_divide_rat: rat > rat > rat).
% 62.96/13.46 thf(func_def_1007, type, divide_divide_real: real > real > real).
% 62.96/13.46 thf(func_def_1008, type, dvd_dvd_int: int > int > $o).
% 62.96/13.46 thf(func_def_1009, type, dvd_dvd_nat: nat > nat > $o).
% 62.96/13.46 thf(func_def_1010, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 62.96/13.46 thf(func_def_1011, type, modulo_modulo_int: int > int > int).
% 62.96/13.46 thf(func_def_1012, type, modulo_modulo_nat: nat > nat > nat).
% 62.96/13.46 thf(func_def_1013, type, unit_f2748546683901255202or_nat: nat > nat).
% 62.96/13.46 thf(func_def_1014, type, zero_n2684676970156552555ol_int: $o > int).
% 62.96/13.46 thf(func_def_1015, type, zero_n2687167440665602831ol_nat: $o > nat).
% 62.96/13.46 thf(func_def_1016, type, suminf_complex: (nat > complex) > complex).
% 62.96/13.46 thf(func_def_1017, type, suminf_int: (nat > int) > int).
% 62.96/13.46 thf(func_def_1018, type, suminf_nat: (nat > nat) > nat).
% 62.96/13.46 thf(func_def_1019, type, suminf_real: (nat > real) > real).
% 62.96/13.46 thf(func_def_1020, type, summable_complex: (nat > complex) > $o).
% 62.96/13.46 thf(func_def_1021, type, summable_int: (nat > int) > $o).
% 62.96/13.46 thf(func_def_1022, type, summable_nat: (nat > nat) > $o).
% 62.96/13.46 thf(func_def_1023, type, summable_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_1024, type, sums_complex: (nat > complex) > complex > $o).
% 62.96/13.46 thf(func_def_1025, type, sums_int: (nat > int) > int > $o).
% 62.96/13.46 thf(func_def_1026, type, sums_nat: (nat > nat) > nat > $o).
% 62.96/13.46 thf(func_def_1027, type, sums_real: (nat > real) > real > $o).
% 62.96/13.46 thf(func_def_1028, type, collect_o: ($o > $o) > set_o).
% 62.96/13.46 thf(func_def_1029, type, collect_complex: (complex > $o) > set_complex).
% 62.96/13.46 thf(func_def_1030, type, collec4429806609662206161d_enat: (extended_enat > $o) > set_Extended_enat).
% 62.96/13.46 thf(func_def_1031, type, collect_int: (int > $o) > set_int).
% 62.96/13.46 thf(func_def_1032, type, collect_list_o: (list_o > $o) > set_list_o).
% 62.96/13.46 thf(func_def_1033, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 62.96/13.46 thf(func_def_1034, type, collec8433460942617342167d_enat: (list_Extended_enat > $o) > set_li5464603477888414924d_enat).
% 62.96/13.46 thf(func_def_1035, type, collect_list_int: (list_int > $o) > set_list_int).
% 62.96/13.46 thf(func_def_1036, type, collec5989764272469232197st_nat: (list_list_nat > $o) > set_list_list_nat).
% 62.96/13.46 thf(func_def_1037, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 62.96/13.46 thf(func_def_1038, type, collec3343600615725829874at_nat: (list_P6011104703257516679at_nat > $o) > set_li5450038453877631591at_nat).
% 62.96/13.46 thf(func_def_1039, type, collect_list_set_nat: (list_set_nat > $o) > set_list_set_nat).
% 62.96/13.46 thf(func_def_1040, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1041, type, collect_nat: (nat > $o) > set_nat).
% 62.96/13.46 thf(func_def_1042, type, collect_num: (num > $o) > set_num).
% 62.96/13.46 thf(func_def_1043, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 62.96/13.46 thf(func_def_1044, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 62.96/13.46 thf(func_def_1045, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1046, type, collec7088162979684241874at_nat: (produc859450856879609959at_nat > $o) > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_1047, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 62.96/13.46 thf(func_def_1048, type, collec6321179662152712658at_nat: (produc3843707927480180839at_nat > $o) > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1049, type, collect_rat: (rat > $o) > set_rat).
% 62.96/13.46 thf(func_def_1050, type, collect_real: (real > $o) > set_real).
% 62.96/13.46 thf(func_def_1051, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 62.96/13.46 thf(func_def_1052, type, collec2260605976452661553d_enat: (set_Extended_enat > $o) > set_se7270636423289371942d_enat).
% 62.96/13.46 thf(func_def_1053, type, collect_set_int: (set_int > $o) > set_set_int).
% 62.96/13.46 thf(func_def_1054, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 62.96/13.46 thf(func_def_1055, type, collec5514110066124741708at_nat: (set_Pr1261947904930325089at_nat > $o) > set_se7855581050983116737at_nat).
% 62.96/13.46 thf(func_def_1056, type, pow_nat: set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1057, type, image_80655429650038917d_enat: (extended_enat > extended_enat) > set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_1058, type, image_int_int: (int > int) > set_int > set_int).
% 62.96/13.46 thf(func_def_1059, type, image_int_nat: (int > nat) > set_int > set_nat).
% 62.96/13.46 thf(func_def_1060, type, image_nat_int: (nat > int) > set_nat > set_int).
% 62.96/13.46 thf(func_def_1061, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 62.96/13.46 thf(func_def_1062, type, image_nat_real: (nat > real) > set_nat > set_real).
% 62.96/13.46 thf(func_def_1063, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1064, type, image_nat_char: (nat > char) > set_nat > set_char).
% 62.96/13.46 thf(func_def_1065, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 62.96/13.46 thf(func_def_1066, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 62.96/13.46 thf(func_def_1067, type, image_real_real: (real > real) > set_real > set_real).
% 62.96/13.46 thf(func_def_1068, type, image_char_nat: (char > nat) > set_char > set_nat).
% 62.96/13.46 thf(func_def_1069, type, insert_o: $o > set_o > set_o).
% 62.96/13.46 thf(func_def_1070, type, insert_complex: complex > set_complex > set_complex).
% 62.96/13.46 thf(func_def_1071, type, insert_Extended_enat: extended_enat > set_Extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_1072, type, insert_int: int > set_int > set_int).
% 62.96/13.46 thf(func_def_1073, type, insert_list_nat: list_nat > set_list_nat > set_list_nat).
% 62.96/13.46 thf(func_def_1074, type, insert_nat: nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_1075, type, insert_num: num > set_num > set_num).
% 62.96/13.46 thf(func_def_1076, type, insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1077, type, insert5050368324300391991at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_1078, type, insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1079, type, insert_rat: rat > set_rat > set_rat).
% 62.96/13.46 thf(func_def_1080, type, insert_real: real > set_real > set_real).
% 62.96/13.46 thf(func_def_1081, type, insert_set_nat: set_nat > set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1082, type, insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1083, type, is_empty_o: set_o > $o).
% 62.96/13.46 thf(func_def_1084, type, is_empty_int: set_int > $o).
% 62.96/13.46 thf(func_def_1085, type, is_empty_nat: set_nat > $o).
% 62.96/13.46 thf(func_def_1086, type, is_empty_real: set_real > $o).
% 62.96/13.46 thf(func_def_1087, type, is_singleton_o: set_o > $o).
% 62.96/13.46 thf(func_def_1088, type, is_singleton_complex: set_complex > $o).
% 62.96/13.46 thf(func_def_1089, type, is_singleton_int: set_int > $o).
% 62.96/13.46 thf(func_def_1090, type, is_sin2641923865335537900st_nat: set_list_nat > $o).
% 62.96/13.46 thf(func_def_1091, type, is_singleton_nat: set_nat > $o).
% 62.96/13.46 thf(func_def_1092, type, is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_1093, type, is_singleton_real: set_real > $o).
% 62.96/13.46 thf(func_def_1094, type, is_singleton_set_nat: set_set_nat > $o).
% 62.96/13.46 thf(func_def_1095, type, remove_o: $o > set_o > set_o).
% 62.96/13.46 thf(func_def_1096, type, remove_int: int > set_int > set_int).
% 62.96/13.46 thf(func_def_1097, type, remove_nat: nat > set_nat > set_nat).
% 62.96/13.46 thf(func_def_1098, type, remove6466555014256735590at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1099, type, remove_real: real > set_real > set_real).
% 62.96/13.46 thf(func_def_1100, type, remove_set_nat: set_nat > set_set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1101, type, the_elem_o: set_o > $o).
% 62.96/13.46 thf(func_def_1102, type, the_elem_int: set_int > int).
% 62.96/13.46 thf(func_def_1103, type, the_elem_nat: set_nat > nat).
% 62.96/13.46 thf(func_def_1104, type, the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1105, type, the_elem_real: set_real > real).
% 62.96/13.46 thf(func_def_1106, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 62.96/13.46 thf(func_def_1107, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 62.96/13.46 thf(func_def_1108, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 62.96/13.46 thf(func_def_1109, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 62.96/13.46 thf(func_def_1110, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 62.96/13.46 thf(func_def_1111, type, set_fo3699595496184130361el_nat: produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o).
% 62.96/13.46 thf(func_def_1112, type, set_or8904488021354931149Most_o: $o > $o > set_o).
% 62.96/13.46 thf(func_def_1113, type, set_or1266510415728281911st_int: int > int > set_int).
% 62.96/13.46 thf(func_def_1114, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 62.96/13.46 thf(func_def_1115, type, set_or7049704709247886629st_num: num > num > set_num).
% 62.96/13.46 thf(func_def_1116, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 62.96/13.46 thf(func_def_1117, type, set_or1222579329274155063t_real: real > real > set_real).
% 62.96/13.46 thf(func_def_1118, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 62.96/13.46 thf(func_def_1119, type, set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1120, type, set_or4662586982721622107an_int: int > int > set_int).
% 62.96/13.46 thf(func_def_1121, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 62.96/13.46 thf(func_def_1122, type, set_ord_atLeast_nat: nat > set_nat).
% 62.96/13.46 thf(func_def_1123, type, set_ord_atLeast_real: real > set_real).
% 62.96/13.46 thf(func_def_1124, type, set_ord_atMost_int: int > set_int).
% 62.96/13.46 thf(func_def_1125, type, set_ord_atMost_nat: nat > set_nat).
% 62.96/13.46 thf(func_def_1126, type, set_or6656581121297822940st_int: int > int > set_int).
% 62.96/13.46 thf(func_def_1127, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 62.96/13.46 thf(func_def_1128, type, set_or5832277885323065728an_int: int > int > set_int).
% 62.96/13.46 thf(func_def_1129, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 62.96/13.46 thf(func_def_1130, type, set_or1633881224788618240n_real: real > real > set_real).
% 62.96/13.46 thf(func_def_1131, type, set_or1210151606488870762an_nat: nat > set_nat).
% 62.96/13.46 thf(func_def_1132, type, set_or5849166863359141190n_real: real > set_real).
% 62.96/13.46 thf(func_def_1133, type, set_ord_lessThan_o: $o > set_o).
% 62.96/13.46 thf(func_def_1134, type, set_or8419480210114673929d_enat: extended_enat > set_Extended_enat).
% 62.96/13.46 thf(func_def_1135, type, set_ord_lessThan_int: int > set_int).
% 62.96/13.46 thf(func_def_1136, type, set_ord_lessThan_nat: nat > set_nat).
% 62.96/13.46 thf(func_def_1137, type, set_ord_lessThan_num: num > set_num).
% 62.96/13.46 thf(func_def_1138, type, set_ord_lessThan_rat: rat > set_rat).
% 62.96/13.46 thf(func_def_1139, type, set_or5984915006950818249n_real: real > set_real).
% 62.96/13.46 thf(func_def_1140, type, set_or890127255671739683et_nat: set_nat > set_set_nat).
% 62.96/13.46 thf(func_def_1141, type, abort_real: literal > (product_unit > real) > real).
% 62.96/13.46 thf(func_def_1142, type, literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal).
% 62.96/13.46 thf(func_def_1143, type, comm_s629917340098488124ar_nat: char > nat).
% 62.96/13.46 thf(func_def_1144, type, unique3096191561947761185of_nat: nat > char).
% 62.96/13.46 thf(func_def_1145, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 62.96/13.46 thf(func_def_1146, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 62.96/13.46 thf(func_def_1147, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 62.96/13.46 thf(func_def_1148, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 62.96/13.46 thf(func_def_1149, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 62.96/13.46 thf(func_def_1150, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 62.96/13.46 thf(func_def_1151, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_1152, type, topolo3100542954746470799et_int: (nat > set_int) > $o).
% 62.96/13.46 thf(func_def_1153, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 62.96/13.46 thf(func_def_1154, type, topolo7531315842566124627t_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_1155, type, topolo2815343760600316023s_real: real > filter_real).
% 62.96/13.46 thf(func_def_1156, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 62.96/13.46 thf(func_def_1157, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 62.96/13.46 thf(func_def_1158, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 62.96/13.46 thf(func_def_1159, type, arccos: real > real).
% 62.96/13.46 thf(func_def_1160, type, arcosh_real: real > real).
% 62.96/13.46 thf(func_def_1161, type, arcsin: real > real).
% 62.96/13.46 thf(func_def_1162, type, arctan: real > real).
% 62.96/13.46 thf(func_def_1163, type, arsinh_real: real > real).
% 62.96/13.46 thf(func_def_1164, type, artanh_real: real > real).
% 62.96/13.46 thf(func_def_1165, type, cos_real: real > real).
% 62.96/13.46 thf(func_def_1166, type, cos_coeff: nat > real).
% 62.96/13.46 thf(func_def_1167, type, cosh_real: real > real).
% 62.96/13.46 thf(func_def_1168, type, cot_real: real > real).
% 62.96/13.46 thf(func_def_1169, type, diffs_complex: (nat > complex) > nat > complex).
% 62.96/13.46 thf(func_def_1170, type, diffs_real: (nat > real) > nat > real).
% 62.96/13.46 thf(func_def_1171, type, exp_complex: complex > complex).
% 62.96/13.46 thf(func_def_1172, type, exp_real: real > real).
% 62.96/13.46 thf(func_def_1173, type, ln_ln_real: real > real).
% 62.96/13.46 thf(func_def_1174, type, log: real > real > real).
% 62.96/13.46 thf(func_def_1175, type, pi: real).
% 62.96/13.46 thf(func_def_1176, type, powr_real: real > real > real).
% 62.96/13.46 thf(func_def_1177, type, powr_real2: real > real > real).
% 62.96/13.46 thf(func_def_1178, type, sin_real: real > real).
% 62.96/13.46 thf(func_def_1179, type, sin_coeff: nat > real).
% 62.96/13.46 thf(func_def_1180, type, sinh_complex: complex > complex).
% 62.96/13.46 thf(func_def_1181, type, sinh_real: real > real).
% 62.96/13.46 thf(func_def_1182, type, tan_real: real > real).
% 62.96/13.46 thf(func_def_1183, type, tanh_real: real > real).
% 62.96/13.46 thf(func_def_1184, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1185, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1186, type, type_d6188575255521822967unit_o: (product_unit > $o) > ($o > product_unit) > set_o > $o).
% 62.96/13.46 thf(func_def_1187, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1188, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1189, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1190, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1191, type, vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1192, type, vEBT_VEBT_high: nat > nat > nat).
% 62.96/13.46 thf(func_def_1193, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1194, type, vEBT_VEBT_low: nat > nat > nat).
% 62.96/13.46 thf(func_def_1195, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1196, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1197, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1198, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1199, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1200, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1201, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1202, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 62.96/13.46 thf(func_def_1203, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1204, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 62.96/13.46 thf(func_def_1205, type, vEBT_vebt_delete: vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1206, type, vEBT_vebt_delete_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1207, type, vEBT_VEBT_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1208, type, vEBT_VEBT_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1209, type, vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1210, type, vEBT_vebt_insert_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1211, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 62.96/13.46 thf(func_def_1212, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1213, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1214, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 62.96/13.46 thf(func_def_1215, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1216, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1217, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 62.96/13.46 thf(func_def_1218, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1219, type, vEBT_V5711637165310795180er_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o).
% 62.96/13.46 thf(func_def_1220, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1221, type, vEBT_VEBT_less_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o).
% 62.96/13.46 thf(func_def_1222, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1223, type, vEBT_VEBT_lesseq_rel: produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o).
% 62.96/13.46 thf(func_def_1224, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 62.96/13.46 thf(func_def_1225, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 62.96/13.46 thf(func_def_1226, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 62.96/13.46 thf(func_def_1227, type, vEBT_V2881884560877996034ft_nat: (nat > nat > $o) > option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1228, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 62.96/13.46 thf(func_def_1229, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 62.96/13.46 thf(func_def_1230, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 62.96/13.46 thf(func_def_1231, type, vEBT_V3895251965096974666el_nat: produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o).
% 62.96/13.46 thf(func_def_1232, type, vEBT_V452583751252753300el_num: produc1193250871479095198on_num > produc1193250871479095198on_num > $o).
% 62.96/13.46 thf(func_def_1233, type, vEBT_V7235779383477046023at_nat: produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o).
% 62.96/13.46 thf(func_def_1234, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 62.96/13.46 thf(func_def_1235, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 62.96/13.46 thf(func_def_1236, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1237, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 62.96/13.46 thf(func_def_1238, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1239, type, vEBT_is_pred_in_set: set_nat > nat > nat > $o).
% 62.96/13.46 thf(func_def_1240, type, vEBT_vebt_pred: vEBT_VEBT > nat > option_nat).
% 62.96/13.46 thf(func_def_1241, type, vEBT_vebt_pred_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1242, type, vEBT_is_succ_in_set: set_nat > nat > nat > $o).
% 62.96/13.46 thf(func_def_1243, type, vEBT_vebt_succ: vEBT_VEBT > nat > option_nat).
% 62.96/13.46 thf(func_def_1244, type, vEBT_vebt_succ_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1245, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 62.96/13.46 thf(func_def_1246, type, accp_nat: (nat > nat > $o) > nat > $o).
% 62.96/13.46 thf(func_def_1247, type, accp_P6019419558468335806at_nat: (produc4471711990508489141at_nat > produc4471711990508489141at_nat > $o) > produc4471711990508489141at_nat > $o).
% 62.96/13.46 thf(func_def_1248, type, accp_P5496254298877145759on_nat: (produc8306885398267862888on_nat > produc8306885398267862888on_nat > $o) > produc8306885398267862888on_nat > $o).
% 62.96/13.46 thf(func_def_1249, type, accp_P7605991808943153877on_num: (produc1193250871479095198on_num > produc1193250871479095198on_num > $o) > produc1193250871479095198on_num > $o).
% 62.96/13.46 thf(func_def_1250, type, accp_P3267385326087170368at_nat: (produc5542196010084753463at_nat > produc5542196010084753463at_nat > $o) > produc5542196010084753463at_nat > $o).
% 62.96/13.46 thf(func_def_1251, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 62.96/13.46 thf(func_def_1252, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 62.96/13.46 thf(func_def_1253, type, accp_P8646395344606611882on_nat: (produc4953844613479565601on_nat > produc4953844613479565601on_nat > $o) > produc4953844613479565601on_nat > $o).
% 62.96/13.46 thf(func_def_1254, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 62.96/13.46 thf(func_def_1255, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1256, type, finite8643634255014194347omplex: set_Pr6308028481084910985omplex).
% 62.96/13.46 thf(func_def_1257, type, finite4251489430341359785d_enat: set_Pr2112562347474612743d_enat).
% 62.96/13.46 thf(func_def_1258, type, finite_psubset_int: set_Pr2522554150109002629et_int).
% 62.96/13.46 thf(func_def_1259, type, finite_psubset_nat: set_Pr5488025237498180813et_nat).
% 62.96/13.46 thf(func_def_1260, type, finite469560695537375940at_nat: set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1261, type, finite4695646753290404266at_nat: set_Pr7459493094073627847at_nat).
% 62.96/13.46 thf(func_def_1262, type, less_than: set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1263, type, lex_prod_nat_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr8693737435421807431at_nat).
% 62.96/13.46 thf(func_def_1264, type, max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1265, type, measure_int: (int > nat) > set_Pr958786334691620121nt_int).
% 62.96/13.46 thf(func_def_1266, type, measure_nat: (nat > nat) > set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1267, type, measure_option_nat: (option_nat > nat) > set_Pr6588086440996610945on_nat).
% 62.96/13.46 thf(func_def_1268, type, measur1827424007717751593at_nat: (set_Pr1261947904930325089at_nat > nat) > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1269, type, measur4922264856574889999at_nat: (set_Pr4329608150637261639at_nat > nat) > set_Pr7459493094073627847at_nat).
% 62.96/13.46 thf(func_def_1270, type, min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 62.96/13.46 thf(func_def_1271, type, pred_nat: set_Pr1261947904930325089at_nat).
% 62.96/13.46 thf(func_def_1272, type, wf_nat: set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_1273, type, wf_Pro7803398752247294826at_nat: set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_1274, type, fChoice_real: (real > $o) > real).
% 62.96/13.46 thf(func_def_1275, type, member_o: $o > set_o > $o).
% 62.96/13.46 thf(func_def_1276, type, member_Code_integer: code_integer > set_Code_integer > $o).
% 62.96/13.46 thf(func_def_1277, type, member_complex: complex > set_complex > $o).
% 62.96/13.46 thf(func_def_1278, type, member_Extended_enat: extended_enat > set_Extended_enat > $o).
% 62.96/13.46 thf(func_def_1279, type, member_int: int > set_int > $o).
% 62.96/13.46 thf(func_def_1280, type, member_list_o: list_o > set_list_o > $o).
% 62.96/13.46 thf(func_def_1281, type, member_list_nat: list_nat > set_list_nat > $o).
% 62.96/13.46 thf(func_def_1282, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1283, type, member_nat: nat > set_nat > $o).
% 62.96/13.46 thf(func_def_1284, type, member_num: num > set_num > $o).
% 62.96/13.46 thf(func_def_1285, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 62.96/13.46 thf(func_def_1286, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 62.96/13.46 thf(func_def_1287, type, member4117937158525611210on_nat: produc4953844613479565601on_nat > set_Pr6588086440996610945on_nat > $o).
% 62.96/13.46 thf(func_def_1288, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 62.96/13.46 thf(func_def_1289, type, member351165363924911826omplex: produc8064648209034914857omplex > set_Pr6308028481084910985omplex > $o).
% 62.96/13.46 thf(func_def_1290, type, member4453595087596390480d_enat: produc1621487020699730983d_enat > set_Pr2112562347474612743d_enat > $o).
% 62.96/13.46 thf(func_def_1291, type, member2572552093476627150et_int: produc2115011035271226405et_int > set_Pr2522554150109002629et_int > $o).
% 62.96/13.46 thf(func_def_1292, type, member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o).
% 62.96/13.46 thf(func_def_1293, type, member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o).
% 62.96/13.46 thf(func_def_1294, type, member1466754251312161552at_nat: produc1319942482725812455at_nat > set_Pr7459493094073627847at_nat > $o).
% 62.96/13.46 thf(func_def_1295, type, member_rat: rat > set_rat > $o).
% 62.96/13.46 thf(func_def_1296, type, member_real: real > set_real > $o).
% 62.96/13.46 thf(func_def_1297, type, member_set_int: set_int > set_set_int > $o).
% 62.96/13.46 thf(func_def_1298, type, member_set_nat: set_nat > set_set_nat > $o).
% 62.96/13.46 thf(func_def_1299, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1300, type, a: nat).
% 62.96/13.46 thf(func_def_1301, type, b: nat).
% 62.96/13.46 thf(func_def_1302, type, deg: nat).
% 62.96/13.46 thf(func_def_1303, type, h: nat).
% 62.96/13.46 thf(func_def_1304, type, info: option4927543243414619207at_nat).
% 62.96/13.46 thf(func_def_1305, type, k: vEBT_VEBT).
% 62.96/13.46 thf(func_def_1306, type, m: nat).
% 62.96/13.46 thf(func_def_1307, type, ma: nat).
% 62.96/13.46 thf(func_def_1308, type, mi: nat).
% 62.96/13.46 thf(func_def_1309, type, na: nat).
% 62.96/13.46 thf(func_def_1310, type, sa: vEBT_VEBT).
% 62.96/13.46 thf(func_def_1311, type, summary: vEBT_VEBT).
% 62.96/13.46 thf(func_def_1312, type, summary2: vEBT_VEBT).
% 62.96/13.46 thf(func_def_1313, type, ta: vEBT_VEBT).
% 62.96/13.46 thf(func_def_1314, type, treeList: list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1315, type, treeList2: list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1334, type, sP0: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1335, type, sP1: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1336, type, sP2: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1337, type, sP3: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1338, type, sP4: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1339, type, sP5: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1340, type, sP6: (option_nat > option_nat > $o) > option_nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1341, type, sP7: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1342, type, sP8: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1343, type, sP9: vEBT_VEBT > nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1344, type, sP10: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1345, type, sP11: nat > nat > $o).
% 62.96/13.46 thf(func_def_1346, type, sP12: nat > nat > $o).
% 62.96/13.46 thf(func_def_1349, type, sP15: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1350, type, sP16: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1351, type, sP17: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1352, type, sP18: option_nat > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1353, type, sP19: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1354, type, sP20: vEBT_VEBT > nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1355, type, sP21: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1356, type, sP22: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1357, type, sP23: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1358, type, sP24: option_nat > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1359, type, sP25: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1360, type, sP26: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1361, type, sP27: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1362, type, sP28: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1363, type, sP29: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1364, type, sP30: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1365, type, sP31: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1366, type, sP32: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1367, type, sP33: option_nat > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1368, type, sP34: vEBT_VEBT > nat > option_nat > $o).
% 62.96/13.46 thf(func_def_1369, type, sP35: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1370, type, sK36: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1371, type, sK37: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1372, type, sK38: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1373, type, sK39: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1374, type, sK40: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1375, type, sK41: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1376, type, sK42: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1377, type, sK43: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1378, type, sK44: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1379, type, sK45: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1380, type, sK46: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1381, type, sK47: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1382, type, sK48: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1383, type, sK49: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1384, type, sK50: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1385, type, sK51: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1386, type, sK52: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1387, type, sK53: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1388, type, sK54: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1389, type, sK55: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1390, type, sK56: vEBT_VEBT > nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1391, type, sK57: vEBT_VEBT > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1392, type, sK58: vEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1393, type, sK59: option_nat > nat).
% 62.96/13.46 thf(func_def_1394, type, sK60: (option_nat > option_nat > $o) > option_nat > option_nat > nat).
% 62.96/13.46 thf(func_def_1395, type, sK61: (option_nat > option_nat > $o) > option_nat > option_nat > nat).
% 62.96/13.46 thf(func_def_1396, type, sK62: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_1397, type, sK63: nat > nat).
% 62.96/13.46 thf(func_def_1398, type, sK64: nat > nat).
% 62.96/13.46 thf(func_def_1399, type, sK65: vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1400, type, sK66: vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1401, type, sK67: vEBT_VEBT > nat > option_nat > nat).
% 62.96/13.46 thf(func_def_1402, type, sK68: vEBT_VEBT > nat > option_nat > nat).
% 62.96/13.46 thf(func_def_1403, type, sK69: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1404, type, sK70: vEBT_VEBT > nat > option_nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1405, type, sK71: vEBT_VEBT > nat > option_nat > nat).
% 62.96/13.46 thf(func_def_1406, type, sK72: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1407, type, sK73: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1408, type, sK74: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1409, type, sK75: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1410, type, sK76: nat > nat).
% 62.96/13.46 thf(func_def_1411, type, sK77: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1412, type, sK78: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1413, type, sK79: vEBT_VEBT > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1414, type, sK80: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1415, type, sK81: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1416, type, sK82: vEBT_VEBT > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1417, type, sK83: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1418, type, sK84: vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1419, type, sK85: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1420, type, sK86: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_1421, type, sK87: nat > nat).
% 62.96/13.46 thf(func_def_1422, type, sK88: nat).
% 62.96/13.46 thf(func_def_1423, type, sK89: nat).
% 62.96/13.46 thf(func_def_1424, type, sK90: vEBT_VEBT > nat > option_nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1425, type, sK91: vEBT_VEBT > nat > option_nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1426, type, sK92: vEBT_VEBT > nat > option_nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1427, type, sK93: vEBT_VEBT > option_nat > nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1428, type, sK94: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1429, type, sK95: vEBT_VEBT > option_nat > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1430, type, sK96: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1431, type, sK97: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1432, type, sK98: option_nat > nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1433, type, sK99: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1434, type, sK100: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1435, type, sK101: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1436, type, sK102: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1437, type, sK103: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1438, type, sK104: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1439, type, sK105: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1440, type, sK106: vEBT_VEBT > option_nat > nat > $o).
% 62.96/13.46 thf(func_def_1441, type, sK107: vEBT_VEBT > option_nat > nat > nat).
% 62.96/13.46 thf(func_def_1442, type, sK108: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1443, type, sK109: option_nat > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1444, type, sK110: option_nat > vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1445, type, sK111: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_1446, type, sK112: nat > option_nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1447, type, sK113: nat > option_nat > vEBT_VEBT > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1448, type, sK114: nat > option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1449, type, sK115: option_nat > vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1450, type, sK116: option_nat > vEBT_VEBT > nat > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1451, type, sK117: option_nat > vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1452, type, sK118: vEBT_VEBT > option_nat > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1453, type, sK119: vEBT_VEBT > option_nat > nat > nat).
% 62.96/13.46 thf(func_def_1454, type, sK120: vEBT_VEBT > option_nat > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1455, type, sK121: vEBT_VEBT > option_nat > nat > nat).
% 62.96/13.46 thf(func_def_1456, type, sK122: vEBT_VEBT > option_nat > nat > nat).
% 62.96/13.46 thf(func_def_1457, type, sK123: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1458, type, sK124: nat > option_nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1459, type, sK125: nat > option_nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1460, type, sK126: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1461, type, sK127: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1462, type, sK128: nat > vEBT_VEBT > option_nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1463, type, sK129: nat > vEBT_VEBT > option_nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1464, type, sK130: nat > vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1465, type, sK131: nat > nat).
% 62.96/13.46 thf(func_def_1466, type, sK132: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1467, type, sK133: vEBT_VEBT > nat > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1468, type, sK134: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1469, type, sK135: nat > vEBT_VEBT > vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1470, type, sK136: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1471, type, sK137: vEBT_VEBT > vEBT_VEBT > nat > $o).
% 62.96/13.46 thf(func_def_1472, type, sK138: nat > nat).
% 62.96/13.46 thf(func_def_1473, type, sK139: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1474, type, sK140: vEBT_VEBT > vEBT_VEBT > nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1475, type, sK141: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1476, type, sK142: vEBT_VEBT > vEBT_VEBT > nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1477, type, sK143: vEBT_VEBT > vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1478, type, sK144: vEBT_VEBT > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1479, type, sK145: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1480, type, sK146: vEBT_VEBT > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1481, type, sK147: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1482, type, sK148: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1483, type, sK149: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1484, type, sK150: vEBT_VEBT > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1485, type, sK151: vEBT_VEBT > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1486, type, sK152: vEBT_VEBT > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1487, type, sK153: vEBT_VEBT > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1488, type, sK154: vEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1489, type, sK155: nat > nat).
% 62.96/13.46 thf(func_def_1490, type, sK156: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1491, type, sK157: vEBT_VEBT > option_nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1492, type, sK158: vEBT_VEBT > option_nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1493, type, sK159: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1494, type, sK160: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1495, type, sK161: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1496, type, sK162: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1497, type, sK163: option_nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1498, type, sK164: option_nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1499, type, sK165: option_nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1500, type, sK166: vEBT_VEBT > nat > nat).
% 62.96/13.46 thf(func_def_1501, type, sK167: (nat > nat > $o) > nat).
% 62.96/13.46 thf(func_def_1502, type, sK168: (nat > nat > $o) > nat).
% 62.96/13.46 thf(func_def_1503, type, sK169: (nat > nat > $o) > nat).
% 62.96/13.46 thf(func_def_1504, type, sK170: (nat > nat > $o) > nat).
% 62.96/13.46 thf(func_def_1505, type, sK171: (nat > nat > $o) > nat).
% 62.96/13.46 thf(func_def_1506, type, sK172: nat > nat).
% 62.96/13.46 thf(func_def_1507, type, sK173: nat > nat).
% 62.96/13.46 thf(func_def_1508, type, sK174: vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1509, type, sK175: vEBT_VEBT > $o).
% 62.96/13.46 thf(func_def_1510, type, sK176: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1511, type, sK177: option_nat > nat > vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1512, type, sK178: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1513, type, sK179: option_nat > nat > vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1514, type, sK180: option_nat > nat > vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1515, type, sK181: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1516, type, sK182: nat > vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1517, type, sK183: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1518, type, sK184: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1519, type, sK185: vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1520, type, sK186: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1521, type, sK187: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1522, type, sK188: vEBT_VEBT > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1523, type, sK189: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1524, type, sK190: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1525, type, sK191: vEBT_VEBT > product_prod_nat_nat).
% 62.96/13.46 thf(func_def_1526, type, sK192: nat > nat).
% 62.96/13.46 thf(func_def_1527, type, sK193: nat > nat).
% 62.96/13.46 thf(func_def_1528, type, sK194: nat > nat).
% 62.96/13.46 thf(func_def_1529, type, sK195: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1530, type, sK196: vEBT_VEBT > option_nat > $o).
% 62.96/13.46 thf(func_def_1531, type, sK197: vEBT_VEBT > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1532, type, sK198: vEBT_VEBT > nat).
% 62.96/13.46 thf(func_def_1533, type, sK199: vEBT_VEBT > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1534, type, sK200: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1535, type, sK201: vEBT_VEBT > option_nat > vEBT_VEBT).
% 62.96/13.46 thf(func_def_1536, type, sK202: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1537, type, sK203: vEBT_VEBT > option_nat > list_VEBT_VEBT).
% 62.96/13.46 thf(func_def_1538, type, sK204: vEBT_VEBT > option_nat > nat).
% 62.96/13.46 thf(func_def_1539, type, sK205: (nat > $o) > nat).
% 62.96/13.46 thf(func_def_1541, type, sK207: option_nat > nat).
% 62.96/13.46 thf(func_def_1542, type, sK208: option_nat > nat).
% 62.96/13.46 thf(func_def_1543, type, ph209: !>[X0: $tType]:(X0)).
% 62.96/13.46 thf(func_def_1544, type, sK210: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1545, type, sK211: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1546, type, sK212: (option_nat > $o) > nat).
% 62.96/13.46 thf(func_def_1547, type, sK213: (option_nat > $o) > nat).
% 62.96/13.46 thf(func_def_1548, type, sK214: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1549, type, sK215: option_nat > $o).
% 62.96/13.46 thf(func_def_1550, type, sK216: nat).
% 62.96/13.46 thf(func_def_1551, type, sK217: option_nat).
% 62.96/13.46 thf(func_def_1552, type, sK218: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1553, type, sK219: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1554, type, sK220: (option_nat > $o) > option_nat).
% 62.96/13.46 thf(func_def_1555, type, sK221: (option_nat > $o) > nat).
% 62.96/13.46 thf(f28214,plain,(
% 62.96/13.46 $false),
% 62.96/13.46 inference(avatar_sat_refutation,[status(thm)],[f27114,f27147,f27301,f27478,f28213])).
% 62.96/13.46 thf(f28213,plain,(
% 62.96/13.46 ~spl206_4 | ~spl206_12 | ~spl206_16),
% 62.96/13.46 inference(avatar_contradiction_clause,[status(thm)],[f28212])).
% 62.96/13.46 thf(f28212,plain,(
% 62.96/13.46 $false | (~spl206_4 | ~spl206_12 | ~spl206_16)),
% 62.96/13.46 inference(subsumption_resolution,[status(thm)],[f28211,f27150])).
% 62.96/13.46 thf(f27150,plain,(
% 62.96/13.46 ((some_nat @ b) != (some_nat @ a)) | (~spl206_4 | ~spl206_12)),
% 62.96/13.46 inference(backward_demodulation,[status(thm)],[f27149,f27087])).
% 62.96/13.46 thf(f27087,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) = (some_nat @ a)) | ~spl206_4),
% 62.96/13.46 inference(avatar_component_clause,[status(thm)],[f27085])).
% 62.96/13.46 thf(f27085,definition,(
% 62.96/13.46 spl206_4 <=> ((vEBT_vebt_mint @ ta) = (some_nat @ a))),
% 62.96/13.46 introduced(definition,[new_symbols(naming,[spl206_4])],[avatar_definition])).
% 62.96/13.46 thf(f27149,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) != (some_nat @ b)) | ~spl206_12),
% 62.96/13.46 inference(forward_demodulation,[status(thm)],[f26690,f27128])).
% 62.96/13.46 thf(f27128,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ k) = (some_nat @ b)) | ~spl206_12),
% 62.96/13.46 inference(avatar_component_clause,[status(thm)],[f27127])).
% 62.96/13.46 thf(f27127,definition,(
% 62.96/13.46 spl206_12 <=> ((vEBT_vebt_mint @ k) = (some_nat @ b))),
% 62.96/13.46 introduced(definition,[new_symbols(naming,[spl206_12])],[avatar_definition])).
% 62.96/13.46 thf(f26690,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) != (vEBT_vebt_mint @ k))),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f1])).
% 62.96/13.46 thf(f1,axiom,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) != (vEBT_vebt_mint @ k))),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_0__092_060open_062vebt__mint_At_A_092_060noteq_062_Avebt__mint_Ak_092_060close_062)).
% 62.96/13.46 thf(f28211,plain,(
% 62.96/13.46 ((some_nat @ b) = (some_nat @ a)) | (~spl206_12 | ~spl206_16)),
% 62.96/13.46 inference(trivial_inequality_removal,[status(thm)],[f28210])).
% 62.96/13.46 thf(f28210,plain,(
% 62.96/13.46 ((some_nat @ b) = (some_nat @ a)) | ($true != $true) | (~spl206_12 | ~spl206_16)),
% 62.96/13.46 inference(superposition,[status(thm)],[f27300,f27737])).
% 62.96/13.46 thf(f27737,plain,(
% 62.96/13.46 ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ b) = $true) | ~spl206_12),
% 62.96/13.46 inference(equality_resolution,[status(thm)],[f27306])).
% 62.96/13.46 thf(f27306,plain,(
% 62.96/13.46 ( ! [X0 : nat] : (((some_nat @ b) != (some_nat @ X0)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) = $true)) ) | ~spl206_12),
% 62.96/13.46 inference(forward_demodulation,[status(thm)],[f27305,f27128])).
% 62.96/13.46 thf(f27305,plain,(
% 62.96/13.46 ( ! [X0 : nat] : (((vEBT_vebt_mint @ k) != (some_nat @ X0)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) = $true)) )),
% 62.96/13.46 inference(trivial_inequality_removal,[status(thm)],[f27303])).
% 62.96/13.46 thf(f27303,plain,(
% 62.96/13.46 ( ! [X0 : nat] : (((vEBT_vebt_mint @ k) != (some_nat @ X0)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) = $true) | ($true != $true)) )),
% 62.96/13.46 inference(superposition,[status(thm)],[f26895,f26937])).
% 62.96/13.46 thf(f26937,plain,(
% 62.96/13.46 ((vEBT_invar_vebt @ k @ h) = $true)),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f16969])).
% 62.96/13.46 thf(f16969,plain,(
% 62.96/13.46 ((vEBT_invar_vebt @ k @ h) = $true)),
% 62.96/13.46 inference(fool_elimination,[status(thm)],[f16968])).
% 62.96/13.46 thf(f16968,plain,(
% 62.96/13.46 (vEBT_invar_vebt @ k @ h)),
% 62.96/13.46 inference(rectify,[status(thm)],[f18])).
% 62.96/13.46 thf(f18,axiom,(
% 62.96/13.46 (vEBT_invar_vebt @ k @ h)),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_17_assms_I2_J)).
% 62.96/13.46 thf(f26895,plain,(
% 62.96/13.46 ( ! [X2 : vEBT_VEBT,X0 : nat,X1 : nat] : (((vEBT_invar_vebt @ X2 @ X1) != $true) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X0) = $true) | ((some_nat @ X0) != (vEBT_vebt_mint @ X2))) )),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f26276])).
% 62.96/13.46 thf(f26276,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : (((some_nat @ X0) != (vEBT_vebt_mint @ X2)) | ((vEBT_invar_vebt @ X2 @ X1) != $true) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X0) = $true))),
% 62.96/13.46 inference(flattening,[status(thm)],[f26275])).
% 62.96/13.46 thf(f26275,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : ((((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X0) = $true) | ((some_nat @ X0) != (vEBT_vebt_mint @ X2))) | ((vEBT_invar_vebt @ X2 @ X1) != $true))),
% 62.96/13.46 inference(ennf_transformation,[status(thm)],[f11441])).
% 62.96/13.46 thf(f11441,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : (((vEBT_invar_vebt @ X2 @ X1) = $true) => (((some_nat @ X0) = (vEBT_vebt_mint @ X2)) => ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X0) = $true)))),
% 62.96/13.46 inference(fool_elimination,[status(thm)],[f11440])).
% 62.96/13.46 thf(f11440,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : ((vEBT_invar_vebt @ X2 @ X1) => (((some_nat @ X0) = (vEBT_vebt_mint @ X2)) => (vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X0)))),
% 62.96/13.46 inference(rectify,[status(thm)],[f31])).
% 62.96/13.46 thf(f31,axiom,(
% 62.96/13.46 ! [X4 : nat,X11 : nat,X5 : vEBT_VEBT] : ((vEBT_invar_vebt @ X5 @ X11) => (((vEBT_vebt_mint @ X5) = (some_nat @ X4)) => (vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X5) @ X4)))),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_30_mint__corr)).
% 62.96/13.46 thf(f27300,plain,(
% 62.96/13.46 ( ! [X0 : nat] : (((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) != $true) | ((some_nat @ a) = (some_nat @ X0))) ) | ~spl206_16),
% 62.96/13.46 inference(avatar_component_clause,[status(thm)],[f27299])).
% 62.96/13.46 thf(f27299,definition,(
% 62.96/13.46 spl206_16 <=> ! [X0 : nat] : (((some_nat @ a) = (some_nat @ X0)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) != $true))),
% 62.96/13.46 introduced(definition,[new_symbols(naming,[spl206_16])],[avatar_definition])).
% 62.96/13.46 thf(f27478,plain,(
% 62.96/13.46 ~spl206_15),
% 62.96/13.46 inference(avatar_contradiction_clause,[status(thm)],[f27477])).
% 62.96/13.46 thf(f27477,plain,(
% 62.96/13.46 $false | ~spl206_15),
% 62.96/13.46 inference(trivial_inequality_removal,[status(thm)],[f27476])).
% 62.96/13.46 thf(f27476,plain,(
% 62.96/13.46 ($true != $true) | ~spl206_15),
% 62.96/13.46 inference(superposition,[status(thm)],[f27297,f26740])).
% 62.96/13.46 thf(f26740,plain,(
% 62.96/13.46 ((vEBT_invar_vebt @ ta @ h) = $true)),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f23888])).
% 62.96/13.46 thf(f23888,plain,(
% 62.96/13.46 ((vEBT_invar_vebt @ ta @ h) = $true)),
% 62.96/13.46 inference(fool_elimination,[status(thm)],[f23887])).
% 62.96/13.46 thf(f23887,plain,(
% 62.96/13.46 (vEBT_invar_vebt @ ta @ h)),
% 62.96/13.46 inference(rectify,[status(thm)],[f19])).
% 62.96/13.46 thf(f19,axiom,(
% 62.96/13.46 (vEBT_invar_vebt @ ta @ h)),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_18_assms_I1_J)).
% 62.96/13.46 thf(f27297,plain,(
% 62.96/13.46 ( ! [X1 : nat] : (((vEBT_invar_vebt @ ta @ X1) != $true)) ) | ~spl206_15),
% 62.96/13.46 inference(avatar_component_clause,[status(thm)],[f27296])).
% 62.96/13.46 thf(f27296,definition,(
% 62.96/13.46 spl206_15 <=> ! [X1 : nat] : ((vEBT_invar_vebt @ ta @ X1) != $true)),
% 62.96/13.46 introduced(definition,[new_symbols(naming,[spl206_15])],[avatar_definition])).
% 62.96/13.46 thf(f27301,plain,(
% 62.96/13.46 spl206_15 | spl206_16 | ~spl206_4),
% 62.96/13.46 inference(avatar_split_clause,[status(thm)],[f27294,f27085,f27299,f27296])).
% 62.96/13.46 thf(f27294,plain,(
% 62.96/13.46 ( ! [X0 : nat,X1 : nat] : (((some_nat @ a) = (some_nat @ X0)) | ((vEBT_invar_vebt @ ta @ X1) != $true) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) != $true)) ) | ~spl206_4),
% 62.96/13.46 inference(forward_demodulation,[status(thm)],[f27293,f27087])).
% 62.96/13.46 thf(f27293,plain,(
% 62.96/13.46 ( ! [X0 : nat,X1 : nat] : (((vEBT_invar_vebt @ ta @ X1) != $true) | ((vEBT_vebt_mint @ ta) = (some_nat @ X0)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ k) @ X0) != $true)) )),
% 62.96/13.46 inference(superposition,[status(thm)],[f26700,f26683])).
% 62.96/13.46 thf(f26683,plain,(
% 62.96/13.46 ((vEBT_VEBT_set_vebt @ ta) = (vEBT_VEBT_set_vebt @ k))),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f3])).
% 62.96/13.46 thf(f3,axiom,(
% 62.96/13.46 ((vEBT_VEBT_set_vebt @ ta) = (vEBT_VEBT_set_vebt @ k))),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_2_assms_I3_J)).
% 62.96/13.46 thf(f26700,plain,(
% 62.96/13.46 ( ! [X2 : vEBT_VEBT,X0 : nat,X1 : nat] : (((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X1) != $true) | ((vEBT_invar_vebt @ X2 @ X0) != $true) | ((some_nat @ X1) = (vEBT_vebt_mint @ X2))) )),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f26374])).
% 62.96/13.46 thf(f26374,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : (((some_nat @ X1) = (vEBT_vebt_mint @ X2)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X1) != $true) | ((vEBT_invar_vebt @ X2 @ X0) != $true))),
% 62.96/13.46 inference(flattening,[status(thm)],[f26373])).
% 62.96/13.46 thf(f26373,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : ((((some_nat @ X1) = (vEBT_vebt_mint @ X2)) | ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X1) != $true)) | ((vEBT_invar_vebt @ X2 @ X0) != $true))),
% 62.96/13.46 inference(ennf_transformation,[status(thm)],[f25750])).
% 62.96/13.46 thf(f25750,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : (((vEBT_invar_vebt @ X2 @ X0) = $true) => (((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X1) = $true) => ((some_nat @ X1) = (vEBT_vebt_mint @ X2))))),
% 62.96/13.46 inference(fool_elimination,[status(thm)],[f25749])).
% 62.96/13.46 thf(f25749,plain,(
% 62.96/13.46 ! [X0 : nat,X1 : nat,X2 : vEBT_VEBT] : ((vEBT_invar_vebt @ X2 @ X0) => ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X2) @ X1) => ((some_nat @ X1) = (vEBT_vebt_mint @ X2))))),
% 62.96/13.46 inference(rectify,[status(thm)],[f30])).
% 62.96/13.46 thf(f30,axiom,(
% 62.96/13.46 ! [X11 : nat,X4 : nat,X5 : vEBT_VEBT] : ((vEBT_invar_vebt @ X5 @ X11) => ((vEBT_VEBT_min_in_set @ (vEBT_VEBT_set_vebt @ X5) @ X4) => ((vEBT_vebt_mint @ X5) = (some_nat @ X4))))),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',fact_29_mint__sound)).
% 62.96/13.46 thf(f27147,plain,(
% 62.96/13.46 spl206_12),
% 62.96/13.46 inference(avatar_split_clause,[status(thm)],[f26695,f27127])).
% 62.96/13.46 thf(f26695,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ k) = (some_nat @ b))),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f11385])).
% 62.96/13.46 thf(f11385,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) = (some_nat @ a)) & ((ord_less_nat @ a @ b) = $true) & ((vEBT_vebt_mint @ k) = (some_nat @ b))),
% 62.96/13.46 inference(fool_elimination,[status(thm)],[f11384])).
% 62.96/13.46 thf(f11384,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) = (some_nat @ a)) & ((vEBT_vebt_mint @ k) = (some_nat @ b)) & (ord_less_nat @ a @ b)),
% 62.96/13.46 inference(rectify,[status(thm)],[f10164])).
% 62.96/13.46 thf(f10164,axiom,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) = (some_nat @ a)) & ((vEBT_vebt_mint @ k) = (some_nat @ b)) & (ord_less_nat @ a @ b)),
% 62.96/13.46 file('/export/starexec/sandbox/tmp/tmp.UjSd3wCyy2/DTF2THF_3077.p',conj_2)).
% 62.96/13.46 thf(f27114,plain,(
% 62.96/13.46 spl206_4),
% 62.96/13.46 inference(avatar_split_clause,[status(thm)],[f26697,f27085])).
% 62.96/13.46 thf(f26697,plain,(
% 62.96/13.46 ((vEBT_vebt_mint @ ta) = (some_nat @ a))),
% 62.96/13.46 inference(cnf_transformation,[status(thm)],[f11385])).
% 62.96/13.46 % SZS output end Proof for DTF2THF_3077
% 62.96/13.46 % (3383)------------------------------
% 62.96/13.46 % (3383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 62.96/13.46 % (3383)Termination reason: Refutation
% 62.96/13.46
% 62.96/13.46 % (3383)Memory used [KB]: 36076
% 62.96/13.46 % (3383)Time elapsed: 0.762 s
% 62.96/13.46 % (3383)Instructions burned: 2442 (million)
% 62.96/13.46 % (3383)------------------------------
% 62.96/13.46 % (3383)------------------------------
% 62.96/13.46 % (3184)Success in time 8.31 s
%------------------------------------------------------------------------------